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Observations of normal pressure on windgenerated sea waves Dobson, Frederick William 1969

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OBSERVATIONS OF NORMAL PRESSURE ON WIND-GENERATED SEA WAVES b y FREDERIC WILLIAM DOBSON B . S c , Dalhousie U n i v e r s i t y , 1959 M . S c , Dalhousie U n i v e r s i t y , 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept th i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I agr e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f PHYSICS The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date AUGUST. 1969 Supervisor: Professor Robert W. Stewart i i ABSTRACT The process by which the wind makes sea waves grow i s not w e l l -understood, p a r t l y because of the lack of adequate observat ional informa-t i o n on the normal pressures which t rans fer energy to the waves. The p r i n c i p a l object of th i s experiment has been to provide some of the miss ing data . A system for making simultaneous measurements of normal pressure and wave height was developed and tested i n the laboratory and i n the f i e l d . The system consisted of a disc-shaped buoy 23 cm i n diameter ( i n which was embedded the pressure sensor) which rode up and down on a v e r t i c a l r o d , which was the wave sensor. Careful a t t en t ion was paid to r e j e c t i n g s o - c a l l e d "dynamic" pressures associated with the d i s t o r t i o n of the a i r flow by the buoy. The r e s u l t s from the experiment are presented as power and cross-spectra of the pressure and wave s i g n a l s . Spectra of Energy (E) and Momentum ( t " w ) f luxes to the waves, and of £ , the f r a c t i o n a l energy increase of the waves per r a d i a n , are also presented. Wave power spectra are found to be normal for the s i t e ; the pressure power spectra cons i s t of a "basic" spectrum s i m i l a r to that observed over land, on which i s superimposed a wave-induced "hump". The phase angle between the waves and the pressure at the frequency of the peak of the wave spectrum i s found to be s h i f t e d from - 1 8 0 ° (pressures high over wave troughs) by amounts which exceed the t h e o r e t i c a l pred ic t ions of Mi les (1957) by an average of 20 ^ 5 ° over a wide range of i i i cond i t ions . The E and 7*w spectra are found to be sharply peaked at or above the frequency of the peak of the wave spectrum. The integrated energy f luxes E show large s c a t t e r , i n d i c a t i n g that the wave generation process var i e s cons iderably i n time (and space) . The integrated momentum fluxes 7T W to the waves show no s i g n i f i c a n t d i f ference from t o t a l f luxes from -3 a i r to water computed assuming a constant drag c o e f f i c i e n t of 1.2 x 10 ; i t appears that about 80% of the t o t a l drag of the water on the wind i s caused by the wave generation process . The ^ s p e c t r a exceed the pred ic t ions of M i l e s ' (1957) theory by fac tors of 5 to 8, i n d i c a t i n g that h i s " i n v i s c i d laminar" model . i s not adequate to exp la in observed rates of wave growth. The present r e s u l t s f a l l c lose to an empir i ca l curve suggested by Snyder and Cox (1966) except at high frequencies , where they are considerably lower. A dimensionless p l o t of C versus the r a t i o of wind speed to wave speed i s presented; the observed data i s f i t t e d by the simple r e l a t i o n f (f) = e a / e w ( u 2 / c - 1 ) , where ^ / ^ w i s the r a t i o of the dens i t i e s of a i r and water, U2 i s the mean wind speed at a height of two meters, and c i s the phase v e l o c i t y of the waves. This formula i s only considered app l i cab le for U 5 / c 4. 6, where U5 i s the mean wind speed at 5 meters he ight . Also presented are the r e s u l t s of a dry - land comparison of the buoy pressure sensor with two other pressure sensors; besides i n d i c a t i n g that the buoy sensor was adequate, th is comparison produced some i n t e r e s t i n g i v p r e l i m i n a r y i n f o r m a t i o n on the v e r t i c a l and h o r i z o n t a l s t r u c t u r e o f the t u r b u l e n t p r e s s u r e f i e l d i n the a t m o s p h e r i c b o u n d a r y l a y e r . 'TABLE OF CONTENTS PAGE Abs trac t . . . . . i i Table of Contents v L i s t of Tables v i L i s t of F igures . . v i i i Acknowledgements x i v Sect ion 1: In troduct ion 1 Sect ion 2: Theories 4 Sec t ion 3: Observations 17 Sect ion 4: Experiment 38 Sect ion 5: Data Ana lys i s and I n t e r p r e t a t i o n 71 Sect ion 6: Results 105 Sect ion 7: Discuss ion of Results 133 Sect ion 8: Conclusions 161 Appendix 1: Spike Removal 168 Appendix 2: The Boundary Bay Experiment 184 Appendix 3: Wave Damping by Adverse Winds 219 B ib l iography 235 NOTE: A d e t a i l e d table of contents precedes each of the Sections and the Appendices. v v i LIST OF TABLES TABLE PAGE 4.1 P r e d i c t e d Phase Lag of Pressure Behind Wave E l e v a t i o n over Wind-Generated Waves . . 40 4.2 Values of K/m for Six Wave Probe C a l i b r a t i o n s 48 5.1 Summary of Information from Test Hand-Dig i t i zed Data . . 78 5.2 Expected Ef fec t s of Backscat ter ing on Wave Power Spectrum 97 5.3 Expected E f f e c t s of Backscat ter ing on Pressure-Waves Cross Spectrum 98 6.1 Summary of Information for Runs with Buoy on Waves: October-November 1967 108 6.2 Further Information for Runs with Buoy on Waves . . . . 109 6.3 Pressure C a l i b r a t i o n s Used: October-November 1967 . . 110 6.4 Coherence Changes Caused by Removal of - f i * 1 Pressure-Waves Cross Spectra 117 6.5 Mean Fluxes of Energy and Momentum from Wind to Waves . 124 7.1 Dimensionless Parameters c / u . , TV , and L for Runs P * with Buoy on Waves 138 7.2 Comparison of Observed Phase Angle of Pressure Re la t ive to Waves at 0.6 Hz, with That Calcu la ted from M i l e s ' I n v i s c i d Laminar Model 142 7.3 Comparison of Observed Values of E with Those Obtained by Kolesnikov and Efimov (1962) 146 7.4 F r a c t i o n of the Wind Stress Supported by the Waves . . 149 7.5 Observed and Pred ic ted T r a n s i t i o n Fetches 152 7-6 Comparison of Observed and Pred ic ted Values of T = E / w E 156 v i i LIST OF TABLES (continued) TABLE PAGE A2.1 Information Summary for Boundary Bay Runs: September, 1968 193 A2.2 Summary of Results of Ana lys i s on Hot-Wire Anemometer Spectra 196 A2.3 Advect ion V e l o c i t y of Pressure-Generat ing Eddies . . . . 204 A2.4 Calcu la ted Pressure Differences Between A i r and Ground Sensors for Run 5e 214 A3.1 Corrected Phase Angles Between the Wave S igna l and Pressure , Sonic "B", and Sonic "w" 225 A3.2 Comparison of T h e o r e t i c a l and Observed Pressure and V e l o c i t y Amplitude over Swell Moving Against the Wind 227 v i i i LIST OF FIGURES FIGURE 1. Map of S i t e of Experiment 2. A v a i l a b l e Fetch at S i t e 3. Photograph of Recording Plat form and Instrument Masts (Looking Northeast) 4. Wave Probe C a l i b r a t i o n : O s c i l l a t o r Frequency vs Immersion Depth 5. Wave Probe C a l i b r a t i o n : (Frequency) ^ vs Immersion Depth 6. Schematic Cross -Sect ion of Microphone 7. Schematic Diagram of Pressure Measurement System 8. Pressure Recording E l e c t r o n i c s : Block Diagram 9. Wir ing Diagrams of Buoy O s c i l l a t o r and A m p l i f i e r 10. FM Tuner Rat io Detector Response Curve 11. Pressure Sensor C a l i b r a t i o n Setup 12. Laboratory C a l i b r a t i o n of the Buoy Pressure Sensor 13. Diagram of the Buoy 14. Pressure D i s t r i b u t i o n over a Planetory E l l i p s o i d S i m i l a r to the Buoy i n Shape 15. Schematic Diagram of Wind Tunnel Setup 16. P r o f i l e of Buoy (To Scale) 17. Aerodynamic C a l i b r a t i o n of Buoy: Pressure vs Distance from Bow for Various Attack Angles 18. Aerodynamic C a l i b r a t i o n of the Buoy: Pressure vs Distance from Bow for ^ 3 0 ° Yaw Angles 19. Aerodynamic C a l i b r a t i o n of the Buoy: F r a c t i o n of Stagnation Head at Pressure Port vs Wind Speed i x LIST OF FIGURES (continued) FIGURE 20. 21. 22. 23. 36. Schematic Representation of the E f f e c t of Attack Angle on the Pressure Measured by the Buoy Ef fec t s on p + ^*£ Phasor of 507o E r r o r i n Pressure C a l i b r a t i o n Phase Correct ions for Time S h i f t of 7£ S igna l Compared with F u l l Correc t ion Curve Ef fec t s on p + fg*^ Phasors of Low and High-Frequency Approx i -mations to F u l l Phase C o r r e c t i o n 'Curve 24. Power Spectra of P S ' P s + W ' t : Run 1 25. Power Spectra of P S > P S + ZM> V. : Run 2a 26. Power Spectra of V P s + w -: Run 2b 27. Power Spectra of P > P + ' : Run 3 28. Power Spectra of p. P + t ^ t > % : Run 4a 29. Power Spectra of p, p + fj<2 ,<£ : Run 4b 30. Power Spectra of v, p + e ^ t > ? : Run 5 31. Power Spectra of P S ' P s # + w > 7 : Run 6 32. Coherence Spectra between p g , ^ g Phase Spectra p s> p s + mv. s : R u n 1 33. Coherence Spectra between p , -w ; Phase Spectra between p ,7? and s *» s s ^ s P S > P S + (V8"ZS: R u n 2 a 34. Coherence Spectra between p , Yi ; Phase Spectra between p ,io and s t s s c s p s ' p s + ^ 3 ^ s : R u n 2 b 35. Coherence Spectra between p , Yf ; Phase Spectra between p ,V s » s s * s s' i s ' Run 3 and P S ' P s + f a 3 ^ s ; Coherence Spectra between Pg> Phase Spectra between p g , and p S ' p s + e j n :• R u n 4 a LIST OF FIGURES (continued) FIGURE 37. Coherence Spectra between p g , i £ , ; Phase Spectra between p g ,y£ and P i P s r s + >^ g ^ _ : Run 4b 38. Coherence Spectra between p , ^ ; Phase Spectra between p , •»£ and P e » P,, + e * n , : R u n 5 39. Coherence Spectra between p^, Phase Spectra between p^ '7s a n d P > P s r s + to"Zs: R u n 6 40. Energy and Momentum F l u x Spectra and Wave Power Spectrum: Run 1 : 41. Energy and Momentum Flux Spectra and Wave Power Spectrum: Run 2a 42. Energy and Momentum Flux Spectra and Wave Power Spectrum: Run 2b 43. Energy and Momentum F l u x Spectra and Wave Power Spectrum: Run 3 44. Energyaand Momentum Flux Spectra and Wave Power Spectrum: Run 4a 45. Energy and Momentum Flux Spectra and Wave Power Spectrum: Run 4b 46. Energy and Momentum Flux Spectra and Wave Power Spectrum: Run 5 47. Energy and Momentum Flux Spectra and Wave Power Spectrum: Run 6 48. Spectra of { : Comparisons with Snyder and Cox, M i l e s : Run 1 49. Spectra of if : Comparisons with Snyder and Cox, M i l e s : Run 2a 50. Spectra of if : Comparisons with Snyder and Coxrj; M i l e s : Run 2b 51. Spectra of f : Comparisons with Snyder and Cox, M i l e s : Run 3 52. Spectra of f : Comparisons with Snyder and Cox, M i l e s : Run 4a 53. Spectra of f : Comparisons with Snyder and Cox, M i l e s : Run 4b 54. Spectra of If : Comparisons with Snyder and Cox, M i l e s : Run 6 5 5 / Dimensionless p l o t of £ vs U,_/c 56 - 63 Power and cross - spec tra of hypothe t i ca l s inuso ida l "pressure" and'Vave" s igna l s p(t ) and * £ ( t ) ; removal of a "spike" once per cyc le i s i n d i c a t e d by the subscr ip t "s". x i LIST OF FIGURES (continued) FIGURE 56. Power Spectrum of P g ( t ) 57. Coherence between P g ( t ) and ^ ( t ) 58. Phase Angle between P g ( t ) and ^ (t) 59. Power Spectrum of ^ (t) 60. Power Spectrum of P g ( t ) 61. Coherence between P g ( t ) and ? £ g ( t ) 62. Phase Angle between p g ( t ) and ^ | t ) 63. Power Spectrum of ^ (t) 64. Comparison of the "^(t) and 7 2 s ( t ) Power Spectra for Run 1 65. Comparison of the ^ ( t ) and ^ g ^ ) Rower Spectra for Run 2b 66. Comparison of the ^ (t) and ^ s(*-) Power Spectra for Run 3 67. Comparison of the ?^(t) and ^ s ^ Power Spectra for Run 6 68. Comparison of p, Y£ Phase Spectra from "Clear" and "Spike-Contaminated" data: Run 1 69. Comparison of p, J£ Phase Spectra from "Clear" and "Spike-Contaminated" data: Run 2b 70. Comparison of p , ^ Phase Spectra from "Clear" and "Spike-Contaminated" data: Run 3 71. Comparison of p,?£ Phase Spectra from "Clear" and "Spike-Contaminated" data: Run 6 72. Comparison of Energy F lux Spectra from "Clear" and "Spike-Contaminated" data: Run 1 73. Comparison of Energy F lux Spectra from "Clear" and "Spike-Contaminated" data: Run 2b 74. Comparison of Energy F lux Spectra from "Clear" and "Spike-Contaminated" data: Run 3 x i i LIST OF FIGURES (continued) FIGURE 75. Comparison of Energy Flux Spectra from "Clear" and "Spike-Contaminated" data: Run 6 76. Map of S i t e Area for Boundary Bay Experiment 77. Photograph of Equipment Deployment P r i o r to a Run 78. Buoy Sensor C a l i b r a t i o n for Boundary Bay Experiment 79. Frequency Response of C a l i b r a t i o n Drum 80. T y p i c a l Power Spectrum of Downwind V e l o c i t y F luc tuat ions 81. Comparison of the Power Spectra from Three Pressure Sensors: Run l a 82. Comparison of the Power Spectra from Three Pressure Sensors: Run 5 a 83. Non-dimensional Pressure Spectrum from Boundary Bay 84. Coherence between "Buoy" and " A i r " Pressure Sensors 85. Phase Angle between "Buoy" and " A i r " Pressure Sensors 86. Coherence between "Buoy" and "Reference" Pressure Sensors 87. Phase Angle between "Buoy" and "Reference" Pressure Sensors 88. Coherence between "Buoy" Pressure Sensor and Hot-wire A i r Speed 89. Phase Angle between "Buoy" Pressure Sensor and Hot-wire A i r Speed 90. Coherence between " A i r " Pressure Sensor and Hot-wire A i r Speed 91. Phase Angle between " A i r " Pressure Sensor and Hot-wire A i r Speed 92. Coherence between "Reference" Pressure Sensor and Hot-wire A i r Speed 93. Chart Recording of Wave, Pressure , and Sonic Anemometer Signals during Passage of 4-Second Swell Group 94. Scale Drawing of Instrument Setup for Run 5 x i i i LIST OF FIGURES (continued) FIGURE 95. Time V a r i a t i o n of U, Q from Sonic Anemometer (Means over Ten-Second Interva l s ) for Run 5 96. Coherence Spectra during Passage of Swell Group 97. Phase Spectra during Passage of Swell Group x i v ACKNOWLEDGEMENTS This work has been part of the A i r - S e a I n t e r a c t i o n research program of the I n s t i t u t e of Oceanography at the U n i v e r s i t y of B r i t i s h Columbia. I t has been supported by the O f f i c e of Naval Research (U. S. A . ) , Grant No. NRO 83-207. During my stay at IOUBC I have been supported by the I n s t i t u t e of Oceanography, a UBC Graduate Fe l lowship , and a Nat ional Research Counci l Postgraduate Scholarship . I would l i k e to thank Dr. R. W. Stewart and Dr. R. W. B u r l i n g for many s t imula t ing d i scuss ions , and Dr. B u r l i n g for h i s painstaking examina-t i o n of my thes i s . A l l of the graduate students and s t a f f of the I n s t i t u t e have helped me i n my work one way or another, and I thank them c o l l e c t i v e l y . In p a r t i c u l a r I am deeply indebted to John G a r r e t t , who spent an inordinate amount of h i s time on my behal f , and to Ron Wilson, who h a n d - d i g i t i z e d a va luable piece of data for me. I a lso thank Jim E l l i o t t , with whom I spent some of my most pleasant and productive s c i e n t i f i c hours at Boundary Bay. L a s t l y I thank my wife Eve lyn , who l o v i n g l y typed th is thesis so many times. F . W. Dobson SECTION 1: INTRODUCTION 1.1 Purposes of the Research 1 1.2 P r a c t i c a l Considerations 2 v SECTION 1: INTRODUCTION 1.1 Purposes of the Research The p r i n c i p a l concern of th is thesis i s with the determination of the c o r r e l a t i o n between f luc tuat ions i n surface pressure and surface e l eva t ion on wind-generated g r a v i t y waves. This permits the est imat ion of two important parameters: the v e r t i c a l f luxes of mechanical energy and of momentum from the wind to the waves. As such the research i s i n the most l i t e r a l sense a study of the a i r - s e a i n t e r a c t i o n , and therefore has been c a r r i e d out as part of the A i r - S e a In terac t ion program at the I n s t i t u t e of Oceanography of the U n i v e r s i t y of B r i t i s h Columbia. A knowledge of the energy f l u x from the wind to the waves provides i n s i g h t into the wave generation process i t s e l f . Even today th is process i s not wel l -understood; the " c l a s s i c a l " theory (the M i l e s - P h i l l i p s theory: M i l e s , 1960) i s now thought to be inadequate to exp la in the observed growth of wind waves at sea. Both Mi les and P h i l l i p s (1957) have stated that the least-known v a r i a b l e i n t h e i r equations i s p ( x , t ) , the pressure at the sea surface . The pressure measured i n th is experiment i s p ( o , t ) , the time v a r i a t i o n of pressure at one l o c a t i o n . The Mi les theory deals with energy t rans fer from the mean flow i n the a i r to the waves v i a a p o s i t i v e feedback process , whereby the mean flow streamlines are modif ied by e x i s t i n g waves so as to increase the energy t rans fer . The end r e s u l t of the theory i s a p r e d i c t i o n for the energy transfer i n terms of the magnitude and phase of the wave-induced normal pressure at the water surface . An object of th i s work i s to 1 2 measure the energy transfer by measuring th is pressure, and to compare i t with the t h e o r e t i c a l p r e d i c t i o n s . Stewart (1961) has predic ted that a large f r a c t i o n of the t o t a l wind s tress on the water surface goes d i r e c t l y into wave momentum. A knowledge of the energy f lux to the waves i m p l i e s , assuming they are almost i r r o t a t i o n a l , a knowledge of the momentum f l u x to them (see, f or instance , Stewart 1961). Therefore a secondary object i s to compare the momentum f lux measured i n this way with that e i ther i n f e r r e d from assumed logar i thmic v e r t i c a l p r o f i l e s of mean wind speed or measured by measuring the Reynolds s tress 7j* = _ putw and thus obta in an estimate of the s i ze of Stewart's f r a c t i o n . In sp i t e of a great deal of recent work (Miles 1957, 1959 a and b, 1960, 1962, 1965, 1967; P h i l l i p s 1957, 1966, 1967; Bryant 1966; Benjamin 1959; L i g h t h i l l 1962; Stewart 1967), no theory ex i s t s today which pre-d i c t s w i th in an order of magnitude the energy transfer from the wind to the waves i t generates. A p r i n c i p a l reason for the f a i l u r e of the theories i s the lack of good experimental information on the flow of the a i r over waves. The p r i n c i p a l object of th i s experiment i s to pro-vide some of the miss ing informat ion, and i n so doing to throw some l i g h t on the bas i c mechanisms by which the wind generates waves on water. 1.2 P r a c t i c a l Considerations The p r a c t i c a l problems of put t ing a pressure sensor on the water surface and of ge t t ing i t to measure normal pressures there are far from n e g l i g i b l e . The f i r s t d e c i s i o n to be made i s whether the buoy carry ing the pressure sensor should be allowed to d r i f t as f r e e l y as poss ib le (Lagrangian measurement) or to be constrained to move only v e r t i c a l l y 3 ( q u a s i - E u l e r i a n measurement). Because the analys i s d i f f i c u l t i e s seem insurmountable i n the former measurement, the l a t t e r i s f e l t to be mandatory. Unfortunate ly the q u a s i - E u l e r i a n measurement i s much more d i f f i c u l t ; i t f a i r l y b r i s t l e s with problems. The p r i n c i p a l one i s keeping water away from the pressure sensing port ; a f i l m of water 1 mm thick causes a spike i n the pressure record larger by a fac tor of f i v e than the larges t a i r pressure amplitudes a c t u a l l y observed i n the data . In a Lagrangian measurement i t would be poss ib le to keep water from the sensor 90% of the time i n a f a i r l y strong wind; with the q u a s i - E u l e r i a n measurement i t i s not . Therefore s p e c i a l data process ing must be used to extract use fu l information from a pressure s i g n a l on which are superimposed numerous water-induced sp ikes . Another major cause for concern, separate from the Lagrang ian-Euler ian problem, i s the e f f ec t of the shape of the v e h i c l e carry ing the sensor on the pressures i t measures. Care must be taken to ensure r e j e c t i o n of the s o - c a l l e d "dynamic" pressure i - P "U'1 , where p i s a i r densi ty and U f. Vet wind speed, caused by the streamline conf igurat ion set up as the a i r flows over the buoy. The l a s t problem to be mentioned i s the one which haunts a l l ocean-ographers (except perhaps the Ivory-Tower breed): that of making s e n s i t i v e electronomics work i n the presence of s a l t water. SECTION 2: THEORIES 2.1 The Kelv in-Helmholtz I n s t a b i l i t y 4 2.2 J e f f r e y s ' S h e l t e r i n g Hypothesis 5 2.3 The S t a b i l i t y Analyses of Wuest and Lock 7 2.4 E c k a r t ' s Model 8 2.5 P h i l l i p s (1957) 9 2.6 M i l e s ' (1957) I n v i s c i d Laminar Model 11 2.6.1 Assumptions 11 2.6.2 Theory . . 12 2.7 M i l e s ' (1962) Viscous Laminar Model 14 SECTION 2: THEORIES From the beginning, man's encounter with the sea has been a stormy one. Extended sea voyages, for a long time only poss ib le with the a id of the winds, were s e r i o u s l y impeded by the waves which those winds generated. That the wind generates sea waves has long been recognised. The quest ion, "How?" has no s a t i s f a c t o r y answer; i n some sense th is ind icates the complexity of the process . In the fo l lowing paragraphs t h e o r e t i c a l progress made up to the time when th i s work was begun w i l l be reviewed. 2.1 The Kelv in-Helmholtz I n s t a b i l i t y The f i r s t to suggest the p o s s i b i l i t y of an i n s t a b i l i t y on a dens i ty d i s c o n t i n u i t y between f l u i d s moving r e l a t i v e to each other was H. Helmholtz (1868). The problem was tackled i n more d e t a i l i n 1874 by S i r W. Thomson (Lord K e l v i n ) , who included the e f fects of surface tens ion. His s o l u t i o n i s known as the "Kelvin-Helmholtz I n s t a b i l i t y " , and was the f i r s t theory i n which an attempt was made to p r e d i c t the growth of sea waves. The problem i s taken up i n Lamb (1932, § 268). Both f l u i d s are con-s idered as i n v i s c i d . The boundary layer between the two f l u i d s i s assumed to be of n e g l i g i b l e thickness compared with wave heights to be considered. The mean flow of both f l u i d s i s taken, to be uniform (no v a r i a t i o n i n speed U with distance from t h e i r common boundary), but d i f f e r e n t i n the two media. For the case considered here the a i r speed i s U a and the water speed i s zero. ^ i s a i r densi ty and ^ w i s water densi ty; surface tension T i s inc luded i n the a n a l y s i s . Solut ions are looked for i n the case for which the i n t e r f a c e i s 4 5 deformed with a t r a v e l l i n g wave of the form exp i k ( x - c t ) where the wave number k i s known and the wave phase v e l o c i t y c i s to be solved f o r . The r e s u l t i n g c i s complex i n general: r C = ± J 3 (Cw-g>) . T f c _ fwg>U> j 2 1 The waves w i l l grow exponent ia l ly i f for a given k the a i r speed Ua i s large enough to make the c expression complex; the p o s i t i v e root s ign produces growth. Because water waves have a minimum v e l o c i t y (see, for ins tance , Lamb 1932, § 267) there i s a range of wind speeds which generate no waves of any wave length; s u b s t i t u t i o n of C w into the s t a b i l i t y c r i t e r i o n y i e lds the minimum wind speed at which th i s mechanism can be e f f e c t i v e , about 650 cm/sec. Since n a t u r a l l y - o c c u r r i n g wind -generated waves are i n i t i a t e d at much lower speeds, and s ince i n fac t the assumption of a very th in i n t e r f a c i a l boundary layer i s u n r e a l i s t i c , th is mechanism i s not thought to be very e f f e c t i v e over most of the ranges of wind speeds and wavelengths commonly observed at sea (see Mi les 1959b). The most t e l l i n g argument against the e f f i c a c y of the K e l v i n -Helmholtz mechanism has been given by Stewart (personal communication). I f exponential wave growth i s occurr ing then the bracketed term on the r i g h t hand s ide of equation 2.1 i s pure imaginary. This means that the propagation v e l o c i t y of the waves being generated i s given by Ua. which i s very small for the case of a i r f lowing over water. The wave must grow s t r a i g h t up and hardly propagate at a l l , a h i g h l y u n r e a l i s t i c s tate of a f f a i r s . 2.2 J e f f r e y s ' She l t er ing Hypothesis S i r Harold Jef freys (1925) re-examined the problem and produced 6 another mechanism which could generate waves. He assumed that as the a i r flows over the wavy water surface , separat ion occurs on the leeward side of the wave crests with re-attachment somewhere fur ther down on the leeward slope of the wave. This produces low ambient pressures on the downwind slopes of the waves, and hence a pressure f i e l d coupled to the wave p r o f i l e i n such a way that the c o r r e l a t i o n of pressure and wave v e r t i c a l v e l o c i t y i s p o s i t i v e , producing a p o s i t i v e energy transfer from the a i r flow to the waves. He assumed that transfers caused by shear stresses are n e g l i g i b l e . In h i s c a l c u l a t i o n s he replaced the sea surface with a s ing l e F o u r i e r component, a long-cres ted sine wave. He then ca l cu la ted the work done on th i s s ine wave by the component of normal pressure i n quadrature with 7£ , the wave he ight , which i s given by where s i s less than one arid was c a l l e d by Jef freys a "she l t er ing c o e f f i c i e n t " . He ca lcu la ted s by f i r s t c a l c u l a t i n g the ra te of energy loss due to molecular v i s c o s i t y and then the l ea s t wind U . that can J mm mainta in waves against th i s l o s s , and compared th is with "observed" l eas t winds which seemed jus t capable of generating waves. Since i n J e f f r e y s ' theory, h i s ca l cu la ted s var ie s as (U m ^ n )^j h i s choice of the "correct" U m ^ n was c r i t i c a l ; th i s represents a weak point i n h i s argument. He chose Um±n to be 110 cm sec ~ \ g i v i n g an s of 0.3. U r s e l l (1956) reviews the data a v a i l a b l e at that time on pressure v a r i a t i o n s over s o l i d model waves i n wind tunnels: the works of Stanton et a l (1932), Motzfe ld (1937), and T h i j s s e (1951), and f inds l i t t l e e v i -7 dence to support a value for s as large as that suggested by Je f f reys : "The evidence of the three sets of measurements . . . on the whole favours the conclus ion that the pressure d i f ferences over a s o l i d p r o f i l e com-posed of a number of waves are an order of magnitude smaller than the d i f ferences postulated by J e f f r e y s . " He goes on to point out that no experiments had been done (1956) over l i q u i d p r o f i l e s , and that there-fore the question of the ac tua l s i z e of s i s s t i l l open. There are now some such observations; they w i l l be discussed i n "Observations". I t i s perhaps worth not ing here that the type of separat ion suggested by Je f freys i s not the only type of separat ion which can e x i s t over the waves and produce the necessary phase quadrature between waves and pressures . There i s also the p o s s i b i l i t y , f i r s t c l e a r l y pointed out by L i g h t h i l l (1962), that flow r e v e r s a l can occur over the waves i n the co-ordinate system moving at the wave phase speed, and that th is type of "separation" can produce the required phase quadrature. 2.3 The S t a b i l i t y Analyses of Wuest and Lock Wuest (1949) and Lock (1954) both treated the problem of the laminar flow over a s e m i - i n f i n i t e p l a t e . The ir work i s reviewed by U r s e l l (1956). Although t h e i r a n a l y t i c a l procedures d i f f e r , they both looked for the condit ions necessary for the onset of i n s t a b i l i t i e s i n the flow. They used the same p h y s i c a l model: a i r flows over deep water under the inf luence of g r a v i t y and with surface tens ion; the a i r flow has no pressure gradient . V i s c o s i t y causes boundary layers i n a i r and water. Both layers are assumed to have i n s t a b i l i t i e s which s t a r t at some l o c a t i o n and grow downstream. Outside the boundary layer i n the a i r the flow i s i n i t i a l l y uniform and the water at r e s t . The i n t e r f a c e i s then perturbed . 8 with a small (much smaller than the boundary layer thickness) s i n u s o i d a l disturbance of wavelength X much less than the distance x from the s t a r t of the l a y e r . The problem i s then reduced to the determination of the s t a b i l i t y of th is flow for various wind speeds and values of X and X • Their r e s u l t s , as U r s e l l points out, are hard to r e l a t e to any f i e l d s i t u a t i o n (or for that matter, to any simple laboratory experiment). What was pred ic ted i n f a c t was the condi t ion for t r a n s i t i o n to turbulence from laminar f low, whi le the flow over wind-generated waves i s i n v a r i a b l y turbulent . The models, being determinations of the onset of i n s t a b i l i t y , sa id nothing about the subsequent behaviour of the i n s t a b i l i t i e s i n the a i r or the water. 2.4 E c k a r t ' s Model Eckart (1953) proposed a model which was the forerunner of P h i l l i p s ' (1957) Theory. He assumed an i n v i s c i d open ocean over which a t h e o r e t i c a l c i r c u l a r "storm" e x i s t s . The storm cons is ts of a s ta t ionary random d i s t r i b u t i o n of s i m i l a r "gusts" (regions of r e l a t i v e l y high normal pressure) . The model neglects shear stresses and any d i s t o r t i o n of the flow by wave-produced feedback. The pressure v a r i a t i o n s i n the gusts are small enough so that the equations of motion can be made l i n e a r . The gusts are not d i r e c t l y coupled ( i . e . through a feedback mechanism) to any waves already present; they move over the water at the wind speed, and have been present for long enough that condit ions w i th in the "storm" and outside i t are s t a t i o n a r y . 9 The problem was posed i n terms of F o u r i e r - S t i e l t j e s i n t e g r a l s , and the r e s u l t s are s t a t i s t i c a l . He obtained an expression for mean-square wave height i n terms of mean-square pressure at the "storm" centre , "storm" diameter, "gust" diameter, and distance from the storm. He then used known wave heights from a r e a l storm to see what mean-square pressures h i s theory pred ic t ed . The pressures required are an order of magnitude too large (they would have to be greater than 1 p U . , the f u l l dynamic head for a i r moving at a speed U a ) . This 2 Va leads him to be l i eve that randomly d i s t r i b u t e d normal pressures i n the a i r are probably i n s u f f i c i e n t to generate waves. The weakest point i n h i s conclus ion i s i n the es t imat ion of pressures; at that time, no measurements were a v a i l a b l e of the mean-square pressure over waves i n the open ocean. In the l i g h t of recent measurements, h i s conclusions were c o r r e c t . 2.5 P h i l l i p s (1957) 0. M. P h i l l i p s (1957) has developed a wave generation theory which along with that of Mi les (1957) has rece ived much a t t e n t i o n over the past ten years . Since the pred ic t ions of the theory are not d i r e c t l y a p p l i -cable to the present experimental r e s u l t s , only a b r i e f ou t l ine i s given below. P h i l l i p s considers the subsequent motion on an i n i t i a l l y f l a t water surface a f ter the onset of a turbulent wind. The water motions are assumed to be i n v i s c i d and i r r o t a t i o n a l . Stated s imply, he f inds that as the turbulent pressure f l u c t u a t i o n s i n the a i r are c a r r i e d over the water surface , forced o s c i l l a t i o n s are generated which t r a v e l i n a l l d i r e c t i o n s and which have a l l the wave numbers present i n the a i r pressure 10 spectrum. Of th i s i n i t i a l broad wave spectrum, two components for each F o u r i e r component i n the pressure f i e l d correspond to free g r a v i t y wave o s c i l l a t i o n s : those two which move at angles to the wind such that the i r propagation speed i n the wind d i r e c t i o n equals the "advection v e l o c i t y " of t h e i r generating pressure f l u c t u a t i o n s . These components grow as long as the pressure f luc tua t ions r e t a i n t h e i r phase r e l a t i v e to the waves; that i s , they "resonate" with the pressure f i e l d . The development of the waves f a l l s into two stages, depending on whether the "time scale" for which the atmospheric pressure f luc tuat ions e x i s t and maintain t h e i r phase r e l a t i v e to the waves i s greater or less than the elapsed time from the onset of the wind. In the " i n i t i a l " stage of development, the most prominent waves are those which t r a v e l at the minimum phase v e l o c i t y of g r a v i t y - c a p i l l a r y waves c m = [ 4 j j [ j M (where g i s the a c c e l e r a t i o n of g r a v i t y and T i s V ?w/ the surface tension of the water, and ^ i s i t s d e n s i t y ) . They move at angles to the wind given by.O^s COS*(c«,/U aj) > where Uaa i s the advection v e l o c i t y of the generating pressure f l u c t u a t i o n s . In the " p r i n c i p a l " development stage the phase of a given F o u r i e r component of the pressure wanders r e l a t i v e to the wave phase, and as a r e s u l t the wave amplitude grows asVt , as i n a random walk problem. He derives for the mean-square wave amplitude the equation a 1* f^t/z^ <} U aj , where a i s wave amplitude, p i s atmospheric pressure, p i s water dens i ty , and U,j i s the advection v e l o c i t y of the pressure f i e l d . To v e r i f y h i s theory he estimates the approximate s i ze of the rms pressure f luc tua t ions to be O. lp^ U * 2.3 where >^ i s a i r densi ty and (J* i s mean wind speed at "anemometer. h e i g h t " - - u s u a l l y 10 meters. He f inds that th i s estimate gives order -o f -magnitude agreement of h i s theory with observed wave growth. The v a l i d i t y of the estimate 2.3 i s discussed i n "Discuss ion of Results", p. 139. Mi les (1960) has incorporated the above model in to a composite one i n v o l v i n g i t and h i s own: Mi les (1957). This w i l l be discussed i n the fo l lowing paragraphs. 2.6 M i l e s ' (1957) I n v i s c i d Laminar Model The present work provides a tes t of the wave generation theory pro-posed i n 1957 by M i l e s ; therefore , th is theory w i l l be discussed i n more d e t a i l than the others mentioned so f a r . 2.6.1 Assumptions The a i r i s assumed to be i n v i s c i d and incompress ible , as i s the water; the a i r i s given a prescr ibed mean shear flow ( i n the absence of waves) which var i e s with height above the surface only . I t i s assumed that there i s a feedback from the waves such that wave-induced a i r v e l o c i t y and pressure perturbat ions are two-dimensional and small enougtr to be unimportant i n the nonl inear processes of the equations of motion. Turbulence i n the a i r i s neglected (except for the i m p l i c i t assumption that i t maintains the prescr ibed shear f low) . Because of the l a s t assump-t i o n , the flow i s c a l l e d "quasi - laminar". Mean water currents are assumed to be absent. The water-wave motion i s assumed to be i r r o t a t i o n a l , and wave amplitudes and slopes small enough that l i n e a r wave theory (see, for instance , P h i l l i p s , 1966, £ 3.2) may be used. The magnitude of the speed of the waves i s assumed to be that of free g r a v i t y waves plus a second-order per turbat ion term which i s caused by the component of aerodynamic pressure i n phase with the wave s lope . 2 . 6 . 2 Theory The wave e l e v a t i o n of a given Four ier wave component i s taken to be _7£(x,t) - a e x p [ c t e c x - c t ) ] 2 . 4 , where a i s amplitude, . k= ETT/'X ( X i s wavelength) i s the wave number, x i s h o r i z o n t a l dis tance (the waves are long-cres ted) , c i s the wave phase speed, and t i s time. The per turbat ion aerodynamic pressure assoc iated with the mean flow he takes to be p = ( o l + c|3) p f t U^k^ 2-5> where ^ i s a i r dens i ty , U, i s a "reference speed" ( l a t e r defined as 2.5 {^/ P,g^^ > where t" i s the t o t a l momentum f l u x from a i r to water) , and ok. and |3 are dimensionless c o e f f i c i e n t s . The hydrodynamic equations are then solved for the wave v e l o c i t y c, g i v i n g = Cw + e a/ew (<*• + ^p) u,2* 2 . 6 , where C w i s the speed of free g r a v i t y waves ( C w » Vg/h. . f o r wave number k i n deep water) and £ w i s the r a t i o of the dens i t i e s of a i r and water. He then assumes Xw1 .» e./f w l* + c p ) u t i and expands c about c , keeping only the f i r s t two terms. This gives w 13 c c w n + ^e«/ewi^ + i p ) ( u , / c w ) j 2.7. Subs t i tu t ing 2.7 into 2.4 gives for the surface e l eva t ion ^ C x , t ) & a e x p [ l f . / e w p h c w ( U 1 / c J l t J e « p [ t k ( K - c w t g 2.8, where i t i s assumed that |a + c<3| « e w/e a(cw/u,)* Miles then defines 2.9. 2.10 as the f r a c t i o n a l increase i n wave energy E per r a d i a n . From 2.8 th is r - ( e . / f t , ) P ( u . / e « ) 1 In a l a t e r paper (Miles 1960) the r e l a t i o n i s general ized for a r b i t r a r y wave propagation angle ( r e l a t i v e to the wind) 6 to / = ( ? « / P w ) | 3 (u,COS0/Cw) In the same paper h i s own model i s general ized to inc lude that of P h i l l i p s (1957). He f inds that to obta in JT he must solve the i n v i s c i d form of the Orr-Sommerfeld equation (see, for ins tance , L i n , 1955, Equation 1.3.15). He f inds that the growth rate of a given F o u r i e r component of the wave f i e l d i s a d i r e c t funct ion of the curvature of the v e r t i c a l wind shear at the height (the " c r i t i c a l " height z c ) above the wave where U = C w . The i n v i s c i d Orr-Sommerfeld equation i s integrated numerica l ly (Conte and M i l e s , 1959) for the commonly-observed logar i thmic p r o f i l e 2. U(2) = U, l o g ( z / z o ) = 2 .5 u j o o , (Z/za) 2.11 where U # = .(t'/^ ) * i s the " f r i c t i o n v e l o c i t y " , T the Reynolds s t r e s s , and 14 Z 0 i s a "roughness length" (see, for example, Lumley and Panofsky, p. 103). The equation has a lso been solved for several d i f f e r e n t wind pro-f i l e s and i n c l u d i n g viscous e f fects by Benjamin (1959), using an orthogonal s i n u s o i d a l co-ordinate- system which fol lows the unperturbed flow stream-l i n e s . The r e s u l t of M i l e s ' computations i s a curve (Miles 1959a) r e l a t i n g (5 to C w/(j ( for various values of the wind p r o f i l e parameter Other curves presented i n the same paper represent h i s pred ic t ions of the energy input rate from the wind to the waves and of the phase angle between a i r pressure and wave e l e v a t i o n . Some of the predic ted phase angles have been ca lcu la ted for p r o f i l e parameters expected i n the exper i -ments done at the present s i t e , a n d they have been presented i n Table 4 .1 . 2.7 M i l e s ' (1962) Viscous Laminar Model Mi l e s (1962), i n studying the growth of short-wavelength waves on shallow water, i s l ed to propose a theory whereby energy may be trans-f e r r e d by "viscous Reynolds stresses" from the wind to the waves when the height of the c r i t i c a l layer i s so small that i t i s i n the s o - c a l l e d "viscous sublayer" immediately above the water surface , where the curva-ture of the mean wind p r o f i l e i s l i n e a r and so the i n v i s c i d laminar mechanism, which depends for i t s a c t i o n on the existence of p r o f i l e curvature , i s i n e f f e c t i v e . The term "viscous Reynolds stresses" implies that the energy transfer occurs through the ac t ion of pressures i n quad-rature with the waves. The t h e o r e t i c a l model described i n the paper i s r e f e r r e d to as the "viscous laminar" model. 2.12. 15 The theory considers the growth of two-dimensional waves on the surface of a s l i g h t l y viscous l i q u i d (which we w i l l c a l l the water) of f i n i t e depth which i s subjected to prescr ibed surface s tresses . These normal and tangent ia l s tresses are ca lcu la ted by s o l v i n g the viscous Orr-Sommerfeld equation for the a i r above the wavy water surface i n the orthogonal s i n u s o i d a l co-ordinate system used by Benjamin (1959). By subject ing the equations of motion for the water to boundary con-d i t i o n s which match the ca l cu la ted normal and tangent ia l s tresses at the a ir -water i n t e r f a c e , an eigenvalue equation for the complex wave phase speed i s obtained and the imaginary part of the phase speed i s computed to obta in the growth rate of the waves. The so lut ions to the eigenvalue equation for the water waves are com-posed of two parts : the f i r s t s o l u t i o n gives free surface water waves damped by the a c t i o n of v iscous stresses at the surface and at the bottom; the second s o l u t i o n i s for T o l l m i e n - S c h l i c h t i n g waves i n the a i r perturbed by the wavy motion of the water. Mi les points out that although the two classes of waves are independent at most wind speeds and wavelengths, there i s a p o s s i b i l i t y of resonance between them at a p h y s i c a l l y r e a l i z a b l e combination of wind speed and wavelength (u* = 5 cm sec ^ and \ s 5 cm, where ui( i s the f r i c t i o n v e l o c i t y as defined i n equation 2.11). The growth of the f i r s t c lass of waves i s discussed i n some d e t a i l . Approx i -mate so lut ions are found for the growth rate and growth rate curves are presented. Since the measurements made i n the present set of experiments are almost completely outs ide the range of v a l i d i t y of the viscous laminar model, i t s pred ic t i ons w i l l not be discussed i n any more d e t a i l . The 16 range of v a l i d i t y of the model w i l l be g iven, however, so that the r e s u l t s may be discussed i n terms of the various wave generation theor ies . The thickness of the laminar sublayer i s S s Si 5 D c i / U . » 2.13 where 1)^ i s the kinematic v i s c o s i t y of the a i r . Mi les defines a c h a r a c t e r i s t i c length <5C for the thickness of the c r i t i c a l layer accord-ing to . )/. 5 C = ( V . / U l k ) ' s 1 where U c i s the slope of the mean wind p r o f i l e at the c r i t i c a l height and k i s the wavenumber of the waves. He then defines a non-dimensional height Z = Z c / < $ c and f inds that the viscous laminar model appl ies over the range 0 < 2 < 2 . 3 . He computes numerical values for wave growth assuming a mean wind p r o f i l e which i s l i n e a r i n the laminar sublayer and logar i thmic above i t . His pred ic t i ons cover wavelengths from 1 - 10 cm and values of u^ from 5 - 30 cm sec ^. The energy transfer i s thus appreciable only for values of c / u * less than about 8. This concludes the out l ines of t h e o r e t i c a l progress on wave genera-t i o n . The discuss ions given are f a r from exhaustive, and other theories have been advanced since the date of the 1962 Mi les paper. Any of these newer ideas which are re levant to the present work are mentioned i n con-text , during "Discuss ion of Results". SECTION 3: OBSERVATIONS 3.1 . Sverdrup and Munk (1947) 17 3.1.1 Comparisons with Theories of Mi les (1957) and P h i l l i p s (1957) 18 3.2 F i e l d Measurements of Normal Pressures over Waves . . . . 20 3 .2 .1 Kolesnikov and Efimov (1962) 20 3 .2 .2 Longuet-Higgins , Cartwright and Smith (1963) . . . 22 3 .2 .3 Summary . 25 3.3 Recent Observations 26 3.3.1 Tests of M i l e s ' Viscous Laminar Model T 26 3 .3 .2 F i e l d Tests of M i l e s ' I n v i s c i d Laminar Model . . . 27 3 .3 .2a Snyder and Cox (1966) 27 3.3.2b Barnett and Wilkerson (1967) . . . . . . . 29 3.3.3 Laboratory Tests of M i l e s ' I n v i s c i d Laminar Model 31 3 .3 .3a Wiegel and Cross (1966) . 31 3.3.3b Shemdin and Hsu (1967) 33 3 .3 .3c Bole and Hsu (1969) 34 3 . 4 Summary of Observations 36 SECTION 3: OBSERVATIONS The amount of a v a i l a b l e experimental data on pressures over waves and on rates of wave growth under the ac t ion of the wind has been increas -ing r a p i d l y over the past ten years . Since so large a f r a c t i o n of the t o t a l information has been publ ished i n the l a s t three years , i t has become necessary to d i v i d e the fo l lowing reviews of the experiments most re levant to th i s work into two parts : those which were publ ished p r i o r to the incept ion and design of the present experiment (which was conceived i n e a r l y 1965) and those which appeared i n the l i t e r a t u r e from that time u n t i l l a t e 1968. The l a t t e r form the background against which the f i n a l r e s u l t s of the experiment are d iscussed. 3.1 Sverdrup and Munk (1947) The best summary of r e l i a b l e observations a v a i l a b l e i n 1957 when Mi les and P h i l l i p s publ ished t h e i r now-c las s i ca l wave generation theories was that made by Sverdrup and Munk (1947) to produce a method of p r a c t i c a l wave forecas t ing for the U. S. Navy. This summary drew h e a v i l y on those of Krummel (1911), Patton and Marmer (1932), and Cornish (1934) as wel l as o thers . The relevance of the theory of wave forecas t ing and of the assumptions about how waves grow as discussed by Sverdrup and Munk need not be con-s idered here-- these points were discussed f u l l y by U r s e l l (1956). The authors summarize the observations of wave growth on non-dimensional p lo t s (a) of wave steepness H / L versus wave "age" c / U a , (b) of wave v e l o c i t y c / U a and wave height g H / U a 2 versus fe tch gF /U a 2 and durat ion g t / U a , where H = peak to trough wave height of the " s i g n i f i c a n t " waves, 17 18 L = wavelength of the " s i g n i f i c a n t " waves, c = wave phase v e l o c i t y , U a = anemometer wind speed at 8 m he ight , g = g r a v i t a t i o n a l a c c e l e r a t i o n , F = fe tch: distance over which the wind blows over the water, and t = durat ion: time for which wind has blown. I t should be noted that H and L do not r e f e r to the long-crested s inu-soids considered i n p o t e n t i a l theory and i n the wave generation theor ies . Sverdrup and Munk define them as the "average height and per iod of the o n e - t h i r d highest waves", and as such t h e i r energy budget i s not the same as that of long-crested s inuso ids; they discuss the d i f ferences f u l l y . The f a c t that H and L are not simply r e l a t e d to the a and \ of a simple s i n u s o i d a l wave should be borne i n mind during the d iscuss ions which fo l low. This work, b a s i c a l l y empir i ca l i n nature , and the wave generation theories of Jef freys (1925) and Eckart (1953), were the most recent a v a i l -able theories of wave generation when an a r t i c l e by U r s e l l (1956) turned the a t t e n t i o n of t h e o r e t i c a l f l u i d dynamicists to the complete inadequacy of the e x i s t i n g explanations of the process . As the d i r e c t r e s u l t of the a r t i c l e two theories were publ ished w i t h i n a year, by Mi les (1957) and P h i l l i p s (1957). These two theor ies , as u n i f i e d by Mi les i n 1960, form the " c l a s s i c a l " view of wave generat ion. 3 .1.1 Comparisons with Theories of Mi les (1957, 1960) and P h i l l i p s (1957) P h i l l i p s "(1957), having obtained a value for the mean square wave amplitude a 1 i n terms of the wind durat ion and speed and the mean square 19 f l u c t u a t i o n s i n normal pressure of the wind f i e l d (see pp. 9-11), e s t i -mated the s i ze of p 4 as " ? - 0.1 (e<Xf 3 . 1 , and compared the r e s u l t i n g a*" with that expected from the r e s u l t s i n Sverdrup and Munk's paper; he found the points r e l a t i n g g H / U 2 to g t /U to be i n good agreement with h i s t h e o r e t i c a l curve. By 1960, however, P h i l l i p s concluded that h i s estimate of p was too large by as much as one order of magnitude. Mi les (1959a) made, besides an estimate of the minimum wind speed necessary to r a i s e waves, a comparison with the Sverdrup-Munk data as w e l l . He computed from h i s theory the value of the "she l t er ing c o e f f i c i e n t " s (see J e f f r e y s ' wave generation theory, pp. 5-9) by two d i f f e r e n t methods: f i r s t , by computing the mean energy transfer to the waves us ing h i s theory; and, secondly, by computing the mean t h e o r e t i c a l momentum t r a n s f e r . The l a t t e r computation involved the assumption of a d i r e c t i o n a l d i s t r i b u t i o n 2 for the waves (one propor t iona l to cos 6 i s used). Both assumed a wave spectrum of the f u n c t i o n a l form given by Neumann (1952). Wind p r o f i l e parameters were taken from empir i ca l r e s u l t s quoted by E l l i s o n (1956). _ o From these two computations, h i s estimates for s were 1.1 x 10 and _2 0.9 x 10 ; the value found by Sverdrup and Munk to f i t the observations _ o i s 1 .3 x 10 . Munk (1955) also ca lcu la ted a range of values for s from shearing stresses of the wind on the water i n f e r r e d from some e a r l i e r measurements of Van Dorn, and i t s value o f 0 . 8 - l . 2 x l 0 was also used by Mi les as evidence of the v a l i d i t y of h i s theory. Thus the " c l a s s i c a l " theory of wave generat ion, the M i l e s - P h i l l i p s theory as u n i f i e d by Mi les (1960), r e l i e s h e a v i l y for the experimental 20 proof of i t s v a l i d i t y on the wave growth observations summarized by Sverdrup and Munk i n 1947. At the time when the theories of Mi les and P h i l l i p s were publ i shed , the only information on pressure f luc tua t ions over waves was that contained i n the three experiments done on v a r i a t i o n s i n normal pressure on s o l i d wave models by Stanton ^t al (1932), Motzfe ld (1937), and T h i j s s e (1951), a l l of which were considered by Mi les (1957) to be i r r e l e v a n t to the study of the flow of a i r over water. The r e s u l t s obtained by these three experiments, reviewed by U r s e l l i n 1956, are far from cons i s tent , the f i r s t two experiments g i v i n g measured values for s ranging from 0.006 (Stanton) to 0.11 (Motzfeld); T h i j s s e 1 s work ind icates that the pressure d i s t r i b u t i o n i s such that the wave would grow as expected by J e f f r e y s , i n d i c a t i n g an s nearer 0.27. 3.2 F i e l d Measurements of Normal Pressures over Waves In 1962 two accounts were publ ished of measurements of pressure f l u c t u a t i o n s over waves, both from f r e e l y f l o a t i n g buoys equipped with accelerometers for measuring the wave height s imultaneously with the pressure . Each w i l l be discussed below i n the context of the present work. 3 .2.1 Kolesnikov and Efimov (1962) Kolesnikov and Efimov (1962) used a buoy about 30 cm i n diameter--th is f i gure i s not given i n the paper, but i s obtained from a d e s c r i p t i o n of an e a r l i e r model (Kolesnikov and Kononkova, 1961). I t was c o n i c a l i n shape underwater; the part which was i n the a i r was a segment of a sphere The pressure port was at the top of the segment on the v e r t i c a l axis of the buoy, and was about 5 cm above the water when the buoy was l e v e l . Wave heights were obtained by twice i n t e g r a t i n g the output of a v e r t i c a l 21 accelerometer located i n the buoy. The d e s c r i p t i o n of the experiment i s terse i n the extreme. A diagram of the inner workings of the buoy i s not dimensioned. The reader must r e l y for c r e d i b i l i t y of the r e s u l t s on such statements as, "In measurements of the frequency dependence of pressure , surface , ve lo -c i t y and wave p r o f i l e , the c a l i b r a t i o n curves of the microbarometer and the wave recorder were l i n e a r . " . Fur ther , the statement i s made that an error ex i s t s i n the pressure measurements caused by the f a i l u r e of the buoy to remain p a r a l l e l with the surface of the waves which, for the measurements made at sea, "was about 1270." The probable e f fects of th i s error are discussed more f u l l y below. A c a l c u l a t i o n of the mean input of energy E to the waves i s made for each run using the formula E = ± \ pet) xu-CO d t 3.2, where T i s the time i n t e r v a l over which the run was taken and p(t) and w(t) are the pressure at the surface and the v e r t i c a l v e l o c i t y of the waves. The values of E and of U, the wind speed at 2.5 meters height^, are presented—nothing e l s e - - f o r seven runs. The f i r s t point to be made about the experiment concerns the measure-ment of pressure from a buoy which fol lows the o r b i t a l motions of the waves (Lagrangian measurement). The buoy contained a s ing le accelerometer which measured v e r t i c a l a c c e l e r a t i o n s . Whether the f i r s t i n t e g r a l of the s i g n a l from this accelerometer can be considered an accurate representa-t i o n of the true v e r t i c a l v e l o c i t y of the waves depends on the amount of motion the buoy describes r e l a t i v e to the o r b i t a l motion of the 22 surface water p a r t i c l e s . Fur ther , the pressure v a r i a t i o n along a wave p r o f i l e i s d i s t o r t e d by the a c t i o n of the buoy, which fol lows the wave surface o r b i t a l v e l o c i t i e s . This d i s t o r t i o n probably has l i t t l e e f f ec t on the values of energy transfer to the waves computed by Kolesnikov and Efimov, s ince i t s e f fects are averaged out over a wavelength. The p r i n c i p a l source of error i n the energy f luxes computed from pressure-wave c o r r e l a t i o n s measured by the Kolesnikov-Efimov buoy ar i ses from the contamination of the pressure s i g n a l with dynamic pressures associated with the flow of the a i r over the buoy. I ts s p h e r i c a l above-water shape ind ica tes that the dynamic f l u c t u a t i o n s exerted on the buoy surface vary between about p^ — +1 at the junct ion of the buoy with the water and about -p^ at the top of the buoy where the pressure measurement port i s located ( i f the buoy i s represented by a hemisphere embedded i n a f l a t surface , p o t e n t i a l flow theory gives " ' / ^ f j for the dynamic pressure at the top of the hemisphere); here U i s the mean wind v e l o c i t y near the water surface . This means that as the buoy t i l t s r e l a t i v e to the water surface , an unknown spurious component w i l l be added to the quadrature spectrum between the waves and the pressure , r e s u l t i n g i n the computation of erroneous energy t r a n s f e r s . Without a d e t a i l e d time h i s t o r y of the t i l t f luc tuat ions of the buoy, the spurious pressures could not be extracted from the recorded s i g n a l . In any case, the computed values of the energy f l u x from Kolesnikov and Efimov (1962) are compared with the present r e s u l t s where the wind speed was-; the same, i n "Discuss ion of Results". 's 3.2.2 Longuet-Higgins , Cartwright and Smith (1963) The buoy used by Longuet-Higgins , Cartwright , and Smith (1963) was 23 much larger than that of Kolesnikov and Efimov: i t consisted of an aluminum d i s c about 1.7 meters i n diameter and 0.6 meters deep, which when loaded f l o a t e d with i t s top about 30 cm above the water surface (the 30 cm i s an estimate from t h e i r F igure l a ; no f igure i s quoted). The pressure sensor was r a i s e d 6 cm above the top of the buoy and observed the pressure above a t h i n h o r i z o n t a l d i s c about 45 cm i n diameter ( e s t i -mated from the same f igure) through twelve o r i f i c e s , presumably near the centre of the d i s c . The c r o s s - s e c t i o n of the part of the buoy above water was roughly concave with a f l a t top. The ra i sed d i s c was located at the geometric centre of the buoy. Since the p r i n c i p a l object of the experiment was to measure the d i r e c t i o n a l spectrum of wind-driven waves, the buoy was provided with three accelerometers which responded to rates of change of v e r t i c a l motion, p i t c h and yaw. Thus this experiment t h e o r e t i c a l l y provided enough informa-t i o n , assuming the buoy d id not move r e l a t i v e to the water surface p a r t i c l e s , to correc t the observed pressure s igna l for the e f fec t of the motion of the buoy along the wave p r o f i l e (see p . 2 2 ); th is c o r r e c t i o n was apparently not made. Longuet-Higgins et a l , having obtained d i r e c t i o n a l spectra' for the waves, then f i t a number of t h e o r e t i c a l angular d i s t r i b u t i o n s to them, f i n d i n g that one propor t iona l to cos (V^S) appears to f i t t h e i r data best; r i s found to vary from 7 to about 0.4 as U^/c var ie s from 0.1 to 0.5, where U i = 2.5 u* and u# i s the f r i c t i o n v e l o c i t y estimated from observed mean v e l o c i t i e s . They then compute the jin phase component of the t o t a l pressure predic ted by M i l e s ' theory as general ized by Benjamin (1959) to inc lude v i s c o s i t y and other wind p r o f i l e s besides logar i thmic ones. 24 They f i n d that the observed r a t i o s of t h e i r pressure spectra to t h e i r wave spectra are predic ted moderately w e l l by the t h e o r e t i c a l r a t i o s . This f a c t they take to i n d i c a t e that the pressures measured by the buoy are almost e n t i r e l y produced by "aerodynamical pressure changes due to the flow of the a i r over the undulat ing surface" together with the s t a t i c pressure term ^ <j . They a lso inc lude a table of phase d i f ferences between the pressure and the waves as measured by the buoy for one of the runs. These phase angles become less and less r e l i a b l e at high frequencies but are accurate to about + 1 0 ° below a) = 3 rad/sec; the peak of the wave spectrum i s at 0.6 r a d / s e c . The phase angles they get are about - 1 8 0 ° (pressure lag behind waves) and th i s lag appears to increase with increas ing frequency, although the measurements are deemed untrustworthy at the frequency (3 rad/sec) where the increase begins. Since an increase of pressure lag beyond 1 8 0 ° ind icates that wave damping i s occurr ing as the d i f ference U - c increases , the trend i s i n e x p l i c a b l e i n terms of e x i s t i n g theor ies . The f a c t that the phase of the pressure s i g n a l remains - 1 8 0 ° over the part of the wave spectrum containing most of the wave energy they take as fur ther evidence of the v a l i d i t y of M i l e s ' t h e o r e t i c a l pred ic t ions (see Experiment: Table 4,1) that the phase s h i f t s of the pressure versus wave c o r r e l a t i o n s should remain c lose to - 1 8 0 ° . Where poss ib l e , compari-sons with the data obtained from the present experiment are compared, i n "Discuss ion of Results". As was probably the case with the Russian buoy, i t i s expected that the pressure s i g n a l recorded by the Eng l i sh buoy was contaminated by dynamic pressure f l u c t u a t i o n s . Since Longuet-Higgins e_t a l only report information of the pressure-waves cospectrum, only the e f fects of dynamic 25 pressure contamination on the in-phase component of pressure w i l l be d iscussed. The height at which the E n g l i s h buoy f loa ted r e l a t i v e to i t s d i a -meter i s considerably less than was apparently the case for the Russian buoy, and i t i s therefore expected that dynamic pressure contamination i s of less s i g n i f i c a n c e . Two sources of contamination ex i s t ; that associated with t i l t i n g of the buoy r e l a t i v e to the water surface (p i tching) and that a r i s i n g from the dynamic pressure generated by the por t ion of the a i r flow r e l a t i v e to the buoy which i s coherent with the waves. Unfortunate ly not enough i s known e i ther of the response of the buoy to the wave f i e l d or of the a i r flow which ex i s t s over r e a l waves to estimate the s i ze of e i t h e r e f f e c t . I t i s expected that the f i r s t of the two w i l l be more important, s ince the dynamic pressure at the sensor var ie s as ^ U with the buoy p i t c h angle, where the re levant dynamic pressure for the second source of contamination i s ^ U u ; i n these express ions , U i s the mean wind speed near the water surface and u 1 i s the p o r t i o n of the wind speed r e l a t i v e to the buoy which i s coherent with the waves, which i s at l eas t one order of magnitude less than. U o r d i n a r i l y . 3 .2 .3 . Summary In summary, both the Kolesnikov and Efimov buoy and the buoy used by Longuet-Higgins ^t al appear to introduce into the pressure s ignals they record spurious dynamic pressures which .may be as large as '/g f*^ for the Russian buoy and are somewhat smal ler , but far from n e g l i g i b l e , for the E n g l i s h buoy. The r e s u l t s from the two experiments cannot be com-pared, s ince Kolesnikov and Efimov report information of the pressure-waves quadrature spectrum and Longuet-Higgins ej: al are able to give information 26 only on the cospectrum. Of the two measurements the one by Longuet-Higgins j2t al must be considered the most s i g n i f i c a n t because the buoy they used apparently had a lower p r o f i l e on the water. For th i s reason t h e i r f i n d i n g that the phase of t h e i r pressure s igna l i s near ly - 1 8 0 ° with respect touthe waves suggests that M i l e s ' (1957) i n v i s c i d theory gives at l ea s t a good f i r s t approximation to the a i r flow over the waves. 3.3 Recent Observations A d i s c u s s i o n of the observations published s ince 1962 must d iv ide them into two c lasses: those which are re levant to the i n v i s c i d theory proposed by Mi les i n 1957, and those per ta in ing to the s o - c a l l e d viscous laminar theory (Benjamin 1959; Mi les 1962). In the l a t t e r theory i t i s assumed that the height of the c r i t i c a l layer i s so small that i t i s w i t h i n the viscous sublayer , a t h i n stratum of a i r adjo in ing the water. In the theory, wave generation i s presumed to occur through the ac t ion of v iscous Reynolds stresses ( s e e p . 14 f f . ) . 3 .3.1 Tests of M i l e s ' Viscous Laminar Model Since the flow regime postulated by the l a t t e r theory i s not that commonly observed for the range of wind-generated g r a v i t y waves at sea which contain most of the t o t a l energy, and s ince i t probably never occurred during the present measurements, those experiments dea l ing with i t w i l l be mentioned but not commented on. The fo l lowing l i s t of works publ ished up to la te 1966 has been unabashedly extracted from Miles (1967); l a t e r references are the r e s u l t of a l i t e r a t u r e search. The experiments dea l ing with the viscous laminar theory have a l l been c a r r i e d out i n the laboratory i n wind-water tunnels , where short -27 wavelength waves are generated on smooth water under contro l l ed turbulent boundary l a y e r s . Included i n th is category are the works of Hamada (1963), Holmes (1963), Hanratty and Woodmansee (1965), Cohen and Hanratty (1965), Hidy and Plate-(1965, 1966), P la te and Hidy (1967), Hires (1967), and P l a t e , Chang and Hidy (1969). These were conducted i n "normal" i . e . long, s t r a i g h t , narrow, wind tunnels; a nove l , although e n t i r e l y q u a l i t a t i v e , approach has been taken using c y l i n d r i c a l geometry by Auerbach and Richardson (1967). With the exception of the f i r s t two and l a s t two papers mentioned, a l l show more or less s a t i s f a c t o r y agreement with the viscous laminar theory. The l a t e s t work, that of P l a t e , Change and Hidy produces c o n f l i c t i n g re su l t s - - exper imenta l growth rates are found some of which exceed and some of which f a l l short of the t h e o r e t i c a l pred ic t ions by amounts greater than the expected experimental e r r o r s . 3 .3 .2 F i e l d Tests of M i l e s ' I n v i s c i d Laminar Model The experiments re levant to M i l e s ' i n v i s c i d model can be subdivided into two groups: f i e l d measurements, and those made i n the laboratory i n wind-wave tunnels . The f i e l d measurements w i l l be discussed f i r s t . 3 .3 .2a Snyder and Cox (1966) Since 1962 only two experiments have been publ ished on observations of wave growth at sea under more or less c a r e f u l l y monitored condi t ions . The f i r s t i s that of Snyder and Cox (1966), who towed behind t h e i r vesse l an array of wave detectors tuned to waves of a s ing le wavelength (17 m). I t was towed downwind at the group v e l o c i t y of these waves during offshore winds. The sh ip ' s speed was measured with a t a f f r a i l l o g , and wind speeds with an anemometer on the ship at a height of about 6 m above the *Not a v a i l a b l e to the w r i t e r ; comments are from Miles (1967). 28 water surface . From the r e s u l t i n g information the l i n e a r and exponential growth rates &(kB,t) and |3(fc0,t) of the s p e c t r a l i n t e n s i t y F (k„,t) of the 17 m waves have been ca lcu la ted from (Hasselman, 1960) . dF(k 0 ,t)/dt = a ( k O J t ) + p(feo,t)F(k„,t) . ... 3 .3 . The time h i s t o r y of F ( t , k „ ) was recovered from the observed spectra for a number of runs , and values of (X and 3^ were determined by f i t t i n g th i s time h i s t o r y with a regress ion of the form 3.3. I f the spectrum of pressure f l u c t u a t i o n s i s assumed to be s i m i l a r to one measured by P r i e s t l e y (1965) over mown grass i n V i r g i n i a , then the values of <X found are cons is tent with those pred ic ted by P h i l l i p s ' (1957) theory. The f i t t e d values of ^ are found to increase l i n e a r l y with wind speed, although showing considerable s c a t t e r . The experimental points are found to be f i t t e d quite we l l by the r e l a t i o n P =(e./e„)U-y - «.) 3- 4 ' where k i s the wave number and cja the frequency of the 17 m wave.s, and IT i s the observed mean wind v e l o c i t y . The measured values of ^ are com-pared with the pred ic t ions of the theories of Jef freys (1925) assuming a s h e l t e r i n g c o e f f i c i e n t of 0.27, and of Mi les (1957) assuming a l o g a r i t h -mic p r o f i l e , a p r o f i l e parameter I~l (Equation 2.12) of 3 x 10"^ and a roughness length of lO'^cm. This assumed roughness length appears to be high by a fac tor of about 20 (see P h i l l i p s 1966, or Smith 1967), which makes the A used too large and the t h e o r e t i c a l growth rate computed from Mi les (1957) too small by an unknown f a c t o r . |3 , however, i s only a weak funct ion of jQ ; th is overestimate ofJTI w i l l thus cause an error i n G> which probably does not exceed 257». Therefore the bas i c conclus ion 29 reached by Snyder and Cox remains una l tered . The ir measurements of s p a t i a l growth are about one order of magnitude greater than those pred ic ted by M i l e s ' (1957) theory. They also made an estimate of the t o t a l momentum transfer to the i r wave f i e l d assuming equation 3.4 to hold and us ing a simple empir i ca l spectrum; they obtained a wind drag c o e f f i c i e n t of about 7 x 10 J -cons iderably larger than the value of 1.0 - 1.4 x 10"-* obtained from d i r e c t measurements, e .g . Smith (1967) or Weiler and B u r l i n g (1967). They assumed the discrepancy to be accounted for by the assumptions used to make the est imate, which "tend to give an overestimate of the momentum trans fer by normal s tress . . . " . Because they overestimated C^ by such a large amount, t h e i r conclus ion that the major por t ion of momentum trans-fer at sea goes d i r e c t l y into waves must be regarded as h i g h l y speculat-i v e . 3.3.2b Barnett and Wilkerson (1967) The second f i e l d experiment was that of Barnett and Wilkerson (1967). They measured p r o f i l e s of the sea surface under the ac t ion of a w e l l -e s tab l i shed of fshore wind i n the upwind and downwind d i r e c t i o n , using as t h e i r wave sensor a s e n s i t i v e radar a l t imeter mounted i n an a i r p l a n e , which flew over the water at an a l t i t u d e of 150 m over a distance of 340 km. The a l t imeter averaged heights over a c i r c u l a r area of ocean about 9 m i n diameter, thus f i l t e r i n g out a l l wave frequencies greater than about 0.4 Hz. The large speed of the plane--about 100 m/sec--meant that the e n t i r e wave f i e l d was sampled twice w i t h i n about two hours. Wind speeds near shore were extrapolated from concurrent values at land s t a t i o n s . Those fur ther offshore were obtained by successive f ixes with 30 Loran - C; wind p r o f i l e informat ion, as i n the case of the Snyder and Cox measurements, was not taken. The ir r e s u l t s are presented f i r s t as two p lo t s ( t h e i r Figures 6 and 7): one for the upwind one and one for the downwind run, which show con-tours of equal s p e c t r a l dens i ty on a graph with wave frequency as ordinate and distance from shore as abeissa; these graphs therefore show the development with fe tch of the complete measured power spectrum, and i n themselves present an immense amount of data i n a very ed i fy ing manner. They found, as expected, that the major s p e c t r a l peak moves to lower frequencies as the f e tch increases . They found "considerable" energy i n t h e i r spectra at frequencies below the peak, i n d i c a t i n g a broad-band energy input mechanism which they presume to be that of P h i l l i p s . I f a t t en t ion i s f i xed on the behaviour with time of a p a r t i c u l a r s p e c t r a l component for which U / c > I , i t i s found to grow exponent ia l ly u n t i l i t reaches a maximum and then to lose energy u n t i l i t reaches an e q u i l i b r i u m energy 30 - 707= below that of the maximum: i t "overshoots" i t s maximum values . The f i n d i n g of most i n t e r e s t here i s the f a c t that t h e i r experiment-a l l y determined values of the exponential growth parameter ^ (Equation 2.10, p. 13) are i n c lose agreement, at l ea s t for t h e i r downwind run , with the empir i ca l curve suggested by Snyder and Cox (Equation 3.4, p. 28.) They thus a r r i v e d at the same conclus ion as d id Snyder and Cox. The i n v i s c i d model proposed by Mi les pred ic t s wave growth rates smaller by an order of magnitude than those measured. In t h e i r upwind run, they found p o s i t i v e growth rates at wave f r e -quencies for which the wave phase speed exceeded the wind speed as they measured i t . They suggest poss ib le reasons for t h i s , but there seems l i t t l e j u s t i f i c a t i o n i n making such suggestions i n view of the scantiness of t h e i r wind speed informat ion . 3 .3 .3 Laboratory Studies Three laboratory studies s ince 1962 have attempted to match condit ions i n wind-water tunnels with M i l e s ' i n v i s c i d laminar theory. The f i r s t , Wiegel and Cross (1966), was c a r r i e d out at the Department of C i v i l Engineer ing , U n i v e r s i t y of C a l i f o r n i a , Berkeley; the other two, Shemdin and Hsu (1967) and Bole and Hsu (1969) used the extensive wind-wave f a c i l i t i e s a v a i l a b l e at the Department of C i v i l Engineering at Stanford U n i v e r s i t y . 3 .3 .3a Wiegel and Cross (1966) Wiegel and Cross (1966) measured a i r speed and normal pressure above mechanical ly-generated waves i n three d i f f e r e n t wind-water tunnels , for a v a r i e t y of wave speeds and wind speeds. They measured a i r v e l o c i t y and normal pressure at the same downwind p o s i t i o n as a res i s tance wire wave gauge, us ing a tota l -head tube f o r a i r speed and a p i t o t - s t a t i c tube for the normal pressures . The tubes were placed as c lose as poss ib le to the surface of the waves, but were always above the c r i t i c a l height z c for the waves under study. The s t a t i c holes of the p i t o t - s t a t i c tube were at the same downwind p o s i t i o n as the wave probe and the t i p of the t o t a l head tube. The frequencies of the mechanical ly-generated waves ranged from 1.5 to 2 Hz, g i v i n g wavelengths of 70 to 40 cm and wave speeds of 100 to 80 cm/sec; wind speeds used were 670 to 1340 cm/sec. The water depth i n 32 t h e i r tunnels ranged from 15 to 45 cm. The r e s u l t s are presented i n terms of coherence and phase cros s - spec tra and power spectra of the s t a t i c pressure and wave e l e v a t i o n . Although some incons i s tenc ies were observed ( i n one run observed phase lags of pressure behind wave height were zero throughout the s p e c t r a l range, not 160 - 1 7 0 ° as predic ted by the i n v i s c i d theory) , i n general agreement with the i n v i s c i d theory was good. There are two causes of uncer ta in ty i n t h e i r measurements. The f i r s t has been discussed i n some d e t a i l by Shemdin and Hsu (1966), who found that by measuring pressures over waves with a probe f i x e d i n space above the c r i t i c a l height z e whereU=c, (as was the probe used by Wiegel and Cross) they obtained r e s u l t s which were s i m i l a r to those obtained by Wiegel and Cross but which d i f f e r e d sharply from r e s u l t s obtained when t h e i r probe stayed beneath z c . They took th is to mean that the v a l i d i t y of M i l e s ' (1957) theory could not be checked with pressure measurements made above z c . The second cause of uncer ta in ty i s t h e i r use of a p i t o t - s t a t i c tube to measure pressures i n c lose proximity to the^waves. As the wavelength of the waves used i n the wind-wave tunnel decreased with increas ing f r e -quency, the pressure measured at the s t a t i c ports of the tube would con-t a i n increas ing f r a c t i o n s of the stagnation p r e s s u r e L p u U , where U i s the mean free-s tream a i r speed and u' i s the p o r t i o n of the t o t a l a i r speed f l u c t u a t i o n s which i s coherent with the waves (the same defect besets the buoy measurement system used i n the present experiments, and i t s e f fects are discussed i n d e t a i l i n "Data Analys i s and Interpreta t ion" on p. 91 f f ) . The inf luence of th is dynamic pressure contamination on the 33 r e s u l t s cannot be estimated without a d e t a i l e d knowledge of t h e i r measure-ment system. 3.3.3b Shemdin and Hsu (1967) Shemdin and Hsu (1967) studied the pressure d i s t r i b u t i o n over mechani-c a l l y generated waves i n a wind-water tunnel , us ing a s t a t i c pressure probe which moved v e r t i c a l l y with the water surface a short distance above i t . The probe was a t h i n c i r c u l a r d i s c mounted p a r a l l e l to a v e r t i c a l plane containing the wind d i r e c t i o n . Pressure was measured at the axis of the d i s c . Although such devices are n o t o r i o u s l y s e n s i t i v e to yaw, the small turbulence l eve l s and accurate ly a l igned flow a v a i l a b l e i n the wind tunnel e l iminated yaw e f f e c t s , which might cause large errors i f the same experiment were attempted i n the f i e l d . In order to insure that the c r i t i c a l l ayer for the waves was s u f f i c i e n t l y th ick for the probe to remain w i th in i t , i t was necessary to thicken the boundary layer over the water a r t i f i c a l l y using roughness elements located at a trans-i t i o n p la te where the a i r f i r s t met the water. In th i s way they were able to obta in the required thicknesses for wind speeds up to 600 cm/sec. For comparison of t h e i r r e s u l t s with M i l e s ' (1957) theory, c a r e f u l l y measured v e r t i c a l p r o f i l e s of mean wind speed were made at each mean wind speed; they were found to be c l o s e l y l ogar i thmic , although a system-a t i c d e v i a t i o n from a least -squares f i t was found. This d e v i a t i o n was ignored for the c a l c u l a t i o n of the p r o f i l e parameter/! . (Equation 2.12) required for comparison with M i l e s ' theory. To obta in the phase d i f f erence between pressure and waves, they super-imposed chart recordings of the observed pressure on simultaneous wave record ings , and found least - square f i t s inusoids for each wind speed. 34 The phase information obtained i n each case i s e f f e c t i v e l y equivalent to that which would be contained at one frequency i n a c r o s s - s p e c t r a l a n a l y s i s . The f i n a l r e s u l t s were presented i n a table showing measured and t h e o r e t i c a l (predicted from M i l e s ' 1957 theory) phase s h i f t s , and measured and t h e o r e t i c a l values of the per turbat ion pressure amplitude for 0.4 and 0.6 Hz waves over a range of wind speeds from 120 - 1160 cm/sec ^. The M i l e s ' i n v i s c i d theory was found by the authors to be s a t i s f a c t o r i l y v e r i f i e d by t h e i r observed phase s h i f t s . Mi les himself (1967) i s not so sanguine--he f inds that the i r phase measurements " . . . are i n f a i r agreement wi th , although somewhat larger (to a s t a t i s t i c a l l y s i g n i f i c a n t extent) than, the t h e o r e t i c a l pred ic t ions of the i n v i s c i d laminar m o d e l . . . " . 3 .3 .3c Bole and Hsu (1969) The l a t e s t published account of observations re levant to M i l e s ' i n v i s c i d laminar theory i s that of Bole and Hsu (1969). They made care-f u l observations with capacitance probes of the growth with fe tch under the a c t i o n of the wind of mechanical ly-generated waves. V e r t i c a l wind speed p r o f i l e s were taken at numerous l o c a t i o n s , both with the mechanical ly-generated waves present and absent. They used a modif ied v e r s i o n of the same wind-water tunnel used by Shemdin and Hsu (1967). The ir waves v a r i e d i n frequency from 0.9 to 1.4 Hz; the water depth was about 100 cm. Wind speeds used were 350 - 1350 cm/sec. The ir boundary layer was not thickened (as was that of Shemdin and Hsu) and as a r e s u l t , the thickness of the c r i t i c a l layer was smal l , ranging between .013 and .34 cm. They doubted the existence of a laminar sublayer , not ing that at the 35 wind speeds they used the standard deviat ions of surface roughness of the water caused by wind-generated r i p p l e s s u b s t a n t i a l l y exceeded the estimated thicknesses of both the laminar sublayer and of the c r i t i c a l l a y e r . They a l s o , therefore , doubted the existence of organized vortex motion below the c r i t i c a l l a y e r . These two f a c t s , taken together, appear to e l iminate the d i r e c t a p p l i c a t i o n of t h e i r r e s u l t s to e i ther the i n v i s c i d laminar model or the viscous model of M i l e s . They nevertheless appl ied t h e i r r e s u l t s to the former. They found t h e i r mean v e l o c i t y p r o f i l e s to be logar i thmic c lose to the water s u r f a c e - - t h i s r e s u l t being obtained i n the absence of mechanical ly generated waves. When these waves were present the p r o f i l e s were s i m i l a r , but could no longer be extended as c lose to the water surface . They found evidence that r i p p l e s on the mechanical ly-generated waves were steep and sharp-cres ted , and appeared to "r ide" the wave cre s t s . They computed s p a t i a l wave growth i n a stepwise manner, s ince they observed that the boundary layer evolved with f e t c h . From t h e i r s p a t i a l growth curves they obtained measured values for the growth parameter (Equation 2.8, p. 1 3 ) , and found that the r a t i o s of these to values com-puted from Mi les i n v i s c i d laminar model using measured p r o f i l e parameters v a r i e d from 1 - 1 0 , with a mean of about 3. They took th is as evidence of the t r u t h of Mi les (1957) statement that h i s i n v i s c i d model would only give q u a l i t a t i v e agreement with r e a l i t y when the flow over the water surface could be taken as aerodynamically rough. They go on to suggest that the presence of sharp-crested r i p p l e s on the crests of the waves ind icates the p o s s i b i l i t y that separat ion may indeed occur over t h e i r waves, and that th i s may account for the large r a t i o s of observed to 36 predicted <f . Because of the small c r i t i c a l layer thicknesses e x i s t i n g i n the measurement, i t i s problematical whether the r e s u l t s shown should be applied to the i n v i s c i d or the viscous laminar model. I t should be possible with the data they obtained f o r Bole and Hsu to make the l a t t e r comparison; the r e s u l t s of such a comparison would be e d i f y i n g . 3 . 4 Summary of Observations In summary, there are a v a i l a b l e i n the l i t e r a t u r e only four observa-tions of f l u c t u a t i o n s i n s t a t i c pressure over wind-generated g r a v i t y waves--two, those of Longuet-Higgins et al and Kolesnikov and Efimov--being f i e l d measurements and two, those of Wiegel and Cross and Shemdin and Hsu, being made i n wind-water tunnels. Both f i e l d measurements are suspected of being strongly contaminated by dynamic pressure f l u c t u a t i o n s , the l a t t e r probably somewhat more than the former. The measurements of phase made by the English buoy are not accurate enough to permit the computation of a pressure-wave quadrature spectrum. Of the two wind-tunnel measurements only those of Shemdin and Hsu remain above suspicion, although strong reservations about t h e i r data analysis are expressed i n Bole and Hsu ( 1 9 6 9 ) . Their r e s u l t s are equivocal when compared with Miles' i n v i s c i d laminar theory; phase s h i f t s reported i n Shemdin and Hsu (1967) are only s l i g h t l y greater than those predicted by theory, while some r e s u l t s from the same experiment published l a t e r (Shemdin 1968) apparently indicate growth rates larger by a factor of about two than those predicted by Miles' i n v i s c i d model. The two wave growth measurements made were both done i n the f i e l d , 37 and were those of Snyder and Cox (1966) and Barnett and Wilkerson (1967). They both reached the same conclus ion: observed rates of wave growth exceed those pred ic ted by the i n v i s c i d laminar model by about one order of magnitude. In f a c t , the growth rates they observe are larger by more than a fac tor of two than the l arges t ones reported by Shemdin (1968). Since r e a l waves break, energy f luxes computed from f i e l d measure-ments of s p a t i a l growth should be smaller than those obtained from measurements of pressure f l u c t u a t i o n s over s i n u s o i d a l waves i n a wind tunnel . I t can thus be seen that the experiments described leave our understanding of the mechanism of energy transfer from the wind to sea waves i n some confusion. SECTION 4 : EXPERIMENT 4 . 1 Rat ionale 38 4 . 2 Design C r i t e r i a 39 4 . 2 . 1 The Pressure Measurement System 39 4 . 2 . 2 The Wave Measurement System . 44 4 . 3 The Measurement S i t e . . . 44 4 . 4 The Measurement of Wind Speed 45 4 . 5 The Wave Sensor 46 4 . 5 . 1 C a l i b r a t i o n 46 4 . 6 The Pressure Sensor 47 4 . 7 The Pressure Measurement System 51 4 . 7 . 1 The Waterproofing Diaphragm '51 4 . 7 . 2 The Backup Volume 52 4 . 8 E l e c t r o n i c s 53 4 . 8 . 1 Design C r i t e r i a 53 4 . 8 . 2 D e s c r i p t i o n of System 55 4 . 9 C a l i b r a t i o n of Pressure Sensor . 57 4 . 9 . 1 Laboratory C a l i b r a t i o n s 58 4 . 9 . 2 F i e l d C a l i b r a t i o n s . . . 59 4 . 1 0 The Buoy 60 4 . 1 0 . 1 Underwater Shape 60 4 . 1 0 . 2 Upper Surface: Aerodynamics 62 4 . 1 1 Wind Tunnel Tests 64 4 . 1 1 . 1 The Wind Tunnel 65 SECTION 4: EXPERIMENT (continued) 4.11.2 Experimental Procedure 65 4.11.3 The Dynamic Pressure Rejec t ion Ring . . . . . . 66 4.11.4 The A r t i f i c i a l Turbulent Boundary Layer . . . . . . 67 4.11.5 The Aerodynamic C a l i b r a t i o n of the Buoy . . . . 68 4.11.6 Consequences of Attack Angle V a r i a t i o n 70 SECTION 4: EXPERIMENT The preceding sec t ion has set the stage for th is one; the t h e o r e t i c a l and experimental evidence a v a i l a b l e up to mid-1965 formed the basis for the design c r i t e r i a and experimental procedures described immediately below. These descr ip t ions are fol lowed by d e t a i l e d accounts of apparatus and c a l i b r a t i o n s . 4.1 Rat ionale As mentioned i n the i n t r o d u c t i o n , the f i r s t dec i s i on which had to be made i n the design of the experiment was whether the buoy should be allowed to d r i f t more or less f r e e l y on the waves (Lagrangian measurement), or be constrained to move only v e r t i c a l l y ( q u a s i - E u l e r i a n measurement). Cross spectra between pressure and wave e l eva t ion were requ ired , as wel l as the power spectra of each v a r i a b l e . I f the waves are measured with accelerometers on the buoy, then the r e l a t i v e phase of pressure and waves can be recovered c o r r e c t l y . The true frequency to which a given frequency i n the recorded information i s r e l a t e d cannot be extracted , unless motions of the buoy r e l a t i v e to the waves moving i t are small with respect tor.the wave motions. For th is reason, and s ince two attempts at measuring pressure f l u c t u a -t ions with f r e e l y moving accelerometer-equipped buoys had already been made (see p . 2 0 f f ) , i t was f e l t necessary that the next attempt--the one under d i s cuss ion here--be q u a s i - E u l e r i a n . The experimental technique used was to measure the pressure at the surface of a disc-shaped styrofoam f l o a t with a gymbal-suspended bearing 38 39 on i t s v e r t i c a l axis, which moved up and down on the wave sensor, a teflon-sheathed brass rod clamped v e r t i c a l l y i n the water. 4.2 Design C r i t e r i a 4.2.1 The Pressure Measurement System The en t i r e pressure measurement apparatus was designed so that as precise a measurement as possible could be made of the phase r e l a t i o n between surface elevation and pressure. Rather severe compromises were found necessary with amplitude measurements, as w i l l become clear i n sub-sequent paragraphs, but no such compromises were permitted i n the phase measurements. That the phase measurement i s c r u c i a l can be seen i n Table 4.1. The table shows t h e o r e t i c a l l y predicted phase differences between pressure and wave elevation for representative wind speeds as computed by Miles (1959a) and used to obtain his Figure 6: 6 = t a n ' [ - p d l . / c f / d - a ( U , / c f ) } 4.1 (this i s a modification of Miles' c a l c u l a t i o n made by Longuet-Higgins et a l , 1963; i t includes the e f f e c t of the s t a t i c pressure term p Cj ) . Q i s the phase lead of pressure over wave elevation 1J , C i s the wave phase speed, L)t i s a " c h a r a c t e r i s t i c " v e l o c i t y given by U, -=.0WK) - 2.5 4.2, where U j , i s the f r i c t i o n v e l o c i t y (see, for instance, Lumley and Panofsky, 1964) and K i s von Karman's constant^ a . and p are numerical constants calculated by Miles (1959a) i n h i s theory of wave generation. TABLE 4.1 Predic ted Phase Lag of Pressure Behind Wave E l e v a t i o n over Wind-Generated Waves (Calculated from M i l e s , 1959; the s t a t i c pressure termp f c g»£ has been included i n the pressure, as i n Longuet - Higgins et a l , 1963) Radian .% Frequency U. - a |3 -p(U,/C) Q C r | - a(U,/c)1 (Rad/sec) (Degrees) 0.5 0.025 - - - -1. 0.05 - - - -2. 0.1 7.2 0.42 -0.004 180. 4. 0.2 7.4 3.25 -0.1 174. 6. 0.3 12.0 3.44 -0.15 172. 8. 0.4 13.4 3.4 -0.17 170. 10. 0.5 14.5 3.4 -0.184 169. Wind p r o f i l e values used i n the table are: U » = 20 cm/sec U, = 2.5 a . = 50 cm/sec Zo = 7 x 10" 2 U - * 1 = 3. .5 x 10" 3 cm XI = gze/ U,1 = 3 x 10" 3 (fl ca l cu la ted from given Z 0 , U, values i s 2.8 x 1 0 - 3 ) 41 Using the estimates for the logar i thmic p r o f i l e for "moderate wind over wavy water" i n Mi les (1959a) (u* & 20 cm/sec, z c a 7 * 10" , n ^ 3 x 1 0 - J ) , a and|3 can be c a l c u l a t e d , and the table d i sp lays the r e s u l t s for values of the radian wave-frequency CJ between 0.5 and 10 r a d / s e c . Expected phase s h i f t s are therefore smal l , r e q u i r i n g an accurate phase measurement. A fur ther c a l c u l a t i o n w i l l give some idea of the expected amplitude of the pressure f l u c t u a t i o n s . I f the water surface e l eva t ion i s given by r[ = a sm (ftx - cot) , then the wave slope has amplitude dx I The momentum f l u x to the waves from p', the part of the t o t a l pressure f i e l d i n phase with the wave s lope, i s dx where the overbar denotes a time average. Root mean square slopes on wind dr iven waves have been measured (Cox and Munk, 1954a, b; Cox, 1958), and are (kaf | - 0.1 . I f i t i s assumed that h a l f of the t o t a l wind s tress goes into waves, then for a 5 meter/sec wind, T = f a C D U l - U.«|0"5 « l.5"|0"3« Z5«\0 4 .3 . ~ 0.5" dyne cm - 4. Then jp'| ~ (Y/E)( l /kaj ~ 1.5 dyne cm'Z 4 .4. The s i ze of the expected pressure amplitudes becomes even more important when the e f f ec t of the buoy shape on the a i r flow over i t i s 42 considered. The mean flow streamlines w i l l be d i s t o r t e d by the buoy, the d i s t o r t i o n being re l a t e d to the shape chosen for the part of the buoy above the waterline. This d i s t o r t i o n w i l l cause the presence of dynamic pressures on the buoy surface, and t h e i r s i z e w i l l be a f r a c t i o n F of the f u l l stagnation pressure ! P U , F being a function of posi-2 t i o n on the buoy surface and of the shape of the buoy. If a turbulent wind of speed U + u', where u1 i s the f l u c t u a t i n g part of the t o t a l wind, blows over the buoy, then the f l u c t u a t i o n s i n dynamic pressure at the sensor would be P d = F e a U U 4.5; t y p i c a l values obtained from wind tunnel tests f o r the shape of the buoys used and a 500 cm/sec wind give, assuming u to be given by(a*j for a s i t u a t i o n s i m i l a r to those observed at the measurement s i t e , o = 0.2 x 1.2 x 10" 3 x 60 x 500 4.6. ~ 7 dynes / cm^. This means that to achieve the modest signal to noise r a t i o of ten i t became necessary to devise means for r e j e c t i n g at l e a s t 96% of the dynamic pressure f l u c t u a t i o n s . The siz e of the buoy was of great importance. I t had to be large enough to carry the required e l e c t r o n i c s and yet small enough to respond to high frequency waves beyond the peak of the spectrum. One of the major factors which had to be considered i s the a b i l i t y of the buoy to keep water o f f i t s surface, and i f water gets there to shed i t r a p i d l y . The e f f e c t of the water on the pressure signal has been mentioned ( p. 3 ) . I t remains to note that water, once on the buoy 43 surface , could very e a s i l y then be blown by the wind to the sensor l o c a t i o n . Also surface currents i n the wind d i r e c t i o n produced bow waves at the f r o n t of the buoy which were then blown over the sensor. This cons iderat ion demanded buoy shapes d i a m e t r i c a l l y opposed to the shapes suggested as good for r e j e c t i n g dynamic pressures; therefore , a compromise had to be reached which s a t i s f i e d as c l o s e l y as i s f e a s i b l e each of the two c r i t e r i a . The frequency response of the pressure system was much harder to contro l at low than at high frequencies . The highest frequency at which waves could i n p r a c t i c e be resolved by the wave probe used i n the exper i -ment was not more than 5 Hz, and most pressure measurement devices are quite capable of responding to such frequencies-- indeed.; many, such.as microphones, are not u s u a l l y designed to respond to such low frequencies'. Therefore , the choice of the high-frequency cutof f of the system was one more of convenience than anything e l se . One cons iderat ion was the d i g i t a l sampling r a t e . Frequencies above h a l f th is frequency had to be f i l t e r e d out to avoid a l i a s i n g . The low-frequency cutoff presented more problems. The pressure spectrum i s at i t s l arges t there; i n f a c t , i t becomes thousands of times larger i n the (lower) frequency range associated with the passage of storms and f r o n t a l systems. The lowest frequencies at which wind-driven waves were encountered at the experimental s i t e were about 0.2 Hz, and the usual frequency at which the peak of the spectrum occurred was 0.5 Hz. Because accurate measurement of phase i n the reg ion of wave genera-t i o n was so important, i t became necessary to set the cutoff frequency at l eas t a decade below t h i s , at about 0.05 Hz, thus keeping phase 44 correct ions less than 1 0 ° at 0.5 Hz. Set t ing the cutoff frequency th is low introduced two problems. F i r s t the large , n a t u r a l low-frequency com-ponents i n the pressure s i g n a l introduced d r i f t s ; second, i n t e r n a l tempera-ture v a r i a t i o n s i n the sensor introduced spurious low-frequency noise . Both of these hide information at higher frequencies because of windows used i n the s p e c t r a l a n a l y s i s . 4 .2 .2 The Wave Measurement System Demands on the wave probe were less s t r ingent than those on the pressure measurement system. Expected maximum wave excursions were about 100 cm, and the e n t i r e frequency band of i n t e r e s t w i t h i n which a l l waves more than one or two mi l l imeters i n amplitude were found was 0.1 -3.0 Hz. The probe i t s e l f had to be strong enough so that f l exure caused by the ex tra drag of the buoy was kept smal l , although th i s f l exure d i s -tor ted only the observed shape of the waves; i t d id not a f f e c t the accuracy of the r e l a t i v e phase between pressure and waves. Absolute water depth accuracy was unimportant; only the r e s o l u t i o n mattered. The frequency response of the wave probe used was e s s e n t i a l l y f l a t from DC to a frequency somewhere between 3 and 5 Hz, covering completely the range of i n t e r e s t expected at the experimental s i t e . 4.3 The Measurement S i t e The l o c a t i o n of the measurement s i t e i s shown i n F igure 1. I t i s a t i d a l f l a t adjo in ing Po in t Grey, Vancouver, B. C. The a v a i l a b l e fe tch (distance from shore to observat ion point) as a func t ion of the d i r e c t i o n from which the wind blows i s shown i n F igure 2. On the land to the south-east are apartment blocks ranging i n height above sea l e v e l from 20 to 45 80 meters. Water depth at the s i t e and over most of the sand f l a t ranges from 0 to 4 meters depending on the t i d e . There i s a weak ( < lm/sec) c i r c u l a r t i d a l current i n E n g l i s h Bay. F igure 3 shows the measurement masts themselves and the platform on which the data were recorded. The dis tance from plat form to mast i s 60 meters i n a Norther ly d i r e c t i o n . 4.4 The Measurement of Wind Speed Wind speed was measured i n two d i f f e r e n t ways during the course of the experiment, using cup anemometers, and using a sonic anemometer. For the f i r s t four of the s i x runs presented, mean wind speed and d i r e c t i o n were measured with three Thornthwaite s e n s i t i v e cup anemometers located at heights of 3, 4, and 5 meters above the water surface (anemometers were deployed at lower l e v e l s , but were destroyed jus t p r i o r to the per iod during which the f i n a l l y - u s e d runs were made). Of these three anemometers the one at the 4 meter height read low c o n s i s t e n t l y , and data from i t was not used. Since only two r e l i a b l e determinations of wind speed were therefore a v a i l a b l e , no attempt was made to estimate from them any c h a r a c t e r i s t i c s of the wind p r o f i l e s . For the l a s t two runs presented no cup anemometers were a v a i l a b l e , and wind speed was measured with a f i r s t - g e n e r a t i o n s o l i d - s t a t e K a i j o -Denki 3-dimensional Sonic Anemometer (Mitsuta ^t a l , 1967). This device was used to obta in the mean and f l u c t u a t i n g components of the downwind a i r v e l o c i t y U. From th i s quanti ty the mean wind speed tJ, the mean d i r e c -t i o n 8 , and the wind s tress T = - P ^ U W = ^"1 were extracted . The mean wind speed U5 at 5 meters height;.-' was extra-polated from the value of U at the height of the sonic anemometer (about 46 1.5 meters) assuming a logar i thmic wind p r o f i l e as defined i n Equation 2.11. 4.5 The Wave Sensor The sensor used to measure waves was of the capacitance type, s i m i l a r to that used by G i l c h r i s t (1965). I t consis ted of a \ inch stock (0.635 cm O.D.) brass rod 180 cm i n length covered with a t e f l on d i e l e c t r i c i n the form of spaghett i tubing (0.719 cm O . D . , 0.681 cm I . D . ) , sealed at the bottom and with an e l e c t r i c a l connection to the brass rod at the top. This formed a c y l i n d r i c a l condenser when immersed i n a con-ducting f l u i d such as sea water. I t was connected to the frequency-determing network of a b lock ing o s c i l l a t o r s i m i l a r to that used by B u r l i n g (1955), and Kinsman (1960). The frequency of the o s c i l l a t o r thus v a r i e d with changes i n water l e v e l on the probe. This frequency-modulated (FM) s i g n a l was returned from the sensor l o c a t i o n to the recording p l a t -form v i a c o a x i a l cable . I t was then e i ther recorded d i r e c t l y on analog magnetic tape, or for monitoring purposes was f i r s t demodulated and then ii ii recorded. The demodulator used for this purpose was a Vetter Model 3 three channel FM recorder adaptor. 4 .5 .1 C a l i b r a t i o n The frequency-determing equation of the o s c i l l a t o r was ? = K/(Cp + mH) 4 . 7 > where f i s frequency, K i s a constant i n c l u d i n g r e s i s t i v e elements i n the frequency-determining network, Cp i s a f i xed "padding capacitance", H i s the water depth to which the probe i s immersed, and m i s the (constant) rate of change of probe capacitance with immersion depth. C a l i b r a t i o n s were s t a t i c ; that i s , they were made by measuring the frequency of the 47 o s c i l l a t o r at various f ixed immersion depths i n s a l t water ( s a l i n i t y 4 - 10%). The r e s u l t i n g c a l i b r a t i o n curve was quite nonl inear; a t y p i c a l example i s shown i n F igure 4. The r e s u l t of p l o t t i n g H against t = 1/f i s shown i n F igure 5; from this i t can be seen that the s i g n a l from the probe i s amenable to l i n e a r i z a t i o n during d i g i t a l a n a l y s i s , and th i s was i n f a c t done (see "Data Ana lys i s and Interpretat ion", p. 84). The equation for H, from 4.7, i s H = K / m f - C p / m 4 8 _ The second term i n 4.8 i n the system used had to be adjusted by as much as 307„ from day to day. S ince , however, i t was only height d i f ferences which were r e q u i r e d , these d r i f t s were not important. Changes i n K/m were important, and with th is i n mind, a ser ies of s i x s t a t i c c a l i b r a t i o n s of the probe was made which extended over three days, and for which the s a l i n i t y i n the t e s t ing tank was v a r i e d over a s a l i n i t y range of 2 - 5.5%». A i r temperature was uncontro l l ed but var ied over a range of 2 - 1 3 ° C . H was p lo t t ed a g a i n s t ' ^ f o r a l l the t e s t s , and the slopes (values of K/m) for a l l the l i n e s are given i n Table 4 .2 . On the basis of th is tes t ne i ther a i r temperature nor water s a l i n i t y caused the minor' changes of slope which occurred, and the 957» confidence l i m i t s for wave height d i f ferences observed i n the experiments were +37o prov id ing that the s i g n a l from the probe was l i n e a r i z e d p r i o r to s p e c t r a l a n a l y s i s . 4.6 The Pressure Sensor The pressure sensor was a modified commercial capacitance microphone--TABLE 4.2 Values of K/m for Six Wave Probe C a l i b r a t i o n s Date S a l i n i t y A i r Temperature lCT^K/m (Jan 1968) Time of Day ( ° / o o ) ( ° C ) (cm/sec) Notes: (If (2) 22 a f t 2.0 13.3 1133 23 morn 4.0 10.0 1126 24 a f t 4.5 6.7 1096 24 eve 5.0 .4:4 1126 25 morn 5.2 2.2 1139 26 morn 5.5 3.9 1111 Notes: 1. A i r temperature i s that measured at Vancouver In ternat iona l A i r p o r t . 2. Mean K x 10" 3/m = 1122 cm/sec; S td . Deviat ion = 15.8 cm/sec. 49 a schematic diagram (Figure 6) shows i t s cons truc t ion . The pressure-sensing diaphragm was a gold-coated glass d i s c 1.27 cm i n diameter and about 0.08 cm th ick . I t was clamped to a s t a i n l e s s s t ee l cas ing , and i t s gold coat ing made e l e c t r i c a l contact with the s h e l l . V a r i a t i o n s i n i t s d is tance from a backing p la te in su la ted from the s h e l l caused capac i -tance v a r i a t i o n s which were a measure of the pressure d i f f erence across i t . The microphone was mounted i n the buoy so that the plane of i t s diaphragm was v e r t i c a l , thus minimizing the e f f ec t on the s igna l of v e r t i c a l a c c e l e r a t i o n s . When the microphone was f i r s t obtained, considerable time and e f f o r t were spent i n t r y i n g to get the manufacturer (Al tec Lans ing , Inc . ) to sea l of f a small leak which allowed a i r to pass from the front to the rear of the microphone diaphragm ins ide the s t a i n l e s s s t e e l cas ing . The sea l was deemed necessary i n the present experiment so that the time con-s tant of the microphone could be c o n t r o l l e d . The time constant was s i x seconds for the microphone as rece ived from the manufacturer, with the space behind i t s diaphragm sealed in to a volume of 2 cm , a time constant of about 20 seconds was r e q u i r e d . The attempt had to be abandoned when the manufacturer was unsuccessful i n sea l ing the leak a f ter three t r i e s , each one e n t a i l i n g about a two-month delay. Therefore the time constant had to be lengthened by connecting the space at the rear of the diaphragm to a l arger volume. This volume i s r e f e r r e d to hereafter as the "backup" volume. The pressure measurement port on the surface of the buoy could not be l e f t open. Whenever water was shipped by the buoy i t f looded the area of the port and clogged even 0.5 cm diameter holes for periods up to -r-50 ten seconds "after the f lood". A number of d i f f e r e n t chemical agents for reducing surface tension were used i n an attempt to reduce the s i ze of the smal lest "uncloggable" ho le , but none were p a r t i c u l a r l y success-f u l . Therefore , the hole was sealed with a t h i n rubber diaphragm, sea l ing of f a small forevolume (about 2 cnP) i n front of the microphone diaphragm. Because large ambient low-frequency pressures associated with weather systems exis ted i t became necessary to provide some means for e q u a l i z i n g the i n t e r n a l pressure of the microphone-backup volume system with ambient pressures . This was achieved by venting the backup volume to atmosphere with a slow (period about 50 seconds) leak. This leak was a l so designed to e l iminate as much as poss ib le the e f fects of v a r i a t i o n s of ambient temperature on the pressure i n the backup volume. In p r a c t i c e the l a t t e r e f f e c t was by far the most important, and was the source of the major component of noise at low ( 0 . 1 Hz) frequencies . A simple c a l c u l a t i o n shows i t s e f f e c t . I f the backup volume changes temperature by an amount S T , then the associated pressure change ( i f the backup volume were sealed) would be where PQ i s ambient atmospheric pressure and T Q i s ambient temperature. Put t ing 1000 dyne cm" 2 for p Q , 2 9 0 ° K for T Q , and 0 . 1 ° K for T gives I f th i s &T occurs i n 10 seconds pressure d r i f t s of the order of 2 dyne cm per minute can be expected; these are a small f r a c t i o n of the observed pressure s igna l s at these frequencies , but may be important i n cases where the a i r - s e a temperature d i f ference was large and changing r a p i d l y . 4 .9 , 0.3 dyne cm 51 4.7 The Pressure Measurement System A schematic diagram of the pressure system i s shown i n F igure 7. The enclosed space i n f ront of the diaphragm had a resonant frequency of between 150 and 250 Hz depending on the tension of the rubber waterproof-ing diaphragm. 4.7.1 The Waterproofing Diaphragm Many methods of keeping water out of the microphone were considered, and some were t r i e d . The only success ful one was to sea l the microphone behind the aforementioned rubber diaphragm, and th i s was used i n the experiment. Along with i t s obvious advantages came less obvious d i s -advantages; because i t was mounted h o r i z o n t a l l y , f l u s h with the measure-ment surface , i t caused the sensor to become s e n s i t i v e to v e r t i c a l a c c e l e r a t i o n s . Being th in n a t u r a l rubber i t changed i t s tension, and hence the pressure s e n s i t i v i t y of the system, as i t aged with use. F u r t h e r , treatments to make i t shed water (seebelow) had to be chosen so as to not change i t s tens ion. Because th i s diaphragm sealed the system except for a slow vent to atmosphere v i a the backup volume, any bui ldup of pressure ins ide the system was balanced by an increase i n diaphragm tens ion, with a concomi-tant decrease i n amplitude s e n s i t i v i t y of the instrument. Such pressure buildups occurred a l l too e a s i l y . Changes i n ambient temperature (see p. 50) were t h e i r main source* another was the in termi t t en t passage of water over the diaphragm. This could , i f i t happened frequent ly , cause the mean pressure on the diaphragm to d i f f e r from ambient enough to change the diaphragm tens ion. Because i n p r a c t i s e there was always water on the buoy surface , i t 52 was necessary to prevent the formation of water drops which "nested" i n the diaphragm and prevented pressure measurement. This "nesting" pro-perty of the water i n fac t determined the minimum p r a c t i c a b l e diaphragm tension and hence the maximum s e n s i t i v i t y of the sys tern. This tension was determined by t r i a l and error to be that which reduced the bas i c microphone s e n s i t i v i t y by a factor of 2 - 3. To prevent th i s "nesting", and also to prevent the diaphragm from being wetted by the water (which increased i t s a c c e l e r a t i o n s e n s i t i v i t y and i t s tension by an unknowable amount), i t was treated with a l i g h t s i l i c o n e o i l . The area immediately around i t was, except for small areas, coated with a wetting agent, sodium s i l i c a t e or water g lass . This treatment remained e f f e c t i v e for periods of up to two hours i n condit ions where the diaphragm was being inundated once every 10 - 15 seconds on the average. 4 .7 .2 The Backup Volume The backup volume was an aluminum cy l inder 4 cm long with a 3.5 cm ins ide diameter. I t was connected to the rear of the microphone diaphragm by 10 cm of 1.5 mm I . D . polyethylene tubing. I t was vented to atmosphere by 4 cm of 0.1 mm I . D . s t a i n l e s s s t e e l tubing. Because the buoy was always wet, the vent i t s e l f was sealed into the buoy and an a i r passage provided v i a 10 cm of s t a i n l e s s s t ee l tubing (1 mm I . D . ) which was f ixed v e r t i c a l l y , s o that i t s top d id not often become blocked. The tubing from the backup volume to the microphone casing was cemented f i r m l y i n place i n the buoy to prevent i t from f l e x i n g as the buoy moved on the water. The backup volume was embedded i n the styrofoam buoy. Other than t h i s , i t was not thermally i n s u l a t e d . Although th i s meant that temperature-53 induced pressure d r i f t s introduced some noise at frequencies less than 0.1 Hz, space and weight cons iderat ion precluded any more e laborate des ign. A dess icant was sealed in to the backup volume to prevent con-densation from occurr ing e i ther on the microphone diaphragm or i n the vent to atmosphere. E i t h e r would prevent the system from funct ion ing i n . a p r e d i c t a b l e way; i n f a c t , blockage of the vent proved to be the major source of equipment f a i l u r e during the f i e l d experiments. 4.8 E l e c t r o n i c s The e l e c t r o n i c s system, with the exception of two components (tape recorder , FM tuner) had to be designed s p e c i f i c a l l y for the experiment. The reasons for choosing the p a r t i c u l a r system used are f i r s t ou t l ined; a more d e t a i l e d account of the parts of the system fo l lows . 4 .8.1 The Design C r i t e r i a The design c r i t e r i a to be met by the e l e c t r o n i c s system were as fo l lows: 1. Buoy e l e c t r o n i c s smal l , l i g h t , waterproof; 2. Pressure s e n s i t i v i t y greater than 2 m i l l i v o l t s per dyne cm"*2 change i n pressure . E l e c t r o n i c s noise less than 0.2 mv p-p i n the f r e -quency range of i n t e r e s t (set by the tape recorder noise l e v e l ) ; 3. Frequency and phase response near ly f l a t i n the range 0.1 - 5.0 Hz; 4. Pressure s i g n a l free from contamination caused by the waves. The requirements of No. 1 above are discussed l a t e r i n th is chapter (p. 60ff) . Stated b r i e f l y they are: the p h y s i c a l dimensions of the buoy had to be less than one-half the wavelength of the shortest waves of in teres t - -about 50 cm i n th is experiment; the displacement had to be 54 minimized to reduce the drag of the water on the buoy, which produces i n s t a b i l i t y i n the alignment of the buoy into the wind. The need for waterproofing led to the i n c l u s i o n of the bat tery for the e l e c t r o n i c s i n the buoy rather than b r i n g i n g i n DC power v i a the s i g n a l cable . The l i g h t e s t transducer with the required pressure s e n s i t i v i t y was a capacitance microphone. In order to get the required low frequency response, i t was necessary to use frequency modulation rather than the usual DC s i g n a l condi t ion ing system. Since the capacitance and the modu-l a t i o n of commercially a v a i l a b l e microphones was extremely small ( t y p i c a l l y 5 pf with T-0.02% modulation per microbar pressure d i f ference across the diaphragm) i t was necessary to use a high frequency c a r r i e r . The FM band (88 - 108 megahertz) was chosen, c h i e f l y because components for th is band were e a s i l y bought l o c a l l y . The undesirable contamination of the output s igna l by wave motions proved of major importance i n the design i n two separate areas: f i r s t , i n the choice of methods of data transmission from the buoy; and second, i n the design of s h i e l d i n g for the system. I t was o r i g i n a l l y intended to telemeter the 100 mHz c a r r i e r from the buoy to the hut . A f t e r considerable e f f o r t th i s idea , and even that of t ransmit t ing to the instrument mast had to be given up. The power received from the buoy, small to begin with because of weight l i m i t a t i o n s on the bat tery and s i ze l i m i t a t i o n on the antennae, var i ed over enormous ranges as the conf igurat ion of the water surface (the ground plane) var i ed with wave motions. As a r e s u l t the rece iver was not always f u l l y l i m i t e d , and wave-induced modulations of the amplitude of the c a r r i e r caused s i g n i f i c a n t f luc tuat ions i n the demodulated tuner output. 55 An e f f e c t with a s i m i l a r r e s u l t occurred i n the buoy o s c i l l a t o r . The movement of the ground plane caused large changes i n loading on the c a r r i e r o s c i l l a t o r , which could not adequately be removed (within the l i m i t s on power consumption set by the battery) with a buffer a m p l i f i e r . The problem remained af ter i t was decided to transmit from buoy to mast v i a a coax ia l cable; however, i t was solved by care fu l a t t en t ion to s h i e l d i n g and grounding. The o s c i l l a t o r used i n the experiment was enclosed i n (and grounded to) a box m i l l e d from a block of aluminum, which was i t s e l f grounded only v i a the s igna l cable to the s h i e l d of the radio frequency (RF) ampl i f i e r on the instrument mast. A ground to the sea was considered, but not used when i t was found to have no e f fec t on the performance of the o s c i l l a t o r . 4 .8 .2 D e s c r i p t i o n of System A block diagram of the e l e c t r o n i c s system i s shown i n Figure 8. I t can be separated in to three parts : the transducer e l e c t r o n i c s i n the buoy; an RF a m p l i f i e r on the instrument mast; and an FM r e c e i v e r , match-ing a m p l i f i e r , and tape recorder i n the hut (60 meters from the m a s t -see F igure 3) . Sealed into the buoy were a bat tery (15 v o l t s , carbon-zinc) to power the transducer e l e c t r o n i c s , a small s e l f - l a t c h i n g r e l a y enabl ing the power to the e l e c t r o n i c s to be switched on or of f by applying a DC voltage to terminals on the buoy, and the transducer e l e c t r o n i c s themselves. The e l e c t r o n i c s were potted with beeswax into the small aluminum box to which the microphone was attached. In th is aluminum box, i n shie lded com-partments, were the 110 mHz (Clapp) c a r r i e r o s c i l l a t o r and a small buffer stage cons i s t ing of a low -Q tuned a m p l i f i e r which matched to the s i g n a l 56 cablejwhich was a seven meter length of 50 ohm coax ia l cable (Amphenol RG - 174/u). C i r c u i t diagrams for the o s c i l l a t o r and ampl i f i e r are shown i n F igure 9. Considerable care was taken to make the Q of the o s c i l l a t o r as large as p o s s i b l e . The tank c i r c u i t was temperature-compensated through the use of negative temperature c o e f f i c i e n t capac i tors ; the c o i l was hand wound on a ceramic form ( M i l l e r 4300 - Y) and the c o i l wire was a s t r i p of copper 1 mm wide and 0.05 mm t h i c k . The beeswax pot t ing fur ther reduced temperature e f fects and provided an exce l l ent waterproof-ing for the o s c i l l a t o r and buffer when i t d id ship some water. The t r a n s i s t o r s of both c i r c u i t s were biased for low power d r a i n , a t o t a l of less than 10 ma from the 15 v b a t t e r y . The low-current b i a s i n g used i n the buffer a m p l i f i e r had the added advantage of increas ing i t s i n p u t - t o -output impedence r a t i o , thus improving i t s bu f f er ing c a p a b i l i t y . The RF a m p l i f i e r on the instrument mast was designed to rece ive the 100 mHz c a r r i e r , amplify i t by 25 db, and dr ive a 75 meter length of sh ie lded 300 ohm twin-lead cable . I t was powered by two Mercury c e l l s and used a commercially a v a i l a b l e integrated c i r c u i t (Motorola MC 1110) as a low -Q tuned a m p l i f i e r . I t s input was matched to the 50 ohm coax ia l cable from the buoy and i t s (balanced) output matched to the 300 ohm twin- l ead . The twin- lead cable ran along the sand from the instrument mast to the hut , where i t was connected d i r e c t l y to the antenna terminals of a commercially a v a i l a b l e FM tuner (EICO ST - 97). There the c a r r i e r was demodulated and the voltage analog of the pressure fed v i a an operat iona l a m p l i f i e r to an FM record channel of the tape recorder , which was a 14-57 channel Ampex "CP - 100", or"FR - 1300". The tuner was modified i n two ways to increase the s t a b i l i t y of i t s l o c a l o s c i l l a t o r ; the B + power supply was replaced by a better-regulated one (Dressen - Barnes Model 32 - 101), and the ganged a i r - d i e l e c t r i c tuning capacitors were replaced with g l a s s - d i e l e c t r i c v a r i a b l e capacitors. These modifications and the sp e c i a l care taken with the c a r r i e r o s c i l l a t o r i n the buoy reduced the reproduced noise of the system to less than 1 m i l l i v o l t p-p over a frequency range of 0.01 - 10 Hz. This noise test was of necessity carried out with a fi x e d 5 pf capacitor replacing the microphone, to remove ambient pressure noise. The dynamic range of the enti r e system was li m i t e d by the bandwidth of the Intermediate Frequency amplifier i n the FM tuner/. This amplifier was c a r e f u l l y tuned to give maximum bandwidth and good l i n e a r i t y . The output voltage of the tuner (from a r a t i o detector demodulator) i s plotted against input frequency to the IF amplifier i n Figure 1.0.. 4.9 C a l i b r a t i o n of Pressure Sensor In the paragraphs that follow, descriptions w i l l be given of the c a l i b r a t i o n techniques used i n the determination of the s e n s i t i v i t y of the pressure sensor (the term "pressure sensor" w i l l hereafter r e f e r to the complete pressure measurement system). Both f i e l d and laboratory c a l i b r a t i o n s were done. The f i e l d tests determined amplitude s e n s i t i v i t y only, and were necessary because of day-to-day v a r i a t i o n s i n s e n s i t i v i t y caused by the rubber waterproofing diaphragm covering the pressure port (see the de s c r i p t i o n of the pressure measurement system on p. 5 1 ) . The laboratory c a l i a b r a t i o n s were more precise and more elaborate, giving phase response as well as amplitude response. 58 4 .9 .1 Laboratory C a l i b r a t i o n s The setup for the c a l i b r a t i o n s i s shown i n F igure 11. The e n t i r e buoy was placed ins ide a "clean" ten-ga l lon o i l drum at one end of which was a back p l a t e , through which were led the cable attached to the buoy and a short length of 2.5 mm ID s t e e l tubing, which was led v i a an 0 - r i n g f i t t i n g into one pressure port of a "Barocel" pressure sensor. The other port of the Barocel was at ambient atmospheric pressure . Both the pressure tube to the Baroce l and the buoy cable were sealed c a r e f u l l y to the back p la te of the drum. The other end of the drum was closed with a t h i n , taut rubber mem-brane. Bearing on th i s membrane was a c i r c u l a r aluminum or perspex d i s c about 15 centimeters i n diameter. This d i s c was made to o s c i l l a t e along i t s (hor izonta l ) axis by a l i n e a r d r i v e r (Ling A l t e c v47/3 V i b r a t i o n Generator) , which was c o n t r o l l e d v i a a low-frequency power ampl i f i e r by a s ine wave generator. By su i tab l e adjustment of the pressure exerted by the membrane on the d i s c and of the amplitude and frequency of the s i g n a l to the d r i v e r , s i n u s o i d a l pressures with amplitudes from 10 to 500 dyne c m - 2 (as measured with the Barocel) over a range of frequencies from 0.003 to 10 Hz could be generated i n the drum. Amplitudes used i n ac tua l c a l i b r a t i o n runs were 30 - 100 dyne cm (the reason for choosing these large amplitudes i s discussed on p. 5 9 ) ; the accuracy of the Barocel sensor i s bet ter than * 1% of f u l l s ca l e , which meant* 0.4 dyne cm ^ for the 30 dyne cm amplitude and - 1.3 dyne cm for the 100 dyne cm"2 amplitude, over a frequency range 0 - 10 Hz. The analog outputs from the buoy system and the Barocel were recorded on a two-channel chart recorder (for frequencies below 0.1 Hz) . They were 59 also observed with an o s c i l l o s c o p e , phases being measured by the Lissajous f i g u r e technique. This method proved impractical at frequencies less than 1 Hz. At frequencies below 1 Hz the phase differences were measured d i r e c t l y from the sinusoidal traces on the chart recordings. The p r i n c i p a l source of error i n making the c a l i b r a t i o n s was the v a r i a b i l i t y of ambient atmospheric pressure i n the room where the apparatus was set upV. This was presumably due to f l u c t u a t i o n s i n the wind speed outside, and was on windy days large enough (up to 100 micro-bars) to preclude any c a l i b r a t i o n s at a l l . Hence the c a l i b r a t i o n s were done on calm days and with as large a pressure amplitude as possible i n the drum. This upper amplitude l i m i t was set by the dynamic range of the buoy system, which was i n turn set by the bandwidth of the tuner IF amplifier (See " E l e c t r o n i c s " , p. 5 7 ) . Since most of the noise a r i s i n g from ambient pressure was at low frequencies the accuracy of the c a l i -brations decreased r a p i d l y below about 0.05 Hz, with amplitude accuracy f a l l i n g o f f considerably f a s t e r than did phase accuracy. The laboratory c a l i b r a t i o n of the pressure system used i n the f i e l d i s shown i n Figure 12. The 107o difference at the two amplitudes i n the s e n s i t i v i t y i s ascribed to a non-linear response of the rubber waterproofing diaphragm. 4.9.2 F i e l d Calibrations These were made necessary by the presence of the waterproofing diaphragm. Since i t was natural rubber and was treated with a t h i n f i l m of o i l before the s t a r t of each run, i t was f e l t necessary to carry out f i e l d c a l i b r a t i o n s , and these were done before and preferably a f t e r every run. The method used was simple: the buoy was l i f t e d from the water 60 (or from the bench, i f the c a l i b r a t i o n was made before the r u n ) , and moved v e r t i c a l l y a measured d i s tance , u s u a l l y about 100 centimeters. The voltage change r e s u l t i n g from this pressure change was recorded on the chart of an o s c i l l o g r a p h and th i s d iv ided by f f t 3 ^ h > where ^ i s a i r dens i ty , g i s g r a v i t y , and &V\ i s the height change, gave the s e n s i t i -v i t y . Usua l ly the buoy was moved up and down a number of times and the average voltage change taken. The accuracy (about 107o i n s e n s i t i v i t y ) of th i s method was considerably less than the laboratory c a l i b r a t i o n . Inaccuracies were caused by the i n v a r i a b l e presence of large ambient pressure f luc tua t ions and by the measurement of the short distances over which the buoy could be r a i s e d and lowered. 4.10 The Buoy 4.10.1 Underwater Shape The buoy was made from styrofoam; the bas i c shape was turned on a lathe and a l l fur ther work was done by hand. Af ter f i n a l shaping and c u t t i n g , the e n t i r e surface was coated with epoxy r e s i n . The buoy had a diameter of about 23 centimeters and was about 3 centimeters t h i c k . I t i s shown i n F igure 13. Note i n the f igure that the part of the buoy containing the backup volume was attached to the front s ec t ion . The f i n a l underwater shape was a r r i v e d at p r i m a r i l y by t r i a l and e r r o r . Slow-motion movies were taken of many " t r i a l buoys" i n a v a r i e t y of condi t ions , and most of the information contained i n th i s sec t ion was obtained from these movies. I t was necessary to make the seakeeping c h a r a c t e r i s t i c s of the buoy such that i t would conform to the water surface for as large a percentage of the time as pos s ib l e . The most s t r ingent experimental requirement for the buoy was that i t be constrained to move only v e r t i c a l l y . Since the buoy was free to t i l t with the waves, being attached to the bearing on the wave probe with gymbals, i t was therefore unstable to drag forces exerted on i t by surface currents; that i s , small t i l t s with respect to the water surface were ampl i f i ed . Most e a r l y models of the buoy suffered from th i s i n s t a b i l i t y . Not even the present model was immune, and i t i s to the c r e d i t of the microphone manufacturers that t h e i r product survived "divesV of as much as 10 cm below the surface of the water. The d i v i n g propensi ty was minimized s t a t i c a l l y by put t ing as much buoyancy as poss ib le at the bow of the buoy. This was aided dynamically by making the entry of the bow into the water v e r t i c a l , by " f a i r i n g " the underwater surface , and by p lac ing the gymbals as low as pos s ib l e . I t was discovered ear ly that i n s i t u a t i o n s of ac t ive wave generat ion, the surface water current formed by the wind commonly doubles the o r b i t a l c re s t v e l o c i t i e s of the waves. Thus whereas l i t t l e care needed to be taken with the downwind s ide of the buoy, the shaping of the bow was of i c r i t i c a l importance. The e f f e c t on the buoy of th i s current , besides the d i v i n g propensity a lready mentioned, was the formation of a "bow wave" ahead of i t which could with the a id of the wind be swept up and over the buoy, u s u a l l y reaching the sensor. When th i s happened a large "spike" appeared i n the pressure s i g n a l . Because of the f a i r i n g of the bow the current caused i t to r i d e up on the wave. This e f f ec t was counterbalanced i n two ways: f i r s t , the centre of g r a v i t y of the buoy was kept w e l l forward by judic ious arrange-62 ment of i t s "payload"; second, some of f l o t a t i o n of the f ront h a l f of the buoy was located behind the gymbals, so that t i l t s assoc iated with waves shorter than the radius of the buoy were r e s i s t e d . The f l o t a t i o n under the bow was ended abrupt ly a f ter 4 cm to cause flow separat ion , thus s t a l l i n g the flow beneath the buoy and enhancing the l i f t of the bow f a i r i n g . In th is way the conf igurat ion was designed to keep the bow of the buoy i n a s tate of s t a t i c and dynamic balance over a wide range of water motions. The rear sec t ion of the buoy was hinged to allow the buoy to respond to waves with wave-lengths comparable with i t s r a d i u s . I t had enough f l o t a t i o n to support a h igh-drag f i n , which kept the buoy pointed w i t h i n 2 0 ° of the wind d i r e c t i o n . The sole func t ion of the rear sec t ion of the buoy besides those mentioned, was to make the buoy symmetrical , thus avoiding the p o s s i b i l i t y of r e v e r s a l of the buoy d i r e c t i o n i n cases where the wave o r b i t a l v e l o c i t i e s i n the troughs were greater than the wind-generated surface current . Because short wavelength waves have s i g n i f i c a n t amplitudes i n a wind-generated sea the t i l t i n g parts of the buoy had to be cut away ( ind ica ted by dotted l ines i n F igure 13) on e i ther s ide of the hinges , a l lowing i t to f l e x as much as T 3 0 ° from the h o r i z o n t a l . This contr ibuted g r e a t l y to i t s seakeeping a b i l i t i e s . 4 . 1 0 . 2 U p p e r Surface: Aerodynamics The upper surface of the buoy was a cover beneath which were sealed the pressure sensor and i t s e l e c t r o n i c s , the backup volume, a bat tery which powered the e l e c t r o n i c s , and a small r e l a y which provided a means of switching o f f the bat tery current when the instrument was not i n use. Pressures were transmitted to the sensor through the th in rubber 63 diaphragm, which was glued over a 0.8 cm diameter hole i n the perspex p la te (8.2 cm x 3.9 cm x 0.3 cm--see F igure 13) which sealed the sensor and i t s e l e c t r o n i c s into i t s compartment. The bat tery and the r e l a y were he ld i n place with a s i l i c o n e rubber sea lant . The upwind surface of the buoy was waterproofed with a sheet of t h i n polyethylene which covered a l l j o i n t s . This was he ld securely i n place with a f a s t - s e t t i n g rubber-base cement and masking tape. As a f i n a l p r o t e c t i o n the e n t i r e surface of the buoy except for the diaphragm was spray-painted white, to keep r a d i a t i v e heat transfer to the buoy, and i n p a r t i c u l a r to the backup volume,to a minimum. P r i o r to the design phase of the experiment, i t was r e a l i z e d that the d i s t o r t i o n of the mean streamlines of the a i r flow by the buoy could cause spurious pressure f luc tua t ions at the pressure sensor, which i f not re jec ted i n some way could s e r i o u s l y a f f e c t the v a l i d i t y of the r e s u l t s . To see how great th is e f f e c t would be the s o l u t i o n (see, for ins tance , Lamb (1932), 103) for the p o t e n t i a l flow over a p lanetary e l l i p s o i d of r e v o l u t i o n s i m i l a r to the buoy i n p r o f i l e has been c a l c u l a t e d . The r e s u l t i n g pressure d i s t r i b u t i o n i s shown i n F igure 14. From the f i g u r e , the dynamic pressures exerted on th is shape are asymptotic to -0 .2 times the stagnation pressure i P U" . Although the 2. v * shape'used has a bow ( l e f t s ide of the f igure) which i s much less b l u f f .1 than that used i n the ac tua l buoy, the f igure of -0.1 p^U i s probably w i t h i n 507o of the ac tua l f i gure at the pressure measurement port on the buoy. I f -0 .1 P U i s ca l cu la ted for a 500 cm/sec wind i t gives a value for the dynamic pressure which must be re jec ted at the pressure 64 2 port of about 7 dynes/cm (see p. 4 2 ) . Further c a l c u l a t i o n s show that th is dynamic pressure can be reduced by increas ing the e c c e n t r i c i t y of the e l l i p s e . Thus the best way to minimize shape e f fec t s i s to make the part of the buoy above water as s l i m as p o s s i b l e . I t i s easy to see on the other hand that i f the pressure sensing point i s to be kept r e l a t i v e l y free from water, the best shape would have a b l u f f ( v e r t i c a l ) bow with a small downward slope i n the upwind d i r e c t i o n . The b l u f f bow i s a lso demanded to insure fa s t buoy response to changes i n water l e v e l . Of the two opposing requirements the l a t t e r was dominant. Without a b l u f f bow pressure measurements on the water surface were imposs ible . This meant that the buoy d i s t o r t e d the a i r flow over i t cons iderably , and that large dynamic pressures could be expected at the measurement p o i n t . I t was necessary to r e j e c t these pressures; therefore , a ser ies of wind tunnel tests described i n the fo l lowing sec t ion was c a r r i e d out to f i n d the best method for doing t h i s . 4.11 Wind Tunnel Tests Three ser ies of tests were performed, a l l of them i n the wind tunnel of the Department of Mechanical Engineering of the U n i v e r s i t y . The f i r s t two ser ies were developmental. The f i r s t was concerned with the determining an optimum aerodynamic shape for the buoy w i t h i n the l i m i t a t i o n s stated i n the l a s t sec t ion; the second was taken up with the search for an adequate method for r e j e c t i n g the dynamic pressure caused by the bow of the buoy. The t h i r d ser ies of measurements were used to 65 " c a l i b r a t e " the buoy aerodynamical ly . 4.11.1 The Wind Tunnel The wind tunnel used for the tests was of the c lo sed-re turn type. I t generated wind speeds between 1 and 45 meters /sec , with low (0 .17„) turbulence l e v e l s , and s p a t i a l v a r i a t i o n s i n mean v e l o c i t y i n the tes t sec t ion less than 0.257». The tes t s ec t ion was rectangular and was 0.9 meters wide and 0.7 meters h i g h . I t was provided with tapering f i l l e t s , which p a r t l y compensated for boundary layer growth along i t s length. A i r speed i n the tes t s ec t ion was measured i n terms of the pressure d i f f erence along a 7:1 contrac t ion immediately upstream of the s ec t ion . The speed c a l i b r a t i o n incuded the e f f e c t of v a r i a t i o n s i n mean a i r dens i ty caused by ambient temperature and pressure . The pressure decrease along the contrac t ion was measured with a micromanometer to an accuracy of T20 dynes/cm. This provided values for the free stream wind speed accurate to T17G at 1 meter/sec and T0.57<> at 5 meters/sec . 4.11.2 Experimental Procedure Wind tunnel t e s t ing was done d i r e c t l y on the buoy used i n the f i e l d . The general setup i s shown i n F igure 15. The buoy was recessed i n the f l o o r of the tes t sec t ion at the same l e v e l at which i t f l oa ted on the water; i t was placed at the center of the tes t sec t ion f l o o r on the tunnel a x i s . The bow of the buoy could be set at various p i t c h (attack) angles and yaw (or ientat ion) angles by simple adjustments. In a l l cases, openings i n the s tructure of the buoy and around i t s edges, where i t met the tunnel f l o o r , were c a r e f u l l y sealed with tape. The wave probe which would e x i s t i n the f i e l d measurements was replaced with a laboratory stand 66 of the same diameter, the base of which supported the buoy. Pressures along the upstream h o r i z o n t a l axis of the buoy were measured at f i v e c a r e f u l l y prepared pressure taps (0.75 mm holes). These were d r i l l e d i n a slab of perspex which c l o s e l y resembled the one which sealed the microphone and e l e c t r o n i c s into the buoy (see Figure 13). They were located at 0.53, 0.57, 0.63, 0.66, and 0.69 buoy r a d i i (slant distance) from the bow of the buoy, and are shown i n Figure 16. Pressures were measured r e l a t i v e to the s t a t i c pressure r i n g of the tunnel with a "Barocel" d i f f e r e n t i a l pressure transducer. The l i n e s from the pressure taps and the s t a t i c r i n g to the Barocel were pneumati-c a l l y f i l t e r e d to remove high-frequency pressure f l u c t u a t i o n s present i n the turbulent boundary layer. In spite of t h i s , considerable low-frequency f l u c t u a t i o n s (amplitudes of about 10 dynes/cm) remained; these were found to be smaller on calm days and so were presumed to be the r e s u l t of wind-induced dynamic pressures on the b u i l d i n g housing the tunnel. Therefore the wind tunnel work was done whenever possible on days when the winds were less than a few knots. To reduce these errors a l l measure-ments were averaged over a period (one minute) exceeding the range of periods containing most of the ambient pressure f l u c t u a t i o n energy (0 to 20 seconds). 4.11.3 The Dynamic Pressure Rejection Ring During the e a r l y design stages of the experiment many methods were considered for cancelling the dynamic pressure, which i n this case was a negative (suction) pressure caused by the flow over the boy. The method used i n the actual experiment was by far the best. A thin h a l f -r i n g (see plan view of Figure 13) 4.2 cm I.D. and 0.27 cm high was placed 67 on the surface of the buoy a short distance (1.9 cm) downstream of the pressure por t , with i t s axis normal to the buoy surface and i n t e r s e c t i n g i t jus t upstream of the por t . This obs truc t ion produced a p o s i t i v e pressure at the port which cance l led w i t h i n "tO.l dyne c m - 2 the suct ion pressure from the bow when the buoy was s i t t i n g l e v e l i n the tunnel . Because of i t s symmetry, the r i n g removed almost completely any s e n s i t i -v i t y of the system to yaw (that i s , r o t a t i o n of the buoy about i t s v e r t i c a l a x i s ) . 4 .11.4 The A r t i f i c i a l Turbulent Boundary Layer The a i r flow i n the tunnel was laminar, while that i n the f i e l d , i n the atmospheric boundary l a y e r , was turbulent . Since the atmospheric boundary layer i s so th ick ( t y p i c a l l y 100 - 300 meters) , i t s Reynolds number cannot be matched i n laboratory wind tunnels except at extremely high wind speeds. Therefore an a r t i f i c i a l turbulent boundary layer was created i n the wind tunnel using a t r i p 1.7 meters upstream of the buoy (see F igure 15). The t r i p consis ted of a 2.5 cm square length of wood extending over the f u l l width of the tunnel floory. into which were cut s l o t s about 1.5 cm deep with widths vary ing from 0.5 to 2 cm, at separa-t ions of 1 - 2 cm. V e l o c i t y p r o f i l e s i n the turbulent boundary layer were measured with a small P i t o t tube mounted on a remote ly -contro l l ed t ravers ing mechanism which i s part of the wind tunnel equipment. The mean wind p r o f i l e s produced i n th i s way followed the "Law of the Wal l" : U / u * = f ( z u * / V o ) = f ( y * ) , where uy<. i s the f r i c t i o n v e l o c i t y and Da. the kinematic v i s c o s i t y of the a i r j they e x h i b i t a s t r a i g h t l i n e on a p l o t of U/u V f versus log (yV r) over a range 2 £ log y* 4 3.6. Because the t r a n s i t i o n to turbulence was a r t i f i -68 c i a l l y t r i p p e d , the f r i c t i o n v e l o c i t y was larger and hence the l i n e lay below that found experimental ly for the smooth- trans i t ion Law of the Wall (see, f or ins tance , Wil lmarth and Wooldridge 1962, F igure 3) . 4 .11.5 The Aerodynamic C a l i b r a t i o n of the Buoy Measurements of the pressure at a l l f i v e taps were made over ranges of values of three d i f f e r e n t parameters: wind speed, p i t c h , and yaw. Of these three, wind speed and p i t c h were the most important; symmetry of the r i n g was e f f e c t i v e i n e l i m i n a t i n g almost a l l s e n s i t i v i t y to yaw for measurements up to 1^45°. Three wind speeds: 3 .5 , 4 .3 , and 6.0 i meter sec , and f i v e p i t c h angles: -5 , 0, 5, 10, and 20 degrees, were used throughout the t e s t s . The r e s u l t s , which const i tute the aerodynamic c a l i b r a t i o n of the buoy, are shown i n Figures 17, 18, and 19. These show the pressure d i s t r i b u t i o n over the buoy under the most severe con-d i t i o n s expected i n the f i e l d . F igure 17 shows the d i f ference between ambient pressure and that' measured along the buoy surface as a funct ion of dis tance from the bow i n buoy r a d i i ; the pressure d i f ference i s not normalized to the stagnation pressure so that a bet ter "feel" may be obtained for the s i z e of the dynamic pressures e x i s t i n g on the surface of the buoy (at 6 meters per a. o second the stagnation pressure \ £ a U i s about 220 dyne cm" z ) . The pressure v a r i a t i o n along the buoy i s shown for various attack angles of the buoy, which are taken to be p o s i t i v e i f the bow of the buoy i s t i l t e d upwards. At attack angles less than - 5 ° the bow of the buoy would be sub-merged i f i t were i n the water. The arrow between tap pos i t ions 2 and 3 ind icates the p o s i t i o n chosen for the i n l e t to the pressure sensor used i n the f i e l d experiments. I t might be thought that the arrow should 69 be further from the r i n g than shown i n view of the large s e n s i t i v i t y of the buoy to negative attack angles. This larger s e n s i t i v i t y i s o f f s e t , however, by the response of the buoy to actua l waves; p o s i t i v e attack angles occur more often (and are larger on the average) than do negative attack angles . F igure 18 shows "worst-case" v a r i a t i o n of pressure along the buoy for or i en ta t ions of the buoy with respect to the wind ("yaw" angles) of T30° for an at tack angle of + 5 ° i n a 6 meter sec"^ wind. The t o t a l pressure v a r i a t i o n at the buoy surface for a 6 0 ° change i n yaw angle i s less than 2 dyne c m , or 17o of the stagnation pressure pd = \ U F igure 19 shows the v a r i a t i o n of the pressure at the "optimum" sensor i n l e t l o c a t i o n versus wind speed for various angles of a t tack . I t can be seen immediately i n view of the design c r i t e r i a (dynamic pressure r e j e c t i o n to less than 0.07 p^; see p. 4 2) that attack angles of more than + 1 0 ° are unacceptable; i t can a lso be seen that more than 957o of p j i s re jec ted i f the buoy's attack angle i s less than + 1 0 ° and more than - 5 ° i n the wind speeds as h igh as 7 meters per second (one run was made i n the f i e l d where the wind speed was 8 meter sec"^; from the f igure i t appears that about 6% dynamic pressure contamination occurred i n th i s run) . A t attack angles greater than 1 0 ° , flow separat ion pro-bably occurs at the bow of the buoy at wind speeds less than 6 m sec"''; during the tests separat ion appeared to be present at a l l wind speeds above 3 m sec for a + 2 0 ° attack angle. When s e l e c t i n g runs for sub-sequent a n a l y s i s , a l l runs where the attack angle often exceeded 10 degrees or where the o r i e n t a t i o n angle exceeded 30 degrees were r e j e c t e d . 70 4.11.6 Consequences of Attack Angle V a r i a t i o n F igure 20 shows a schematic representat ion of the e f f e c t on the observed pressure of l i k e l y v a r i a t i o n s i n the angle of attack of the buoy as i t r i d e s over a wave on an e x i s t i n g pressure s i g n a l i n quadrature with the wave e l e v a t i o n . Although the shape of the s i g n a l , and hence i t s amplitude, i s changed cons iderably , i t s phase i s almost una l tered . The p r i n c i p a l e f f ec t of angle of attack v a r i a t i o n s appears to be the i n t r o d u c t i o n in to the observed pressure s i g n a l of harmonics of the wave frequency, i n p a r t i c u l a r of the f i r s t harmonic. SECTION 5: DATA ANALYSIS and INTERPRETATION 5.1 Ana lys i s . . 71 5.1.1 Analog Precondi t ion ing 71 5 .1 .1a Pressure 71 5.1.1b Waves 73 5.1.1c Playback and S e l e c t i o n of Data 73 5 .1 . Id Rerecording 74 5.1.2 D i g i t i z a t i o n 75 5 .1 .2a Hand D i g i t i z a t i o n 75 5.1.3 The IOUBC Fast F o u r i e r Transform Package 79 5 .1 .3a FTOR: The Fast F o u r i e r Transformation Program 80 5.1.3b The Spec tra l Ana lys i s Program SCOR . . . . 81 5.1.4 S p e c i a l Programming 84 5.1.4a The Wave Height S igna l 84 5.1.4b The Pressure S ignal 85 5.1.4c The Sonic Anemometer Signals 86 5.1.4d Spec ia l Programming i n SCOR 87 5.2 I n t e r p r e t a t i o n of the Spectra 88 5.2.1 D i s t o r t i o n s Introduced by the Sensors Themselves . 88 5 .2 .1a The P i t c h i n g A c t i o n of the Buoy 88 5.2.1b The F i n i t e Size of the Buoy 91 5.2.1c Backscat ter ing 94 5.2.2 D i s t o r t i o n s Introduced during Preparat ion and Analys i s of Data 98 5 .2 .2a F i l t e r i n g of the Wave S igna l 98 5.2.2b The Pressure Signals 99 5.3 Summary 104 SECTION 5: DATA ANALYSIS AND INTERPRETATION 5.1 Ana lys i s The ana lys i s of the data w i l l be considered from the point of view of what i s done to the s ignals between the time they o r i g i n a t e at the sensors and the time when the f i n a l computer pr in tout of spec tra , cross-spec tra , e tc . i s generated. For th i s reason, analog p r e - c o n d i t i o n i n g and subsequent d i g i t i z a t i o n w i l l be discussed f i r s t . This w i l l be followed by a b r i e f resume of the d i g i t a l computer programs made a v a i l a b l e to the A i r - S e a I n t e r a c t i o n Group through the labours of some of i t s members, which incorporate the Fast F o u r i e r Transform technique of Cooley and Tukey (1965) into a p r a c t i c a l scheme for obta in ing d i g i t a l spectra from d i g i t i z e d analog voltage s i g n a l s . Next, the programming s p e c i f i c a l l y designed for th is projec t (and which i s hence the work of the author) w i l l be discussed i n some d e t a i l . F i n a l l y , methods of pre-sentat ion of the s t a t i s t i c s of the spectra w i l l be mentioned, as an a id i n the i n t e r p r e t a t i o n of the spec tra . 5.1.1 Analog Precondi t ion ing  5 .1 .1a Pressure The pressure recording system i t s e l f performed the f i r s t f i l t e r i n g of the s i g n a l . The amplitude of the s i g n a l "seen" by the microphone was reduced by about a fac tor of two by the presence of the waterproofing diaphragm; the frequency response of the microphone-waterproofing diaphragm combination i s known to have been f l a t up to 10 Hz and was probably f l a t at 100 Hz, s ince the na tura l resonance of the volume enclosed by the waterproofing diaphragm and that of the microphone occurs at 300 Hz. 72 The combination of the leak around the microphone diaphragm and the backup volume acted as a high-pass pneumatic f i l t e r . Its measured low--f-frequency cutof f was at 50 _10 seconds. A l l of these b u i l t - i n f i l t e r s caused the pressure measurement system to have an amplitude and phase response which v a r i e d with frequency. These responses were measured during c a l i b r a t i o n (see Experiment, F igure 12), and were corrected for i f necessary i n the a n a l y s i s . The FM pressure measurement system employed a phase-sens i t ive r a t i o detector for demodulation of the modulated 100 mc frequency which o r i g i n a t e d at the microphone. The slope of the frequency response curve of the r a t i o detector set the voltage s e n s i t i v i t y of the system and determined the voltage output from the FM tuner for a given input pressure amplitude. The frequency response of the demodulator was found to be f l a t over the e n t i r e frequency range of i n t e r e s t . The s i g n a l from the demodulator was then fed through a s ing le operat iona l a m p l i f i e r , which acted as a buf fer stage between the demodulator and the analog tape recorder and provided a gain which was v a r i a b l e from 0 - 2.5. This a m p l i f i e r allowed optimum use to be made of the dynamic range of the tape recorder; i t s gain was set at the maximum l e v e l cons is tent with keeping the pressure s i g n a l w i t h i n the l i m i t s ( t l . 5 volts) of the tape recorder system. Two tape recorders were used; Ampex models "CP - 100" and"FR -1300". The FM record mode was used, at tape speeds of 1 7/8 or 3 3/4 inches per second ( i p s ) . Both recorders were set up to reproduce the s i g n a l as recorded--no a m p l i f i c a t i o n was attempted. 73 5.1.1b Waves The wave s i g n a l was recorded as described under Experiment (p. 46) i n one of two ways: e i ther the FM output of the wave probe b locking o s c i l l a t o r was recorded i n the D i r e c t mode as an analog s i g n a l or i t was f i r s t demodulated with a separate instrument and then recorded i n the FM mode. In both cases i t was played back i n the FM mode. 5 .1.1c Playback and S e l e c t i o n of Data The signals were played back at the I n s t i t u t e to a chart recorder; the r e s u l t i n g chart records were compared with monitor chart records produced during the run . I t i s on the bas is of these chart recordings that port ions of data were chosen as s u i t a b l e for further a n a l y s i s . Because the s e l e c t i o n c r i t e r i a for the runs were to some extent subjec t ive , being sometimes the r e s u l t of "feel ings that the run looked r i g h t " , they cannot be spe l l ed out completely. The best way seems to be to enumerate reasons why ac tua l runs were thrown out: 1. One of the f a u l t s of the pro tec t ive diaphragm was that i t some-times "bulged out" or became "sucked in" under abnormal tension due to f a i l u r e of the pressure e q u a l i s a t i o n system. The r e s u l t was a r e c t i f i e d pressure s i g n a l , s ince the diaphragm i n th is condi t ion could respond only to pressures which tended to decrease the pressure d i f f erence across i t . Any runs i n which th i s was known to occur or which showed any such r e c t i f i c a t i o n were d i s q u a l i f i e d . 2. I f a drop of water "nested" i n the diaphragm, a c h a r a c t e r i s t i c pressure s i g n a l at the bobbing frequency (2 Hz) of the buoy r e s u l t e d . Runs where th is happened were e a s i l y recognized and thrown out. 74 3. Because of the unavoidable presence of large pressure spikes caused by water splashing on or over the diaphragm, a l l runs except one calm-water case had to be exmined c l o s e l y . Only those with the fewest pressure "spikes" were analysed. In f a c t , ana lys i s was postponed for some months while a (vain) attempt was made to gather more data, on the grounds that some of the best runs presented i n th is thesis were considered by the author (not h i s supervisor) as not worth analys ing! Eventua l ly the only runs thrown out because of spike problems were those where water remained on the diaphragm long enough to change the mean pressure i n the backup volume, leading to the presence of large steps i n the pressure records . 4. The l a s t large source of "bad runs" was d r i f t i n the pressure system. Usua l ly the gain of the buf fer amplifer at the demodulator out-put was kept as high as p o s s i b l e , to maximize the s i g n a l to noise r a t i o . The p r a c t i c a l upper l i m i t to the gain was determined by d r i f t , which i f the gain was too high caused the s i g n a l to exceed the l i m i t s of the tape recorder FM system. Thus runs were re jec ted i f the monitored pressure s igna l s were even p a r t i a l l y bur ied i n noise or i f vol tage l i m i t s were exceeded during part of a ten minute p e r i o d . The remaining runs (there were s i x analysed, of which two were analysed i n two parts ; see "Results") were then prepared for d i g i t i z a t i o n . One run which was h a n d - d i g i t i z e d w i l l be discussed separate ly (p . 75 f f . ) . 5 .1 . Id Rerecording The runs were rerecorded at twice the tape speed (thus reducing noise and time used for ana lys i s ) a f ter precondi t ion ing cons i s t ing of a m p l i f i c a t i o n and high-pass f i l t e r i n g . The low cutof f frequency (0.02 Hz) 75 of the f i l t e r was such that n e g l i g i b l e ( T l ° ) phase s h i f t occurs at the lowest frequencies (0.05 Hz) of i n t e r e s t . To minimize the amount of data l o s t due to r i n g i n g i n the f i l t e r s a DC voltage equivalent to the mean s i g n a l value near the s t a r t of the run was appl ied to each f i l t e r ; then when the run began the DC voltages were replaced by the s igna l s them-se lves . This method e f f e c t i v e l y removed spurious low frequency components from both the wave and pressure s i g n a l s . The a m p l i f i e r gains were set to make optimum use of the dynamic range of the tape recorder . 5.1.2 D i g i t i z a t i o n A l l s igna l s except those processed by hand were d i g i t i z e d on a " D i g i t a l Equipment" A n a l o g - t o - D i g i t a l (A/D) converter, which i s capable of d i g i t i z i n g as many as ten channels of data . This instrument i s i n t e r -faced with a small computer (Control Data 8092 Teleprogrammer) which forms part of the computing f a c i l i t y of the U . B . C . Computing Centre. The 8092 was programmed to wr i te d i g i t a l (A/D) tapes for subsequent process-ing on the main computer at the Computing Centre, which was an IBM 7044 at the time the analyses were done. The d i g i t i z a t i o n was c a r r i e d out i n the same room i n which the 8092 and the 7044 were housed. Signals from the analog tape recorder were passed through low-pass l i n e a r phase s h i f t f i l t e r s with a cutoff f r e -quency of 6 Hz p r i o r to d i g i t i z a t i o n . A sampling frequency of approxi -mately 50 Hz was used for a l l runs. In most cases a short sec t ion of the composite noise of the p r e a m p l i f i e r , f i l t e r s , and tape recorders was also d i g i t i z e d . 5 .1 .2a Hand D i g i t i z a t i o n Runs 4a and 4b were d i g i t i z e d by hand from a chart paper recording 76 of the pressure and wave s igna l s which was taken during the actua l run made on October 30, 1967. The recorder used was a Sanborn Model 320 two-channel recorder . The d i g i t i z i n g was done by Mr. J . R. Wilson, a f e l low graduate student, on a "Thomson Electronics" Model PF-10 P e n c i l Fol lower i n the Canadian Oceanographic Data Centre i n Ottawa. Two separate pieces of data were analysed. The f i r s t of these (run 4a) was s p l i t in to two parts i n order that i t might be analysed for s t a t i o n a r i t y . Since there proved to be no s t a t i s t i c a l l y s i g n f i c a n t d i f ference between the parts they were subsequently merged, and are pre-sented together as run 4a. In both runs , the spikes i n the pressure s ignals were smoothed by hand before d i g i t i z a t i o n by drawing smooth curves j o i n i n g the "good" data on e i ther s ide of them. Two sources of error i n the measurement of time must be considered: those which a f f e c t both channels equal ly and those which do not . The f i r s t type of e rror inc ludes , besides chart v a r i a t i o n s , inaccurac ies i n the d i g i t i z e r . The recorder chart dr ive speed of 5 mm/sec var ie s less than 1"47<,; the claimed r e s o l u t i o n i n the d i g i t i z e r i s T0.1 mm, which corresponds to TO. 02 seconds or T27=; so the t o t a l expected absolute time r e s o l u t i o n i s T4.57o. The second type of error i s the more serious of the two; i t involves processes which produce time s h i f t s between the data d i g i t i z e d from d i f f e r e n t channels at d i f f e r e n t times. There are two sources of such e r r o r s : those associated with the chart recorder and those a r i s i n g from the d i g i t i z a t i o n procedure. The chart dr ive of the recorder i s arranged so that the paper i s p u l l e d over a r idge with s u f f i c i e n t s t r a i n to keep i t taut; heated s t y l i bearing on the r idge produce the trace . The 77 s p e c i f i e d maximum misalignment i s TO.25 mm across the paper which i s 5 cm wide; th i s causes poss ib le misalignments between the channels equi -va lent to T.025 seconds. This sets an upper l i m i t of t l 8 ° at 2 Hz for errors due to th i s source. Another source of error may a r i s e from di f ferences i n the high frequency response of the two s ty lus d r i v e r a m p l i f i e r s . Since the wave s i g n a l i s e s s e n t i a l l y zero by 10 Hz, and s ince the recorder s p e c i f i c a t i o n s i n d i c a t e that when proper ly compensated the a m p l i f i e r responses are only down 3 db at 100 Hz, th i s error should be smal l . However a m p l i f i e r compensation had not been adjusted for some time previous to the run; a lso the pressure of the s ty lus recording the waves increased markedly towards the end of the ac tua l recording (which extended 40 minutes beyond the end of run 4b) causing an at tenuat ion i n i t s response which was v i s i b l e at frequencies as low as 1 - 2 Hz. These fac t s led to a more care fu l cons iderat ion of the e f fects of the frequency response of the chart recorder . The wave s igna l s at e a r l i e r parts of the chart record were v i s u a l l y compared with those at times l a t e r than the two data sect ions (runs 4a and 4b) chosen for ana lys i s ; s ince the h igh-frequency p o r t i o n of the s i g n a l seemed i d e n t i c a l i n appearance both before and a f ter the pieces chosen i t was assumed that the e f fected of s ty lus pressure d i d not set i n u n t i l w e l l beyond the end of run 4b. Moreover the s p e c t r a l shapes computed l a t e r for these runs were i n no obvious way d i f f e r e n t at high frequencies from those for other runs. A r e a l i s t i c estimate of the phase error introduced because of excess s ty lus pressure i s d i f f i c u l t to make. I ts ac t i on was that of a low-pass f i l t e r , and i t v i s i b l y attenuated frequencies of 1 Hz 40 minutes beyond the end of run 4b; i t had no apparent e f f ec t , however, 20 minutes beyond the end of run 4b. This means that i t s cutof f frequency was 78 decreasing with time. An ex trapo la t ion assuming the e f fec t began at the beginning of run 4a gives 75 Hz for the cutoff frequency at the end of run 4b. I f the f i l t e r associated with excess s ty lus pressure tan be assumed to have a transfer funct ion s i m i l a r to that of a simple RC low-pass f i l t e r , then the phase lag produced by such a f i l t e r i s less than 1 ° over the whole frequency range of i n t e r e s t i n th is work (0.05 - 5 Hz) and hence w i l l be ignored as a source of phase e r r o r . The d i g i t i z a t i o n procedure introduces errors through misalignments of the time axis by the operator; for th i s reason, great care was taken to insure accurate alignment. As a fur ther check a sum of three s ine waves of d i f f e r e n t frequencies covering the range of i n t e r e s t was generated on a computer and p lo t t ed twice side by s ide by a Calcomp p l o t t e r to resemble the data as i t appeared on the chart paper. These s igna l s were then h a n d - d i g i t i z e d with the same care and analysed i n the same way as the data. The phase between channels from one cross-spectrum computed from the two h a n d - d i g i t i z e d test s ignals are given below i n Table 5.1. TABLE 5.1 Summary of Information from Test Hand-Dig i t i zed Data Frequency of Spectra l Peak (Hz) at peak Spectra l Estimate (dyne c m " 2 ) 2 Hz" Re la t ive Phase between Channels 0.59 0.3 1.27 32.5 36.0 23.0 - 2 . 0 ° - 2 . 6 ° - 0 . 8 ° 2.54 10.5 4.2' o *Note: The s igna l i s m u l t i p l i e d by the same c a l i b r a t i o n factor used for the pressure i n runs 4a and 4b. 79 This i s considered the best estimate of the phase errors caused by the d i g i t i z a t i o n . I t ind icates that i t i s considerably less than the "i"0.1 mm, or "to.02 seconds, r e s o l u t i o n s p e c i f i e d for the d i g i t i z e r . The errors from a l l sources produce a maximum expected phase error of about (18 2 + 6 2 + 4 2 )^ , or about 3 2 0 ° at 2 Hz; the error decreases l i n e a r l y as the frequency i s decreased. Amplitude errors i n the wave s i g n a l other than those associated with the probe and i t s c a l i b r a t i o n are caused by errors i n the gain of the chart recorder . The gain of both channels of the recorder has been checked i n the f i e l d and on ne i ther channel were errors greater than *27o. This i s then the expected maximum error a t t r i b u t a b l e to the hand-d i g i t i z a t i o n process i n the waves and pressure s i g n a l . I t should be noted that while the error i s about equal to c a l i b r a t i o n errors for the waves s i g n a l , the large uncer ta in t i e s (*207o) i n the pressure c a l i b r a t i o n swamp this error and i t can sa fe ly be ignored. 5.1.3 The IOUBC Fast F o u r i e r Transform Package This package was w r i t t e n by J . F . G a r r e t t and J . R. Wilson, students at the I n s t i t u t e . I t i s a ser ies of programs, some i n FORTRAN IV, and some i n assembler language which i n s t r u c t s the IBM 7044 computer to read the tapes generated by the 8092, store the data i n the computer memory, and perform a Fast F o u r i e r Transform (FFT) on them; the F o u r i e r c o e f f i -c ients r e s u l t i n g from the FFT process are used to compute spectra for a l l channels and cross - spec tra between se lected pa irs of channels. Included i n the package are rout ines for p l o t t i n g the d i g i t i z e d s ignals on a Calcomp p l o t t e r , c a l c u l a t i n g and p r i n t i n g out the d i s t r i b u -t i o n of d i g i t i z e d values over the range of the A/D converter for a l l 80 channels and the f i r s t four moments of these d i s t r i b u t i o n s ( th i s program i s run r o u t i n e l y on a l l d i g i t i z e d data and i s extremely useful for detect ing d i g i t a l e r r o r s ) , and for p l o t t i n g out the d i s t r i b u t i o n of Four i er c o e f f i c i e n t amplitudes versus frequency for a l l channels. Other more s p e c i a l i s e d programs, not used i n the present i n v e s t i g a t i o n , are also a v a i l a b l e . 5 .1 .3a FTOR: The Fast F o u r i e r Transformation Program This program c a l l s an assembler language program which causes the data tape generated by the A/D converter to be read. The maximum number of data points which can be read into the computer memory at a time i s 10,240. Since the FFT subprogram PKFORT (Cooley and Tukey, 1966) accepts blocks of data i n i n t e g r a l powers of two, the maximum number of data 13 points per data block for a s ing le channel i s 8192 or 2 ; i f the number of data points per channel i s made less than th i s more channels can be analysed s imultaneously, up to a maximum of ten. Thus the program reads i n for each channel to be analysed a group of 2^  data points hereaf ter c a l l e d a "data block", where j i s set by the user. The only r e s t r i c t i o n s are the maximum number of points which can be simultaneously stored i n the computer memory and that the data blocks must contain a number of points which i s a m u l t i p l e of two. When one data block has' been stored i n memory PKFORT i s c a l l e d . This subroutine replaces the data points i n memory with 2^  complex F o u r i e r c o e f f i c i e n t s which are t h e - b u i l d i n g blocks for the spectra . Each c o e f f i c i e n t consis ts of a r e a l and an imaginary p a r t , and hence contains amplitude and phase information present i n the s i g n a l at a given frequency. The highest frequency i s the Nyquist frequency f g / 2 ,where f s 81 i s the A/D sampling frequency for the data, and the lowest frequency i s 1 / T > where T i s the block length i n seconds and equals 2^/fs. These Fourier c o e f f i c i e n t s are then written on a " c o e f f i c i e n t " tape along with pertinent information such as an i d e n t i f i c a t i o n number, block number, sampling frequency, etc. Following this the next block i s read i n from the A/D tape; the sequence i s repeated for as many blocks as requested on the input data cards or u n t i l an e n d - o f - f i l e mark i s encountered on the A/D tape. 5.1.3b The Spectral Analysis Program SCOR This program reads the c o e f f i c i e n t tape generated by FTOR. Data cards provide information on number of channels and number of blocks to be done, block number at which the analysis i s to begin, channels to be analysed and which are to have cross spectra, whether l i n e a r or l o g a r i t h -mic (approximately one-half octave^ bandwidths are to be used, phase corrections, etc. The program then reads i n Fourier c o e f f i c i e n t s a block at a time. Provision i s made i n SCOR for smoothing the Fourier c o e f f i c i e n t s before the spectra are computed; smoothing i s carried out at this point by a method (see, for example, Bingham and Tukey, 1967) often referred to as' "hanning"; a three-point running average with weights -\, \, and -% i s applied to the Fourier c o e f f i c i e n t s . This i s equivalent to convolving the spectra with a s p e c t r a l window 3 2 T T 4 / ( 3 [ 4 T T 2 - (2 TT f) 2 T 2 ] 2 ) 5.1, where f i s the duration of the data block i n seconds and f i s frequency. Since the spectra are already convolved with a spectral window associated 82 with the f i n i t e length of the data block which var ie s as &£~ , where 6"f i s the frequency i n t e r v a l from the centre frequency (see Appendix 1, Equation A l . l l ) , hanning of the c o e f f i c i e n t s has the e f f ec t of pro-ducing a composite s p e c t r a l window which f a l l s o f f as & f ^ . I t also has the e f f ec t of spreading the energy of sharp peaks i n the unhanned spectra and of conf in ing to the two lowest frequencies of the smoothed spectra any energy associated with large mean values i n the data. The F o u r i e r c o e f f i c i e n t s of a l l the spectra analysed i n th i s experiment have been hanned. This procedure should insure that the large f a l l o f f rates ( Sf""^ approximately) observed i n the wave spectra are r e a l , and that the energy introduced into the low-frequency regions of the spectra by large mean values or d r i f t s i n the o r i g i n a l data are confined to the f i r s t three of four s p e c t r a l est imates. In f a c t i t w i l l be seen when the r e s u l t s of the s p e c t r a l ana lys i s of the f i e l d data are presented that the four- lowest- frequency s p e c t r a l estimates are u n r e l i a b l e . I f the i n d i v i d u a l F o u r i e r c o e f f i c i e n t s for two channels are R-^  + i l ^ and R2 + i l 2 > where i = ( - 1 ) 2 and f equals the block length i n seconds (and equals l / A f where A f i s the bandwidth between F o u r i e r c o e f f i c i e n t s ) then the power spectra are given by <J>n<f) = T ( R 2 + i 2 ) R + I 1 2 1 2 2 A f and $ 2 2 ( f ) = T ( r 2 + i f ) 2 1 5.2; 2 2 A f the cross spectra are C o 1 2 ( f ) = T ( R 1 R 2.+ I 1 I 2 ) R 1 R 2 + I 1 I 2 5.3. 2 2 Af 83 Q u 1 2 ( f ) = X ( R 2 X 1 " R i I 2 ) = ( R 2 X 1 " R 2 I 1 ) 5.3. 2 2 Af The fac tor of two i n the above equations makes the i n t e g r a l under the power spectrum over p o s i t i v e frequencies equal to the s i g n a l var iance . Phase correct ions are made at th i s point by c a l c u l a t i n g the phase 0 ( f ) = tan"-*- / Qu(f) A , correc t ing th i s angle for instrumental responses, I Co(f) / and then r e c a l c u l a t i n g Co and Qu from 0 -Vc - 2 Co(f) = Co( f ) + Qu (f) c o s O e , 5.4 and Qu(f) = V c o 2 ( f ) + Qu 2 ( f ) s i n 8 t , where Bc i s the corrected phase and the uncorrected estimates are under the square root s ign . A f t e r a l l the required power and cross spectra for a given block and other required information on the block s t a t i s t i c s have been s tored , the sequence i s repeated u n t i l a l l the blocks asked for have been pro-cessed or ani iend-of - f i l e mark i s encountered on the c o e f f i c i e n t tape. Then at each frequency the program averages the s p e c t r a l estimates over the number of blocks processes and computes the standard error of each mean and the average trend over the b locks . The standard error of the mean i s computed from I ( S N " « ' * 5.5 I N - l for N r e a l i z a t i o n s (blocks) of the quanti ty Also computed for each frequency are _ [ Co2(f) +,,Qu2(f)1 " L $ 1 1 ($22 1 Coherence = /ft" N ' 2 5.6 84 and Phase = Qu(f) /Co(f)] 5.7. The program also computes the variance of the zero harmonics (the means of the i n d i v i d u a l blocks) and t h e i r trend. This information i s then pr in ted out i n tabular form. 5.1.4 Spec ia l Programming The programming which was designed s p e c i f i c a l l y for th is pro jec t can conveniently be grouped into two sectors: that concerned with the computation of F o u r i e r c o e f f i c i e n t s (FTOR), and that concerned with spectra (SCOR). The f i r s t sector i s by far the larges t ; i t performed a l l d i g i t a l condi t ion ing of the s i g n a l . In the d e s c r i p t i o n of the programs actual computational d e t a i l s are not discussed; instead an attempt i s made to descr ibe what the programs d id to the s i g n a l s . Since the d i g i t i z e d data were brought into the computer i n b locks , the condi t ion ing was done on one block at a time; the d e s c r i p t i o n that fol lows i s of what was done to a s ing le block of data. Condi t ioning of s igna l s from three sensors--wave he ight , buoy pressure , and sonic anemometer--will be descr ibed. The subroutine i n which th is condi t ion ing was c a r r i e d out i s named FIDDLE. 5 .1 .4a The Wave Height S igna l The wave s i g n a l on transfer from FTOR consisted of v a r i a t i o n s i n tape recorder output voltage about some mean vo l tage . In FIDDLE i t had appl ied to i t the nonl inear wave probe c a l i b r a t i o n given i n Equation 4.8 . I t emerged from th i s c a l i b r a t i o n as deviat ions i n centimeters about the known immersion depth of the wave probe. Two separate wave signals were returned from FIDDLE to FTOR; the f i r s t was the wave signal as described above and the second was the same signal m u l t i p l i e d by a "Spike Function" which was one except during the times that spikes occurred i n the pressure s i g n a l , i n which case i t was set to zero..-The reasons for analysing these two wave signals are outlined i n Appendix 1. 5.1.4b The Pressure Signal In FTOR, a c a l i b r a t i o n factor which includes a l l analog gains, inversions and instrument amplitude c a l i b r a t i o n s was applied to the pressure s i g n a l which was then stored as va r i a t i o n s i n pressure i n units of dyne cm . FTOR generated i f required an extra channel ca l l e d S j f o r Spike Function, a l l the data points of which were at this stage set equal to 1 .0. The pressure signal was next transferred from FTOR to FIDDLE along with the S channel. In FIDDLE the f i r s t operation performed on the pressure s i g n a l was to add to each of i t s data points an amount R-fa3**,i where R i s a factor which could be preset to any number including zero, p i s a i r density, g i s the acceleration due to gravity, and \ is the value of the wave signal at the time of zero, one or two samples i n advance of the pressure s i g n a l . The number of samples by which 7£ preceded the pressure signal could be preset. The reason for choosing an e a r l i e r wave si g n a l i s that i n the actual measurements the wave probe was downwind of the pressure sensor; the advance i s an approxi-mation to the phase correction required by the s p a t i a l separation of the sensors. This correction i s discussed i n some d e t a i l i n "Data 86 Ana lys i s and In terpre ta t ion" , on p. 102 f f . The pressure s i g n a l , i f i t had spikes i n i t , was next transferred from FIDDLE to the spike removal subroutine SPKSKP; the S channel of ones was a lso transferred at th i s time. The actual mechanics of SPKSKP are somewhat complicated and t h e i r d i scuss ion i s deferred to Appendix 1, i n which the e f fec t s of spike removal on the pressure spectra i s a lso discussed. SPKSKP detected the presence of spikes on the bas is of rate of change of the pressure s i g n a l ; information was prev ious ly read into a data card on threshold values and number of points to be skipped a f ter the s i g n a l drops below the threshold . SPKSKP replaced the pressure data points and .the ones i n the S channel by zero during spikes . I f more than a preset f r a c t i o n of the data points i n a block were replaced by zeros , SPKSKP caused the "defective" block to be ignored i n the ana lys i s of "clear" b locks , which may however be separated widely i n time, and was used to inves t iga te the e f f ec t of the spikes on the pressure spectra: see Appendix 1, p. 178 f f . Fo l lowing process ing by SPKSKP the pressure s igna l with spikes removed was averaged and the Four i er transform performed on deviat ions from this average. 5 .1.4c The Sonic Anemometer Signals The sonic anemometer produced three s igna l s : v e r t i c a l v e l o c i t y (w) and two h o r i z o n t a l v e l o c i t i e s (A and B) . These are r e l a t e d t r i -gonometrical ly to u and v , the f l u c t u a t i n g components of the wind v e l o c i t y i n d i r e c t i o n s along and at r i g h t angles to U, the mean wind 87 vector for the run (see Experiment, p. 45 ); the two v e l o c i t i e s were used i n FIDDLE to compute u and v. 5.1.4d Spec ia l Programming i n SCOR The cross - spec tra between the pressure and the wave e levat ions were used as described i n the fo l lowing paragraphs to compute the mean f lux of energy E and of momentum from the a i r to the waves v i a f l u c t u -at ions i n pressure . I t i s these ca l cu la t ions which were s p e c i a l to th is program, and the m o d i f i c a t i o n of SCOR which performed them was c a l l e d SCORF. The mean f l u x of energy from the wind to the waves i s given by the covariance ~~T~ E = p ( , t;a^ u ; / -a t 5.8. SCORF f i r s t computed the spectrum of. energy f l u x E(f) and then integrated under th is to get E . Because the pressure and wave s ignals were composed almost e n t i r e l y of noise above 3 Hz (see p.104 of th is sect ion) the i n t e g r a t i o n was truncated at th i s frequency. The computation done i n SCORF was 3.0 3.0 <E> = Y~ E ( f i ) : A f , = X " 2lt f i Q u , (f ) A f . 5.9, f i = 105 .05 where the lower frequency l i m i t of 0.05 Hz i s the frequency for which t h e r e l i s one complete cycle i n a 1024-point data b lock;< E } i s the integrated energy f l u x for one b lock , A f ^ i s the bandwidth of the s p e c t r a l est imates, and Qupj"^ (^±) t ' i e quadrature spectrum of the spike-contaminated pressure and wave s i g n a l s . The quant i t i e s shown i n Equation 5.9 were computed separate ly for each data block. The mean 88 energy f lux E over the number of blocks i n the run and the standard error i n th is mean were then computed. The spectrum at a frequency f^ of the momentum f l u x from the wind to the waves i s given by * ^ w ( f i ) = - = ( f i ) / C i = 2 f f f . E ( f i ) / g 5.10, where = g / 2 T T f i i s the phase v e l o c i t y of the wave component of frequency f^. SCORF computed, for each data b lock , < T w ) = i / s i ^ 4 T T 2 f 2 i Q u p ^ s ( f ± ) A f i 5.11; the i n t e g r a l under * £ * w ( f i ) w a s truncated at the same frequencies as the i n t e g r a l under E(f^) . As for E ( f ) , SCORF computed t , the mean of '^"J** w ^ over the b locks , and the standard error i n th is mean for the run . 5.2 I n t e r p r e t a t i o n of the Spectra This sec t ion contains f a i r l y de ta i l ed discuss ions of the ef fects on the spectra presented i n "Results" of various instrumental short -comings, and also gives b r i e f accounts of the f i l t e r i n g processes per-formed on the pressure and wave data p r i o r to and during s p e c t r a l a n a l y s i s . The purpose of the sec t ion i s to describe processes which may introduce s i g n i f i c a n t d i s t o r t i o n s i n the spectra and give wherever poss ib le a quant i ta t ive estimate of t h e i r e f f e c t , so that these spectra may be in terpre ted o b j e c t i v e l y . 5.2.1 D i s t o r t i o n s Introduced by the Sensors Themselves 5 .2 .1a The P i t c h i n g A c t i o n of the Buoy The forced response of the buoy (which i s hinged so that i t can 89 respond to steep, short waves"--see F igure 13) to the motions of the water surface i s a complex motion which i s not e a s i l y described a n a l y t i c a l l y . Observations of the movements of the buoy have been made with the a id of slow-motion movies, and these ind ica te that most of the changes i n the angle that the bow of the buoy makes with respect to the instantaneous water surface ( th i s angle i s r e f e r r e d to hereafter as the "pitch'angle") are associated with o s c i l l a t i o n s of the f ront sec t ion (bow) of the buoy about i t s hinge. A resonance i n th is p i t c h i n g mode occurs at 2 T 0.2 Hz. I t i s thought that the larges t d i s t o r t i o n s of the pressure s igna l (and to some extent the wave s ignal ) which can be d i r e c t l y a t t r i b u t e d to the motion of the buoy are the r e s u l t of these o s c i l l a t i o n s . The pressure s igna l i s af fected by the o s c i l l a t i o n s s ince the e f f i c i e n c y of the dynamic pressure r e j e c t i o n arrangements var i e s with the p i t c h angle of the buoy (see f igure 19); p o s i t i v e p i t c h angles (bow t i l t e d up) produce spurious negative dynamic pressures to appear at the measurement l o c a t i o n , so that p i t c h angle and pressure are 1 8 0 ° out of phase. Because of the resonance at 2 Hz the e f f ec t of the p i t c h i n g of the buoy var ies with frequency. At resonance the p i t c h i n g w i l l be i n phase with the waves; consequently the spurious pressures produced should be i n antiphase with the waves and w i l l increase the magnitude of the p, ^ cospectrum as wel l as producing a "bump" i n the pressure power spectrum. The e f f ec t on the wave spectrum should be n e g l i g i b l e . At lower frequencies , the p i t c h f luc tuat ions of the buoy can be expected to lag the waves by 9 0 ° . This case has been mentioned 90 ("Experiment"; p. 70); the expected quadrature pressures generated by such f l u c t u a t i o n s are shown i n F igure 20. I t can be seen that the most important e f f e c t i s the generation of a large spurious f i r s t harmonic of the pressure s i g n a l coherent with the f i r s t harmonic of the waves. This spurious pressure s i g n a l w i l l cause a f a l s e observed correg l a t i o n at twice the fundamental frequency of the waves. I t w i l l be larges t at 2 f , which i s twice the frequency of the peak of the wave spectrum. Because of the phase r e l a t i o n between the n a t u r a l ("Stokes'") f i r s t harmonic i n the waves and the fundamental, the spurious pressure s i g n a l w i l l lead the Stokes' harmonic and i t s e f fects w i l l be an apparent s h i f t i n the observed phase towards - 1 8 0 ° and a decrease i n the E(f ) and T (f) spectra at 2 f . w p I t i s poss ib le to estimate the s i ze of th i s e f f e c t . The amplitude 2 of the Stokes' f i r s t harmonic i s ka / 2 , where k and a are the wave-number and amplitude of the fundamental; here the fundamental frequency w i l l be taken to be f . Run 2a i s taken as a "typical' , ' run. For f P P J"s 0.6 Hz, kp cs 1.4 x 10 2 cm and the wave power spectrum Cj)^(f) -2 -1 at f i s 40 cm Hz . I f these waves occupy a bandwidth of 0.1 Hz P they are equivalent to a s ing l e s i n u s o i d a l wave at 0.6 Hz of amplitude ^ 2 - 2 (2 x 40 x 0 . 1 ) 2 ^ 3 cm; therefore ka /2 ^ 6 x 10 cm. The p r e d i c t i o n for the c o n t r i b u t i o n to 0 ^ ( 1 . 2 Hz) of the Stokes' harmonic i s therefore -2 2 -2 2 -1 (6 x 10 cm) / (2 x 0.1 Hz) s 2 x 10 cm Hz The maximum expected amplitude of the pressure harmonics generated by the p i t c h f luc tuat ions of the buoy i s about 2.5 dyne cm 2 ; i f a^pressure^'ampl i tude (for a 3 -2 cm amplitude wave) of 1 dyne cm i s assumed and i f i t i s fur ther assumed that about one h a l f of th is i s i n quadrature with the Stokes' f i r s t harmonic of the waves, then the contribution to the p , ? £ quadrature spectrum i n a 0.1 Hz bandwidth caused by th i s e f f ec t i s given by Qu__ (f) = { (2 x l O - 2 cm^ x 1 dyneA / ( 2 x 0 . l H z ) X >• \ Hz 2 cm 2 / =s 0.2 dyne cm" 1 H z " 1 . The observed s i z e of Q u p ^ ( f ) at 1.2 Hz i n run 2a i s 3 dyne c m - 1 ^ - 1 ; i t i s therefore expected that the e f f e c t of th is contamination of the energy and momentum fluxes to the Stokes harmonics by the p i t c h i n g ac t ion of the buoy may be 5 - 10% of the t o t a l p,->£ quadrature spectrum at 2 f p . This may show up i n the Q u p ^ ( f ) as a s l i g h t reduct ion i n t h e i r magnitudes near 2 f p ; as mentioned e a r l i e r i t may also appear as a s l i g h t "hump" at 2 f p i n the pressure power spectra . Since the magnitude of the quadrature spectra i s reduced by the e f f e c t , the E(f) a n d £ " w ( f ) spectra would a lso be reduced at 2 f p , and th i s should be looked for i n these spec tra . 5.2.1b The F i n i t e Size of the Buoy The second source of contamination of the spectra i s caused by the f i n i t e s i ze of the buoy. The wavelength of the high-frequency waves becomes comparable with the s i ze of the buoy at frequencies above 1 Hz. At 1.2 Hz the r a t i o of buoy radius to wavelength i s 0.1; i t reaches 0.33 at 2 Hz and 1 at 3.7 Hz. As a r e s u l t , the scales of v a r i a t i o n s i n the wind speed associated with the wave motion are comparable with the buoy, and so the wind speed at the bow of the buoy may d i f f e r s i g n i f i -cant ly from that at the pressure port and at the r i n g downwind of the por t . This means that the f r a c t i o n of the stagnation pressure appearing at the port due to the a i r flow over the bow may not be exact ly cancel led by the e f f e c t of the r i n g (see "Experiment", p. 66). I f the f l u c t u a t i o n s i n wind v e l o c i t y associated with the waves are assumed to be approximately those of the o r b i t a l v e l o c i t i e s of the water p a r t i c l e s at the surface of the waves then t h e i r amplitude i s kac, where k i s wave number, a i s wave amplitude, and c wave phase speed. There-fore i n a mean wind speed U Q the stagnation pressure var i e s along the wave p r o f i l e ( i n the x d i r e c t i o n ) according to Ps - ^ ?a ' 2 u o k a c s i n ( k x ) 5.12. Some f r a c t i o n F of p s appears at the pressure port of the buoy as a suc t ion pressure; i n an a i r flow over the buoy,which i s uniform i n the x d i r e c t i o n , t h i s i s cancel led by a p o s i t i v e pressure Fp^ due to the presence of the r i n g downwind from the port . I f a wind i s present which var ie s as s i n kx over the buoy and i f d and e are r e s p e c t i v e l y the distances from the r i n g and the bow to the pressure port , then the pressure at the port i s given approximately by Pp ~ p r i n g + Pbow - F ?a. ^o kac ( s i n k (x-d) - s i n k (x + e) ) = F £ a U Q kac ( 6 2 + V2)% s i n c ( k x - f ) 5.13, where h = s i n kd + s i n ke Y = cos kd - cos ke , and 5.14. C = t a n ' 1 ( 6 / T ) That g ives , for s u f f i c i e n t l y small values of k, d, and e P p = - F p a U Q k 3 ac (d + e ) 2 cos kx 5.15, 93 i n d i c a t i n g that the spurious pressures caused by the buoys f a i l u r e to completely cancel out the s tagnat ion pressures w i l l be i n quadrature with the waves, lagging them by 9 0 ° . Run 2a i s chosen to estimate the s i ze of the e f f e c t . The estimate w i l l be made at two frequencies: 0.6 Hz, and 2 Hz. For the buoy F ~ 0.1 (see "Experiment", F igure 16), d 2s 2 cm, and e — 7.5 cm. 2 -1 For the run , D Q s 3 x 10 cm sec The wave slope ka w i l l be taken to be ka 21 0.1; c = g/w - 2.6 x 10 cm sec , and k = co 2 / g cs 1.45 x 10" 2 c m - 1 . Taking d + e a 10 cm g ives , from Equation 5.15, p p = -2 .0 x 10" 2 dyne c m - 2 . I f th is i s spread over a bandwidth of 0.1 Hz, the r e s u l t i n g power s p e c t r a l estimate for Pp gives (t>Pp (0.6) = (2 x 1 0 " 2 ) 2 / (2 x 0.1) =i -2 x 1 0 ' 3 (dyne c m " 2 ) 2 H z " 1 . To f i n d the observed quadrature pressure s p e c t r a l estimate coherent with the waves, the square of the ppj quadrature spectrum must be d iv ided by the wave s p e c t r a l estimate at 0.6 Hz: 0pQ (0.6) = Q u 2 . , , (0.6) / 0 , ( 0 . 6 ) 5.16 = (22)2/40 =s 12 (dyne c m " 2 ) 2 H z " 1 . Since (J)pp (0.6) / (J) P Q (0.6) < 2 x 1 0 - 4 the e f f ec t of the f i n i t e s i z e of the buoy can be neglected at 0.6 Hz. At 2 Hz ne i ther kd nor ke can be assumed smal l , and so Equation 5.13 must be used. At 2 Hz, Q(ip^(2) 2* -.04 dyne cm"--Hz"1, and (J)^ (2) 2s 0.063 cm Hz ; using Equation 5.16 th i s gives Cj)po (2) 2s -2.5 x 1 0 - 2 (dyne c m - 2 ) 2 H z " 1 . Equation 5.13 gives for 0p p (2) (J)pp (2) 2s -6.5 x 1 0 - 3 (dyne c m - 2 ) 2 H z " 1 . Thus (J)pp (f) i s about 307o of (J)pQ (f) at 2 Hz, which i s the upper frequency l i m i t beyond which errors i n the measurement of the phase between pressure and waves become large . Therefore the probable d i s t o r t i o n s introduced by the f i n i t e s i z e of the buoy are small at f (the frequency at the peak of the wave spectrum); they increase with frequency and may add as much as 307c, to the magnitude of the p,i£ quadrature spectrum by 2 Hz. S ince , however, the waves and hence the pressures coherent with them at th i s frequency are quite sma l l , i t i s not expected that the e f f ec t w i l l show up i n the pressure power spec tra . The E andZ"*w spectra w i l l be af fected by the same percentage as Q u p ^ ( f ) , s ince they are der ived from i t ; they w i l l tend to be too large at high frequencies . 5.2.1c Backscat ter ing The l a s t source of d i s t o r t i o n i n the spectra to be deal t with i s backscat ter ing of waves from the mast which supports the wave probe, which i s 10 cm i n diameter. The backscattered waves t r a v e l against the wind and are hence damped by i t . The pressure d i s t r i b u t i o n over the inc ident wave i s inf luenced by that over the r e f l e c t e d wave; the r e s u l t i n g spurious phase s h i f t i n the pressure can be i n the d i r e c t i o n of e i t h e r damping or added generation (towards - 1 8 0 ° or towards - 9 0 ° ) depending on the r e l a t i v e phase of the inc ident and r e f l e c t e d waves. This r e l a t i v e phase depends on the length \ of the waves being sca t tered , and hence on t h e i r frequency through the d i s p e r s i o n r e l a t i o n f 2 = gk = g 5.17. 4TT 2 2TT X The mast which supports the wave probe i s a v e r t i c a l c y l i n d e r ; the backscat ter ing from such an object can be ca l cu la ted from p o t e n t i a l flow theory (Havelock,1940). Assuming that the inc ident and scat tered waves are the only waves present and that the inc ident wave i s a plane wave proceeding i n the -x d i r e c t i o n with the p o t e n t i a l ( | ) . = A e i ( « t + k x ) e k z = fa e i c o t e k z 5 > 1 8 where x i s the h o r i x o n t a l and z the v e r t i c a l dimension and the motion i s two-dimensional (uniform i n the t h i r d d i r e c t i o n ) . The p o t e n t i a l for the flow i n the presence of a ^ c a t t e r e r can then be w r i t t e n ^ = $ i + ^ r = e i w t e k z ( 4>i + 4>r) 5.19. In po lar c y l i n d r i c a l coordinates (J)i can be expanded as an i n f i n i t e sum of Bessel funct ions and § r a s a n i n f i n i t e sum of Hankel funct ions of the second k i n d . The o r i g i n of the coordinate system i s taken to be 96 at the geometric centre of the s c a t t e r e r . Introducing the boundary condi t ion that the v e l o c i t i e s normal to the sca t terer must vanish r e s u l t s i n an expression for the r a t i o (J>i/ (J) r i n terms of the above-mentioned sums of Bessel and Hankel funct ions . The wave e l eva t ion ">£ i s given (to the f i r s t order) by z = o = i w ( d>i + < j > r ) e l w t e k z therefore V. r / V i = 4>r/ <J>i 5.20. A computer program wr i t t en by J . F . G a r r e t t , a fe l low student, has been used to compute 4>i and <t>r i n terms of the above-mentioned sums of Bessel and Hankel functions for various values of k, for the p a r t i -cu lar radius of the mast used and the (f ixed) distance from the mast to the wave probe. The Garre t t program truncates the sums of Bessel funct ions at a point where the truncat ion error becomes n e g l i g i b l e . I t computes three quant i t i e s : <4>i> , <<]>r> , and<4>r> / <4>i) . The quantity <\CL) i s <OL> = M a x | a e i r | f o r 0 4 T £ 2 T r ; thus the ^ y denote "maximum value over a f u l l cycle". < 4 > i > and ^ 4 ^ r ^ are complex and vary with the wave number k; the r e l a t i v e phase of the inc ident and r e f l e c t e d waves can be determined from them. The t o t a l amplitude of r e f l e c t e d plus backscattered waves can also be com-puted and compared with the amplitude of the inc ident wave. I t must however be noted that such c a l c u l a t i o n s assume that the inc ident and backscattered waves are coherent. I t has been found by Garre t t (personal communication) that at the Spanish Banks s i t e i n a southeast wind the waves become e s s e n t i a l l y incoherent at frequencies greater than 1 Hz at downwind wave probe separations of 2 m (2 m i s twice the distance from the wave probe to the supporting mast). I f the inc ident and r e f l e c t e d waves are coherent, t h e i r t o t a l ( inc ident plus re f l ec ted) energy must be found by adding them v e c t o r i a l l y before squaring; i f they are incoherent t h e i r t o t a l energy i s given by the sum of the squares of t h e i r amplitudes. The r e l a t i v e phase of the inc ident and r e f l e c t e d waves has been ca l cu la ted at a number of frequencies; the r a t i o of t o t a l to inc ident energy has also been c a l c u l a t e d , on the assumption that inc ident and r e f l e c t e d waves are coherent at frequencies below 1 Hz and incoherent above 1 Hz. The energy r a t i o for coherent waves R C - ( < $ i > + < $ r > ) 2 / ( < < t > i > ) 2 5 . 2 1 i s d i sp layed for frequencies from 0.35 to 1.7 Hz i n Table 5.2; the corresponding r a t i o for incoherent waves Ric=(|<<t>i')| 2 + |<*r>| 2 ) / |<<Dl>| 2 5 . 2 2 i s d i sp layed i n the same table for frequencies from 0.85 to 2.0 Hz. From the table i t i s c l ear that at frequencies near 1 Hz spurious wave energy amounting to as much as 157, of the inc ident wave energy may be present i n the wave spectrum. The e f f ec t of the backscat ter ing on the pressure power spectrum cannot be estimated without a knowledge of the pressure d i s t r i b u t i o n over the r e f l e c t e d waves, which are being damped. TABLE 5.2 Expected Ef fec t s of Backscat ter ing on Wave Power Spectrum f (Hz) Notes: 0.35 0.5 0.85 1.0 1.2 1.4 1.7 2.0 R c (1) .998 .982 1.06 $.91 .872 .923 1.4 100(RC-1) (%) 0.2 2.0 6.0 -9 .0 •14.0 -8 .0 40.0 v i c (2) 1 0 0 ( R i c - l ) (%) 1.00 1.00 1.01 1.02 1.03 1.03 0 0 1 2 3 3 Notes: 1. R c i s computed from Equation 5.21; i t i s the r a t i o of t o t a l to inc ident energy for coherent inc ident and r e f l e c t e d waves. 2. R^ c i s computed from Equation 5.22; i t i s the corres -ponding energy r a t i o for incoherent inc ident and r e f l e c t i n g waves. A q u a l i t a t i v e estimate of the e f fec t of the backscat ter ing on the cross spectrum between pressure and waves can be made. As the wave number k = 2 T T / X of the waves changes by 1 m~\ the r e l a t i v e phase 8 i r of the inc ident and r e f l e c t e d waves changes 1 rad ian ( 5 7 . 3 ° ) . The r e l a t i v e phase at any frequency can thus be found i f the phase at one frequency i s computed from the backscat ter ing formula. The r e l a t i v e phase of inc ident and r e f l e c t e d waves was ca lcu la ted and p lo t t ed versus wave number to f i n d the wavenumbers where the phase was 0 ° , 9 0 ° , 1 8 0 ° and 2 7 0 ° ; these wavenumbers were then converted to frequencies us ing the d i s p e r s i o n r e l a t i o n for water waves, given by Equation 5.17. The q u a l i t a t i v e e f f e c t on C o p ^ ( f ) , Q u ^ ^ ( f ) , and Phase dp^(f) = tan'^CQUp^ ( f ) /Co (f) ) of a damped r e f l e c t e d wave i s determined at each of the four phase angles mentioned above, assuming that the pressure over the r e f l e c t e d wave leads i t by 9 0 ° . The r e s u l t s of the determination are d isp layed i n Table 5.3 i n conjunct ion with a tabu la t ion of the frequencies at which each of the phases <9 i r occur. A "+" i n the Co, Qu and Q (f) columns means the re levant spectrum i s increased by the e f f ec t of the backsca t ter ing , and a means a decrease (thus, s ince Co, Qu and 0 ( f ) are commonly negat ive , the magnitudes of Co and Qu are increased and the angle of © (f) moved towards - 1 8 0 ° i n the columns where " - " signs occur) . A zero means "not af fected". The backscat ter ing e f fec t i s assumed to be unimportant at frequencies above 1 Hz, s ince at th is frequency inc ident and r e f l e c t e d waves are assumed to be e s s e n t i a l l y incoherent . 5.2.2 D i s t o r t i o n s Introduced during Preparat ion and Analys i s of Data  5 .2 .2a F i l t e r i n g of the Wave S igna l The wave s i g n a l i s f i l t e r e d i n three ways. F i r s t , i t i s f i l t e r e d TABLE 5.3 ected Ef fec t s of Backscat ter ing on Pressure-Waves Cross Spectrum S i r Co(f) Qu(f) 6 ( f ) (Degrees) (dyne cm" •'"Hz-1) (dyne cm"•'"Hz"1) (Degrees) (1) (1) (2) 90 + 0 + 180 0 + 270 - 0 0 0 - + 90 + 0 + 180 0 . + 270 - 0 0 0 + 90 + 0 + A ' V i n the Co or Qu columns indicates that p o s i t i v e energy i s added to the re levant spectrum by the backscat ter ing . A + i n the 8 (f) column ind icates that the phase angle between pressure and waves i s increased by the e f fec t of the backsca t t er ing . 99 by the buoy i t s e l f which r ides v e r t i c a l l y on the wave probe. The exact form of d i s t o r t i o n introduced i s not known; i t i s f e l t that i t only becomes s i g n i f i c a n t above the frequency (about 2 Hz) where the wave-length of the waves i s comparable with the diameter of the buoy. Because of the p h y s i c a l dimensions of the bear ing , errors caused by surging of the water between the bearing and the probe are not expected to exceed 1 mm i n observed wave amplitude. T y p i c a l l y the standard dev ia t ion of the wave s i g n a l for frequencies above a low-frequency cutoff of 2 Hz exceeds 1 mm, so the error caused by the presence of the bearing on the wave probe can almost c e r t a i n l y be neglected at frequencies less than 2 Hz. Second, the wave s igna l i s analog f i l t e r e d , as mentioned e a r l i e r , p r i o r to d i g i t i z a t i o n ; i n a d d i t i o n a d i g i t a l f i l t e r i s appl ied during analys i s by taking an unweighted three po int running mean of the sample p o i n t s . The l a t t e r f i l t e r , i s designed p r i n c i p a l l y to reduce high frequency noise when the wave s i g n a l i s d i f f e r e n t i a t e d d i g i t a l l y ; i t i s app l i ed to a l l the waves s ignals for the sake of consistency. The wave power spectrum i s thus the convolut ion of the o r i g i n a l wave spectrum with the transfer funct ions of the above three f i l t e r s , and of the r e s u l t i n g unsmoothed spectrum with the hanning window as given by Equation 5.1 , p.81. 5.2.2b The Pressure Signals Since the pressure s ignals p s ( t ) = p ( t ) S ( t ) , where S(t) i s defined by Equation A l . 2, Appendix 1, and P s ( t ) + f0,9T?.( t) > t n e pressure plus s t a t i c head, are treated d i f f e r e n t l y i n some parts of the a n a l y s i s , they w i l l be discussed separate ly . 100 As mentioned e a r l i e r , the "raw" pressure s igna l recorded at the time of the run i s subjected to some f i l t e r i n g and condi t ion ing p r i o r to d i g i t i z a t i o n . During rerecord ing i t i s passed through a high-pass f i l t e r with a 50-second time constant, which removes DC of fsets and slow d r i f t s from the data . During d i g i t i z a t i o n the s i g n a l i s passed through a l i n e a r phase s h i f t low-pass f i l t e r with a 3 db point at 6 Hz and a f a l l -o f f rate of 12 db per octave. During d i g i t a l condi t ion ing p r i o r to F o u r i e r transformation sect ions of data points where spikes occur are replaced with zero, and any r e s i -dual mean value i s removed from the s i g n a l . I t should be noted that i n run 4y, the h a n d - d i g i t i z e d run, the spikes were removed by hand by draw-ing smooth curves from the beginning to the end points of the spikes . A f t e r the F o u r i e r transformation has been performed the r e s u l t i n g F o u r i e r c o e f f i c i e n t s are smoothed by a d i g i t a l f i l t e r which i s equi -va lent to hanning the spec tra . The p g + Za.^ s i g n a l i s generated during the d i g i t a l a n a l y s i s , a f t er a l l f i l t e r i n g and averaging on the pressure and wave s ignals i s complete but before the F o u r i e r transform i s computed. ^ i s computed from ^ = 1.29 (223) ( P ) 5.23, T 1000 where T i s mean a i r temperature i n degrees K e l v i n , and P i s mean sea. l e v e l pressure i n m i l l i b a r s . T and P are a v a i l a b l e for most of the runs (see Table 6.1); where they are not reasonable estimates are used. There are two important sources of e rror i n the computation of 101 P s + f»9 \ • F i r s t , over the range of frequencies where the s i ze of p s and fa<3'"£ are comparable any c a l i b r a t i o n errors i n p s and v[ are propor t ionate ly greater i n t h e i r d i f f e r e n c e . This error i s found to be s i g n i f i c a n t i n both the power spectra and the cross spectra . The second source of error (in p s + Pag">j ) involves the phys i ca l separat ion of the pressure and wave sensors; the l a t t e r i s 3.8 cm downwind of the former. Because of th i s the phase of the wave s i g n a l must be advanced i n time d i g i t a l l y p r i o r to transformation. The d i g i t a l time s h i f t has the e f f ec t of in troduc ing a phase s h i f t i n the wave s i g n a l which increases l i n e a r l y with frequency. The phase error which the time s h i f t i s meant to correc t var i e s with the r a t i o of the sensor separat ion to the wave-length of the waves; th is r a t i o var i e s as the frequency squared through the d i s p e r s i o n r e l a t i o n for g r a v i t y waves. Therefore the time s h i f t w i l l only correc t the ^ phase over a range of frequencies , on e i ther s ide of which large phase errors i n the p s + fa<") s igna l may occur. The poss ib le e f fects of each of these two sources of error w i l l be discussed i n the fo l lowing paragraphs. F igure 21 shows the e f f ec t on the p + f^g^ phasor of a 507o underestimate i n the pressure c a l i b r a t i o n for two s i t u a t i o n s which commonly occur i n the data: | p | » fag 1*^1 (top) and | p | ~ p^ g \v(\ (bottom). In the f igure the phasor P i s the hypothe t i ca l "correct" pressure mea-sured 'by an i d e a l sensor at the exact l o c a t i o n of the wave probe. P m i s the "correct measured" phasor, i . e . that measured by the ac tua l sensor located about 4 cm upwind of the wave probe, and for which the c a l i b r a t i o n i s c o r r e c t . P m u i s the measured pressure with the under-estimated c a l i b r a t i o n . i s the measured wave phasor and ?£ c i s the 102 wave phasor corrected for the 4 cm sensor separat ion so that i t has the proper phase with P m ; The diagrams show that i f \ P | » t n e ampl i -tude of the p + f^q^ s igna l i s most s trongly affected by the c a l i b r a -t i o n e r r o r , while i f |p ( c- the amplitude of p + Pa<j ^ i s a f fected l i t t l e , but the phase error i s l arge . The f u l l phase c o r r e c t i o n curve for the pressure sensor, which includes both the e f f e c t of the phase response of the sensor (Figure 12) and the f u l l nonl inear e f f e c t of the s p a t i a l separat ion of the pressure and wave sensors as discussed on the preceding page, i s exh ib i ted i n Figure 22. I t i s th i s curve which must be approximated by the time s h i f t of the ^ s i g n a l p r i o r to the a d d i t i o n of pjj ^ to the pressure . For th is reason the ( l inear ) phase c o r r e c t i o n curves for an advance of the wave s i g n a l i n time by one and two samples (at a sampling frequency of 50 Hz the advances are 0.02 and 0.04 seconds) are included i n Figure 22. During a n a l y s i s , the s i ze of the advance to be used was chosen separate ly for each run on the basis that the phase c o r r e c t i o n be as small as poss ib le near the peak i n the spectrum of energy f lux to the waves. The e f f e c t of the phase errors introduced by approximating the f u l l pressure phase c o r r e c t i o n curve by a s t r a i g h t l i n e i s shown i n F igure 23 for four d i f f e r e n t s i tua t ions which appear i n the r e s u l t s : wave generation (p lags* ig by 9 0 ° ) and wave damping (p leads by an" angle of 9 0 ° to 1 8 0 ° ) , for | p | » P^\^\ and \ p \ & fa3 1^ 1 • On a l l the diagrams P and P m are , as for F igure 21, the pressures observed by 103 i d e a l (at the wave probe) and ac tua l (3.8 cm upwind of wave probe) sensors r e s p e c t i v e l y . r£ i s the wave phasor corrected using the f u l l pressure phase c o r r e c t i o n curve and i s that corrected using the l i n e a r approximation r e s u l t i n g from the simple time advance of the wave s i g n a l s . I t w i l l be seen that at high frequencies the approximate phase c o r r e c t i o n underestimates the ac tua l one while i t overestimates i t at low frequencies , as ind ica ted i n F igure 22 (at very low frequencies the approximation underestimates the phase c o r r e c t i o n , but th i s i s not considered here ) . The f i r s t point to be not iced i s that at both high and low f r e -quencies the phase of p + and i t s amplitude are af fected s trongly i f | p | s.Pa<"||^ "|> while ne i ther the amplitude nor the phase of p + P a 3*£ are g r e a t l y af fected i f | P | >) fa<3 • The second point to-note i s that i n a l l cases the e f f e c t of the approximation on the phase angle between i£ and p m + f a i S ^ c (which i s the required phase d i f ference) i s to underestimate i t s magnitude. Since i n general i n the r e s u l t s | p | ~ f 0^l*jlat the lower frequencies and 1P| » f ^ l l l a t t n e higher frequencies , the o v e r a l l e f f ec t on the s i g n a l of the approximation of the proper phase c o r r e c t i o n by a l i n e a r time s h i f t ofy i s at low frequenc ies , to cause an overest imation of i t s power spectrum and an underestimate of any damping or generation which may occur. At h igh frequencies ne i ther the p +f ag»£ power spectrum nor the phase with respect to ^ are af fected grea t ly by the approximation. I t should perhaps be noted that the phase angles given by the cross-spectra between p and i£ and between p + P A3^ and ^ d i f f e r by the errors 104 mentioned on the preceding page, s ince the phase of p i s corrected using the f u l l c o r r e c t i o n curve for the cross-spectrum between p and J£ . With the a p p l i c a t i o n of the f u l l c o r r e c t i o n curve, the expected accuracy of the p,>£ phase angles to be presented i s ± 5 ° from 0.05 to 0.1 Hz, i 1 ° from 0.1 to 1 Hz, and i" 5 ° from 1 to 2 Hz (see Experiment, p. 58 f f ) . The accuracy of the (p + ?£ ) > phases i s harder to def ine , s ince scat ter caused by low coherences i s a lso present . A reasonable estimate of the accuracy also depends on the expected accuracy of the pressure c a l i b r a t i o n s (these are given i n Table 6 .3) , which vary from run to run . Therefore only a rough estimate can be given: * 10 to 2 0 ° i n the frequency range from 0.1 to 1 Hz, and larger outs ide th i s range. These estimates do not inc lude the expected e f fects of the sensors themselves mentioned on p. 88 f f . 5.3 Summary The data analys i s i s seen to be f a i r l y s tra ight forward; not so the i n t e r p r e t a t i o n of the r e s u l t s i n some frequency ranges. The safest course (that chosen i n d i scuss ing the r e s u l t s ) i s to treat only circum-spec t ly the frequency ranges 0.05 to 0.1 Hz and 1.0 to 2.0 Hz i n a l l spectra except perhaps ^)^(f ) , ex trac t ing from these "suspect" ranges only corroborat ive informat ion . I f i t i s necessary to examine the data i n these frequency ranges more c l o s e l y , then the "Interpretat ion" sec t ion should be r e f e r r e d to extens ive ly . SECTION 6: RESULTS 6.1 Introduct ion 105 6.2 Method of Presentat ion 105 6.3 Summary of Runs 106 6.4 The Power Spectra 107 6.5 The Cross-Spectra 113 6.5.1 The Pressure-Waves Cross-Spectra 114 6 .5 .1a Coherence 115 6.5.1b Phase 118 6.5.2 The Fluxes of Energy and Momentum to the Waves . 122 6 .5 .2a Summary of Mean Values of E and *£*w . . 122 6.5.2b The Spectra of Energy and Momentum F l u x 125 6.5.3 The Spectra of £ 130 SECTION 6: RESULTS 6.1 Introduct ion The p r i n c i p a l r e s u l t s of the thes is are presented i n th i s s ec t ion . They cons i s t of power spectra of wave e l eva t ion and normal pressure at the sea surface; they contain information heretofore not a v a i l a b l e on the processes of wave growth and momentum transfer from the a i r to the sea. The measurements from which the above-mentioned spectra have been computed were made under condit ions which taxed the design and use of the instrumentat ion to t h e i r l i m i t s ; they cannot be repeated e a s i l y . In order to ensure confidence i n the r e s u l t s the data have been analysed i n several d i f f e r e n t ways, and a comparison experiment has been made ( c a l l e d the "Boundary Bay" experiment and discussed i n d e t a i l i n Appendix 2) i n which the performance of the buoy pressure sensor was tested against that of other types of pressure sensors. The comparison experiment turned up no serious d iscrepancies between the r e s u l t s from the buoy sensor and those from the sensors against which i t was compared, i n the frequency range considered to be of i n t e r e s t i n the r e s u l t s to be presented below (0.1 to 2 Hz, as ind ica ted i n "Data Ana lys i s and In terpre ta t ion" , p. 104). 6.2 Method of Presentat ion The data w i l l be presented i n various ways to b r i n g out d i f f e r e n t po in t s . F i r s t , a summary of the important facts about each of the runs w i l l be g iven. This w i l l be fol lowed by the power spectra of waves and 105 106 pressure . Then the cross spectra w i l l be shown, f i r s t as coherence and phase p lo t s of the c o r r e l a t i o n s of pressure versus wave e l eva t ion and pressure plus s t a t i c head versus wave e l e v a t i o n . F i n a l l y , the spectra E(f ) and T w ( f ) of computed energy and momentum fluxes from the wind to the waves w i l l be presented, along with spectra of the negative damping r a t i o £ = E / u E , defined by Mi les (1960) as the f r a c t i o n a l increase i n wave energy per r a d i a n . Where poss ib le the mean momentum f l u x *7j'w found from these computations w i l l be compared with that ( X-S) obtained from measured uw covariances . Where th is i s not p o s s i b l e , the mean wind speed w i l l be used to ca l cu la te an approximate mean momentum f l u x from T c = f a C ^ 5 2 6.1 where C D , the drag c o e f f i c i e n t , w i l l be taken as 1.2 x 10" 3; U^, the mean wind speed at 5 meters, was measured .with cup anemometers. 6.3 Summary of Runs Most of the accessory information for the s i x runs i s presented i n order of increas ing wind speed i n Tables 6.1, 6.2, and 6.3. The runs cover a range of wind speeds (extrapolated to 5 m height) of 150 - 800 cm s e c - 1 and a range of values of (U"5 - c ) , the d i f f erence between wind speed at 5 m and wave phase v e l o c i t y at the peak of the wave spectrum, of -150 to 535 cm s e c - 1 . At l eas t two and sometimes three cup anemometers were present for the f i r s t four runs; during the l a s t two runs a sonic anemometer was used. The accuracy of wind speed ex trapo la t ion was not maintained to bet ter than ^570; hence the wind speeds at 5 meters, which are extrapolated from assumed logar i thmic p r o f i l e s , are only meant to be 107 th is accurate . In every case except one (run 1) the winds blew from the S-E quadrant; the v a r i a t i o n of fe tch with azimuth for th i s quadrant has been computed by G i l c h r i s t (1965; h i s F igure 2) and i s reproduced i n F igure 2. The f e t ch i n the wester ly d i r e c t i o n was about 40 km. Currents were measured by t iming t i s sue paper f l o a t i n g 1 - 10 cm below the water surface over known dis tances . In general the observed mean t i d a l d r i f t current was opposing the wind; th is arose from the f a c t that the buoy only operated succes s fu l l y i n these condi t ions . I f the t i d a l current was s lack or a i d i n g the wind, the small "bow wave" formed by the buoy was c o n t i n u a l l y dr iven by the wind over i t s surface to the pressure diaphragm, causing an unacceptable number of spikes i n the data . With one exception (run 5) the a i r was 0.5 - 1 . 5 ° C (3 0 . 4 ° ) cooler than the water at a depth of 5 - 10 cm, i n d i c a t i n g unstable or n e u t r a l l y s tab le condi t ions . The runs were taken i n the f a l l of the year at the s t a r t of coasta l B r i t i s h Columbia's "winter monsoons"--successions of lows reaching the coast from the west and dropping t h e i r moisture against the coas ta l mountains. For th i s reason skies were overcast for a l l r u n s - - i t was r a i n i n g l i g h t l y during run 6. 6.4 The Power Spectra Three power spectra are presented for each run i n Figures 24 - 31. P s denotes the pressure measured by the buoy and contaminated by sp ikes , the wave he ight , and p s + fa<J^  the pressure with f^1^ the s t a t i c pressure head caused by the v e r t i c a l excursions of the buoy, added to i t . The l a t t e r provides an i n d i c a t i o n of the spectrum which would be measured by a hypothe t i ca l sensor f ixed at the mean water l e v e l . A l l the power spectra have logar i thmic (roughly ha l f -oc tave) band-TABLE 6.1 Length U 9 U w Run Time/Date of Run Fetch _ 1 ( c m_^ No. 1967 (min.) (km) (cm sec ) ( true) sec ) tes: (1) (3) (4) 5 1545/XI/17 12.0 1.0 150(s) 135 1 1940/X/16 16.3 40. 220(c) 260 5(SW) 2b 2030/X/19 15.5 2.4 310(c) 115 18(E) 2a 2010/X/19 11.0 2.4 320(c) 115 18(E) 3 1435/X/20 16.8 6.7 340(c) 80 5(W) 6 1225/XI/23 10.5 1.6 570(s) 120 8(W) 4b 1200/X/30 3.1 2.4 700(c) ESE 30(W) (2) 4a 1150/X/30 4.0 2.4 800(c) ESE 30(W) 108 T T P a w ( ° C ) ( ° C ) (mb) (5) (5) (6) 10.2 8.9 1015.2(f) 9.8 11.2 1027.6(u) 9.8 11.2 1023.9(f) 10.0 11.2 1023.9(f) 10.4 11.0 1010.7(f) 7.8 8.2 1015.9(f) 1020.8(f) 1020.8(f) Comments Tide s lack . Water depth 3 . 0 m . Tide s lack . Water depth 2.4 m. Just a f ter sunset. Tide f a l l i n g . Water depth 2.0 m. Dark. Tide f a l l i n g . Water depth 2.1 m. Dark. Tide r i s i n g . Water depth 2.8 m. T r i s i n g . 9. Tide s lack . F ine r a i n . Water depth 3.4 m. Tide f a l l i n g . Water depth 2.7 m. Tide f a l l i n g . Water depth 2.7 m. TABLE 6.1 (continued) Notes: 1. Time/Date months i n Roman numerals. 2. Run 4 wind speed and water current information from memory; o r i g i n a l data destroyed. 3. U^: (c) fo l lowing wind speed means "measured with cup anemometers"; three cups were used of which two were considered accurate enough to use t h e i r r e s u l t s . (s) fo l lowing wind speed means "measured with 3-component sonic anemometer". 4. U : observed surface d r i f t and t i d a l current; d i r e c t i o n i n brackets . w 5. T , T : mean a i r and water temperatures, measured to 1*0.2°C with a thermistor bead at the a w r recording p lat form. 6. P: pressure at time of run as measured at Vancouver In ternat iona l A i r p o r t , corrected to mean sea l e v e l ; trends are (u) unchanged, (f) f a l l i n g , and (r) r i s i n g . 109 TABLE 6.2 U 5 c p f k 2 c p Run No. (cm sec"l) (cm sec~l) (Hz) (cm) Notes: (1) (2), (4) (3), (4) 5 150 332 1.25 >10 4 1 220 370 0.93 >10 4 2b 310 275 0.62 145 2a 320 270 0.58 83 3 340 225 0.54 9.7 6 570 270 0.29 1.2 4b 700 305 0.22 0.7 4a 800 265 0.19 0.2 Notes: 1. i s wave phase v e l o c i t y at the peak of the l oca l ly -genera ted wave spectrum. 2. i s the frequency at which the phase speed c of the waves i s equal to the mean wind speed at 1/k = X/2K above the mean water l e v e l . 3. z c p i s the M i l e s ' c r i t i c a l he ight for the peak of the l oca l ly -genera ted wave spectrum: the height were c p = U. 4. The mean wind speeds used i n the c a l c u l a t i o n of f c and z c p are extrapolated values from logar i thmic p r o f i l e s with C n , the drag c o e f f i c i e n t , set equal to 0.0012. 110 TABLE 6.3 Pressure Estimated C a l i b r a t i o n Accuracy of Run Used C a l i b r a t i o n (mv/dyne cm"2) (%) Notes (1) 5 1.32 20 1 4.0 10 2b 4.0 10 2a 4.0 10 3 3.8 10 6 1.67 20 4b 1.75 30 4a - 1 . 7 5 30 Notes: 1. A l l c a l i b r a t i o n s used (Column 2) were made i n the f i e l d immediately fo l lowing the run for which they are used. 2. The same waterproofing diaphragm was used for a l l the runs; i t s laboratory c a l i b r a t i o n i s shown i n Figure 12, and was 2.4 mv/dyne cm" . I l l widths. The spectra have been smoothed by hanning so that they are not d i s t o r t e d i n regions which change less r a p i d l y than f-* or f-"' (the window f a l l s o f f as 5f~^). In a d d i t i o n a l i a s i n g has been avoided by analog f i l t e r i n g p r i o r to d i g i t i z a t i o n . For more d e t a i l e d discuss ions of these points and those fo l l owing , see "Data A n a l y s i s and In terpre ta -t ion" , pp. 71 f f . The wave spec tra , marked with an i n the f i g u r e s , appear normal for the s i t e ; they show high-frequency slopes between -4 .5 and - 5 . One i n t e r e s t i n g feature i s the presence i n some of the spectra of two separate "peaks"; these are p a r t i c u l a r l y not iceable i n runs 2a, 2b, and 3. The low-frequency peaks occur near 0.2 Hz, i n d i c a t i n g a phase speed i n deep water of about 8 m s e c - 1 . These "peaks" near 0.2 Hz were almost c e r t a i n l y not l o c a l l y generated, and were probably caused by marine t r a f f i c i n the approaches to Vancouver harbour; many f r e i g h t e r s were observed passing the experimental s i t e during the per iod when the runs were made. The most s t r i k i n g feature of the p g spectra i s the presence of regions of excess pressure energy at frequencies where the wave spectra are large; i n runs where the wave spectra show two peaks, the p s spectra show corresponding "humps" at both frequencies . This i s not so i n the p s + PJ]^ spectra; the removal of - P a 3 ^ has the e f f e c t of almost com-p l e t e l y e l i m i n a t i n g the low-frequency regions of excess pressure energy. The higher-frequency "humps" i n the pressure spectra remain, but t h e i r center of mass i s moved to s l i g h t l y higher frequencies . The lowest-frequency parts of the pressure spectra (near log (f) =-1) are the only regions of the e n t i r e 0.05 - 2.0 Hz bandwidth covered by the pressure spectra presented which appear to contain information not d i r e c t l y assoc iated with waves. The small number of low-frequency spec-t r a l estimates which are a v a i l a b l e here suggest that the pressure spectra have a shape near f , at var iance with the f form suggested by s i m i l a r i t y considerat ions (Stewart, R. W.; personal communication). The r e s u l t s of the Boundary Bay experiment i n d i c a t e that the f"^ shape may i n f a c t be spurious , s ince the low-frequency shapes of the power spectra obtained from the two sensors used i n the comparison are c loser to f than f . For th i s reason the shapes of the pressure spectra shown i n Figures 24 - 31 can be assumed to be at best h e a v i l y contaminated with no i se , and are not discussed f u r t h e r . From 0.2 Hz to the upper design frequency of the pressure system at 3 Hz the pressure spectra appear to be dominated by the waves. The e f f e c t on the spectra of the removal of - P ^ ^ from the pressure s i g n a l v a r i e s d r a m a t i c a l l y with frequency. In some of the wave spectra a large peak occurs at 0.2 Hz, presumably caused by swel l (see p.111) . Corres-ponding excursions of the pressure spectra above the ir 'base l e v e l " are almost completely removed by removing - f ^ ^ from the pressure s i g n a l . This ind icates that the wave-influenced part of the pressure i s l a r g e l y i n antiphase with the waves and at most a small part i s i n quadrature; hence wave generation or damping must be small at these frequencies (some damping i s u s u a l l y observed at low frequencies; th is i s described i n the sec t ion on c r o s s - s p e c t r a ) . I t i s worth not ing at th i s point that removal of - has i m p l i c a -t ions important to the assessment of the accuracy of pressure c a l i b r a t i o n s during f i e l d use of the sensor. Since the s e n s i t i v i t y of the sensor has 113 been known on other occasions to change by as much as a fac tor of two during f i e l d use, every a v a i l a b l e check on the value and s t a b i l i t y of the s e n s i t i v i t y during each run i s made. Removal of - p^ cj v£ can at most reduce the pressure spectrum to the l e v e l which i s independent of waves. I t has been noted above that the regions of excess energy i n the pressure power spectra caused by swel l are s u b s t a n t i a l l y reduced by removing - f ^ ^ ', i t w i l l be pointed out l a t e r that the -f^*^ removal has the e f f e c t of reducing the pressure-waves coherence at the swel l frequencies to low values . This i s only poss ib le i f the s e n s i t i v i t y of the sensor i s quite wel l -determined. The estimates of pressure s e n s i t i v i t y accuracy given i n Table 6.3 are based p a r t l y on the above cons iderat ions . At frequencies near and above the l o c a l l y - g e n e r a t e d peak i n the wave spectrum the s i t u a t i o n i s d i f f e r e n t . A f t e r the removal of ~ fj}^ a s u b s t a n t i a l "hump" u s u a l l y remains i n the pressure spectrum, which i s often s h i f t e d towards higher frequencies . This ind icates that a wave-dominated part of the pressure spectra remains which cannot be i n a n t i -phase with the waves; i t fol lows that at these frequencies wave genera-t i o n i s o c c u r r i n g . Further d i s cuss ion on th i s large wave-dominated reg ion of the pressure spectrum i s postponed, as i t can be more f r u i t f u l l y studied by r e f e r r a l to the pressure-waves c r o s s - s p e c t r a . The spectra at frequencies above 1.5 Hz must be viewed with caut ion; a number of experimental and a n a l y t i c a l d i f f i c u l t i e s a r i s e at and above 2 Hz, which are discussed i n d e t a i l i n "Data In terpre ta t ion" , p. 88 f f . 6.5 The Cross-Spectra Presented i n th is sec t ion are three sets of spec tra . F i r s t to be presented are the cross - spec tra between pressure and wave e l e v a t i o n . 114 Second, the spectra E(f ) a n d £ * w ( f ) , which are r e s p e c t i v e l y the energy and momentum f luxes from the wind to the waves, are presented. Last to be descr ibed are the spectra (f) , the f r a c t i o n a l increase of wave energy per rad ian (see p. 13). The l a t t e r three sets of spectra are der ived from the pressure-waves c r o s s - s p e c t r a . 6.5.1 The Pressure-Waves Cross-Spectra For each run three spectra are presented: the coherence and phase spectra between pressure p g and waves i£ , and the phase spectrum between p s + ("^^ and J£ . The subscr ip t "s" means that the spectrum concerned i s convolved with the spike f u n c t i o n . The reasons for present ing the cross spectra between p s and rather than that between p g and r£ have been o u t l i n e d i n Appendix 1. A l l the spectra presented are corrected for phase errors introduced by the pressure sensor and for the s p a t i a l separat ion of the pressure and wave sensors, according to the methods described i n "Data In terpre ta t ion" , p. 101 f f . The cros s - spec tra are presented i n Figures 32 through 39. Shown on each f i gure are the frequencies fp of the peak of the wave spectrum and f^, where the phase v e l o c i t y c equals the wind speed at the height z where kz = 1, so z = X/2TI = g/4Tf 2 f 2 . i s extrapolated from cup anemometer wind speeds assuming a logar i thmic p r o f i l e and a drag c o e f f i c i -ent of .0012 i n runs 1, 2, 3, and 4, and from sonic anemometer measure-ments i n runs 5 and 6. Where large deviat ions i n the spectra can be a t t r i b u t e d d i r e c t l y to no i se , breaks are l e f t i n the s p e c t r a l curves (for example near 1.2 Hz i n F igure 32). Two h o r i z o n t a l reference l i n e s are drawn on each f igure at phase angles of 1 8 0 ° and -90?. In a l l cases negative phase angles mean the pressure lags the waves. 115 6 .5 .1a Coherence The coherence between p s and r^^ behaves i n a general way as expected, being h igh at frequencies where appreciable wave energy i s present and low elsewhere. The s t a t i s t i c a l s i g n i f i c a n c e of the pressure-waves c o r r e l a t i o n becomes n e g l i g i b l e i n p r a c t i c e f or the lengths of run considered here when the coherence drops below 0.3. In two cases, runs 4b and 5, the genera l ly regular f a l l i n coherence as the frequency increases above f p i s temporari ly arres ted , r i s i n g to a subs id iary "peak" at about 1.6 Hz and re turn ing to low values by 1.8 Hz. To avoid c l u t t e r e d graphs the coherence spectra between p s + (> a3»2 and have been omitted. The general e f f e c t of removing - (^ ""f^  f r o m t n e pressure s i g n a l i s to lower the coherences between p and Y) at a l l s t J except the highest frequencies . In some runs there i s a low-frequency p o r t i o n of the t o t a l wave energy present which i s not the r e s u l t of l o c a l wave generation; th i s can l e g i t i m a t e l y be c a l l e d "swell". The swel l i s best observed i n the phase spectra between p and 77 ; i n S w 5 regions where i t i s present the phase i s near ly 1 8 0 ° ( i f the phase i s p o s i t i v e , then wave damping i s occur ing) . I t i s i n t e r e s t i n g toscompare the e f f e c t of removing -f g ^ from p g on the low-frequency coherence of the runs where swel l appears to be present: runs 1, 2a, 3, and 5. Table 6.4 d i sp lays the percentage of the coherence by which the ( p s +P{t3'"£) coherence i s decreased at the low frequencies where swel l i s assumed to be present , and at the frequencies of the wave power spectra a r i s i n g from loca l l y -genera ted waves. Also included are U 5 , Q (mean wind d i r e c t i o n ) , c^ and c p , the wave phase speeds at the swel l and l o c a l l y generated s p e c t r a l peaks r e s p e c t i v e l y 116 and £ n , the frequency above which the coherences of the two pressure spectra with . ^ a r e t n e same s i z e to within 5%. A s t r i k i n g feature of the table i s the large percentage coherence drop at the swell peak frequency of run 1. This can be understood by r e c a l l i n g the p r e d i c t i o n of the pressure at the water surface given by p o t e n t i a l flow theory; this i s (Equation 13, Appendix 3): p = - p a n t 1 + ( 1 - v c ) 2 ] 6 - 2 -If U o and C have the same sign the pressure w i l l vary between - Z P ^ j ^ (U 0«c) a n d ( U 0 & c ) . for the low frequency waves; i f they are opposite i n sign the pressure w i l l vary between - 2P^ Cj 7£ (I U0| 4<x) and -D^gi^ (IU0I » c ) where D >*> 2. This means that i f the wind and waves were t r a v e l l i n g i n the same d i r e c t i o n and |P g| a n <3 f ^ i ^ l w e r e about the same s i z e , as was the case i n run 1, then the removal should drop the coherence to low values. No information i s a v a i l a b l e on the d i r e c t i o n of t r a v e l of the low-frequency waves i n runs other than run 1; some inferences, however, can be made (see Appendix 3). Thus i f the swell were t r a v e l l i n g with the wind the coherence would be halved i f U 0 « C and reduced to a small value i f U 0 ~ C; i f on the other hand the swell were moving against the wind the coherence would again be halved i f JU Q| « C, but would only be reduced by 207. i f JU0| C. I f the swell was running at some other angle to the wind, then the expected drop i n coherence would l i e between the extremes of 207« and 1007„. From the observed drops i n coherence at low frequencies, the swell could have been t r a v e l l i n g i n almost any direction, i n run 2a, and was probably from the Northwest i n runs 3 and 5. The run 5 p r e d i c t i o n i s the same as was i n f e r r e d from the 117 TABLE 6.4 Coherence Changes Caused by Removal of ""^"J"^ i n Pressure-Waves Cross Spectra U 5 C n c p f P s ^ 7o drop i n 7„ drop at -"-n Run No. (cm sec~l) swel l l o c a l peak (Hz) 5 150 340 - SE 40 no peak 1.2 1 220 350 - W 80 no peak 1.7 2b 310 - 280 ESE no peak 25 1.2 2a 320 990 280 ESE 50 40 1.4 3 340 900 220 E 40 65 1.7 6 570 - 275 SE no peak 50 0.9 4b 700 - 320 ESE no peak 12 1.2 4a 800 800 270 ESE no peak 10 1.3 118 veloci ty-wave phase r e l a t i o n discussed i n Appendix 3. 6.5.1b Phase The behaviour of the phase spectra i s best explained i n terms of the energy f l u x (pdfl/dt) from the wind to the waves. This var ie s as s i n (- 0 ) , where 6 i s the phase lead of the pressure ahead of the wave s i g n a l . As has been mentioned e a r l i e r , 6 = - 1 8 0 ° i s pred ic ted by p o t e n t i a l flow theory and no generation occurs i n th i s case (except an i n s i g n f i c a n t amount by the Kelv in-Helmholtz mechanism). The most e f f i c i -ent generation occurs when d = - 9 0 ° (pressure lagging the waves by 9 0 ° ) , and damping occurs i f + 9 0 ° ^ 6 A 1 8 0 ° . So far l i t t l e has been said about the r e l a t i v e importance of the phase angles between p s and ^ , and between p g + p CJ1£ and . The wave surface experiences the former, while a sensor f ixed i n the a i r would make an E u l e r i a n measurement of the l a t t e r . T h e o r e t i c a l pred ic t ions such as those of Mi les (1960) are made i n terms of the phase d i f ferences presented here as the ( p s + fa^ ^ ), 7j"s spectra; therefore these w i l l be discussed below as "the phase angle between the pressure and the waves". The phase spectra (Figures 32 - 39) for a l l the runs show a s i m i l a r pat tern . At frequencies below f^ (where C = U at z =A/£TT) the phase angles are mainly p o s i t i v e , i n d i c a t i n g wave damping. At frequencies near or somewhat lower than f^ the phase crosses 1 8 0 ° , i n d i c a t i n g the onset of wave growth. The s h i f t with increas ing frequency i s u sua l ly sudden, the phase changing from 1 8 0 ° to anywhere from - 1 6 0 ° to - 1 1 0 ° over a frequency range of 0.2 to 0.3 Hz. At higher frequencies i t changes more s lowly, genera l ly cont inuing to s h i f t toward - 9 0 ° and reaching i t as often as not . At s t i l l higher frequencies i n most runs there i s a 119 suggestions (much c l earer i n the p j , 1 ^ phases) that the phase may s h i f t back towards 1 8 0 ° ; i t can only be c a l l e d a suggestion, however, s ince coherences at the higher frequencies (greater than 1.6 Hz) are so low that the phase i s not w e l l - d e f i n e d . Runs 1, 2b, 3, and 6, which had the most spikes i n the pressure data, d i s p l a y larger coherences at these higher frequencies than those with few or no sp ikes , i n d i c a t i n g that i n those runs much of the phase information at high frequencies i s from the spikes themselves and i s hence spurious as far as th i s d i scuss ion i s concerned. The e r r a t i c behaviour of the (p g + pa<j 1£ ) , phase spectrum at low frequencies i n run 1 (Figure 32) i s caused by the fac t that from 0.05 to 1 Hz the removal of - Pj^^ has reduced the , r£ coherences wel l below 0.3. A large noise peak at 1226 Hz appears to have unfortu-nate ly co inc ided with the frequency at which the crossover from damping to wave growth occurred. The pressure s igna l i n run 2a (Figure 33) i s r e l a t i v e l y uncontaminated with sp ikes , and the pressure c a l i b r a t i o n i s considered as known to w i t h in ^107o. I t i s the only run where the high-frequency phase does not show a tendency to drop back towards 1 8 0 ° from - 9 0 ° . The only other runs which are comparatively free from spikes and i n which ac t ive generation occurs, so that a comparison can be made with run 2a, are runs 4a and 4b (which were h a n d - d i g i t i z e d ) ; i n these the spikes were smoothed out of the pressure s ignals by hand p r i o r to d i g i t i z a t i o n . This smoothing pro-cess should not a f f ec t the high frequencies . Subject to the expected errors noted on p. 79, the high frequency phases i n these runs can be compared with those of run 2a. They both i n d i c a t e a trend of the phase 120 at h igh frequencies to r e t u r n toward 1 8 0 ° . Thus i t i s run 2a which must be treated as the exception. A large 0.2 Hz swel l peak i s present i n the wave spectrum of th i s run (Figure 15) which i s absent i n run 2b taken 20 minutes l a t e r . The "swell" was thus probably caused by marine t r a f f i c . In th i s run and i n run 2b the sharp phase s h i f t occurs at a frequency 0.3 Hz below f^; th is i s a larger f^ - f p frequency d i f ference than ex i s t s i n any of the other runs. Run 2b';shows phase s h i f t s of only 6 0 ° from 1 8 0 ° , as opposed to the 9 0 ° s h i f t s occurr ing i n run 2a. The wind speed had dropped s l i g h t l y for t th i s run fo l lowing run 2a; th i s might account for the less e f f i c i e n t energy t r a n s f e r s . The frequency of the wave peak has r i s e n s l i g h t l y , from 0.56 Hz i n run 2a to 0.58 Hz i n run 2b. The pressure s igna l i n run 3 i s also quite "spikey". In th i s run U5 was s l i g h t l y larger than i n runs 2a and b and the fe tch longer (see Table 6 .1) . Swell near 0.2 Hz was present , but as noted on p.116 i t s d i r e c t i o n i s not c l e a r . The low-frequency phases (Figure 27) show that i t was being damped. The phase "peak" at 0.38 Hz i s spurious , being associated with a low coherence between p s ,+ •>£ and 1£ . The s h i f t from damping to generation i n run 3 occurs at a frequency about 0.2 Hz above the peak of the wave spectrum and almost exact ly at f . The behaviour of the phase between 1.1 and 1.4 Hz i s e f f i c i e n t l y masked by noise; the phase at higher frequencies , although considered u n r e l i a b l e , shows a tendency to d r i f t back toward 1 8 0 ° from i t s maximum s h i f t of 9 0 ° ( 1 8 0 ° to - 1 1 0 ° ) at 0.8 Hz. The two h a n d - d i g i t i z e d runs, 4a and 4b, were separated by 10 minutes, 4b being the later.;: of the two. During them the wind speed was higher than 121 for any of the other runs. In both cases the frequency of the peak of the wave spectrum i s we l l above f .^, i n d i c a t i n g that the waves were far from saturated and that therefore during ne i ther run were wave states s t a t i o n a r y . The process of hand d i g i t i z a t i o n i s be l ieved to introduce phase errors less than 2 0 ° below 1.6 Hz; see the d i s cuss ion i n "Data Ana lys i s and In terpre ta t ion" , p. 79. The phase d i f ference between p%,i£i and ( p s + f a < 3 * 2 ) » ^ a r e smaller i n runs 4a and 4b than i n other runs, i n d i c a t i n g that the por t ion of p s corre la ted with the waves was much larger than | ^ | . The sharp phase change associated with the onset of wave growth occurs at a f r e -quency 0.2 Hz above f^ i n run 4a and 0.1 Hz above f^ i n run 4b. In run 4a the phase at frequencies above the frequency of the i n i t i a l sudden change behaves r e g u l a r l y u n t i l 1.3 Hz, when i t returns from near - 9 0 ° to near - 1 8 0 ° w i th in 0.1 Hz. This sharp change of phase i s associated with a very low coherence, i n d i c a t i n g that i t i s not s t a t i s t i c a l l y s i g n i f i c a n t . Roughly the same phase behaviour i s observed i n run 4b except that the r e t u r n i n phase towards 1 8 0 ° i s associated with a large peak i n the p,-^ and (p + f o , " " c o h e r e n c e s near 1.6 Hz. This means that the change has some s i g n i f i c a n c e , whereas the w i ld phase f luc tua t ions at f r e -quencies above 1.6 Hz do not, being associated with low coherences. The reason for the d i f ference i n the coherence spectra above 0.8 Hz i n runs 4a and 4b, separated as they are by only ten minutes, must for lack of any other explanation be put down to v a r i a b i l i t y i n the condit ions of the wind and wave f i e l d s . The d i f f erence only becomes large at f r e -quencies above 0.8 Hz, and as w i l l be seen l a t e r contributes l i t t l e to the energy and momentum fluxes of e i ther run. 122 Run 5 must be viewed with a knowledge of i t s time h i s t o r y . The wind was for the most part weak and v a r i a b l e (about 80 cm/sec) only r i s i n g to 150 - 200 cm/sec for the l a s t t h i r d of the run . The waves i n the frequency band from 0.1 - 0.8 Hz were not l o c a l l y generated, and as has been shown (Appendix 3) they were t r a v e l l i n g against what wind there was. The phase remains p o s i t i v e (+150 to 1 8 0 ° ) i n most of the frequency range for which the coherence exceeds 0.3; hence th i s "swell" was being damped. The phase v a r i a t i o n s above 1 Hz are considered to be random and w i l l not be discussed f u r t h e r . The pressure s igna l i n run 6 i s contaminated by a considerable number of sp ikes . The wave spectrum for the run i s very smal l , presumably because the f e tch (1.6 km) was less than for the other runs. As was noted i n Table 6 .1 , a l i g h t r a i n was f a l l i n g during the run . The sharp low-frequency phase change occurs at a frequency 0.2 Hz lower than f^ and 0.4 Hz below the frequency of the peak of the wave spectrum. The phase between P s + f^ q,^  a n d V£ reaches - 9 0 ° near f^ . and then f a l l s gradual ly back towards 1 8 0 ° , apparently reaching i t at 2 Hz before the coherence drops below 0.4. In general i f a smooth curve i s drawn through the computed phase points of th i s run , the scat ter about this curve i s quite small due to the high coherences present over the en t i re frequency range from 0.2 to 2 Hz. 6.5.2 The Fluxes of Energy and Momentum to the Waves 6 .5 .2a Summary of Mean Values of E and fw Table 6.5 shows, along with and 8 , the in t egra l s under the E and TW spectra for a l l runs. These i n t e g r a l s have been truncated at 3.0 Hz, s ince any pressure and wave s ignals which occur above 3 Hz are observed to be incoherent ( th i s truncat ion i s p a r t i c u l a r l y important i n the 'T'w 123 2 spec tra , s ince they contain an CO i n the i r numerator--see p .88) . In a d d i t i o n the s tresses computed i n th i s way ( 7 J W ) are compared with those ( T$ ) measured by the three-component sonic anemometer, when a v a i l a b l e , and with those computed from 7= 2 C c = 0.0012 ^>(jLU5 6 .3. Also shown are drag c o e f f i c i e n t s C^y ca l cu la ted from CDW = <Tw/ e„ u 5) 6 - 4 -A'K! A c o r r e c t i o n must be appl ied to the '~C<V>/ spectra to account for the f i n i t e width of the d i r e c t i o n a l d i s t r i b u t i o n of the waves. The wave probe records waves from a l l d i r e c t i o n s ; s ince momentum input to the waves only occurs i n the wind d i r e c t i o n , the momentum withdrawn from the a i r by a wave t r a v e l l i n g at an angle © with the wind w i l l vary as cos S A d i r e c t i o n a l d i s t r ibut ion for the waves must be assumed so the correc-t ion may be ca l cu la ted; G i l c h r i s t (1965) f inds that i n East winds at the Spanish Banks s i t e , one which var ie s as cos © i s c lose to the observed one for waves at frequencies higher than the s p e c t r a l peak. Therefore the p r o b a b i l i t y of f i n d i n g a given wave d i r e c t i o n at an angle between 8 and d© to the wind w i l l be given by P(6)d0 = 2 cos 2 ed6 6 > 5 < This gives for the corrected momentum f l u x spectrum Ft 2 (f)p( e ) c o s © de w -It/2 /2 X (f) = 2 \ r 2 C w ( f ) \ c o s 3 6 d e w -it 12 124 TABLE 6.5 Run No. U 5 cm sec" 1 e degrees (true) erg cm ^ sec"l « 1 dyne cm" 2 T c dyne -2 cm T dyrie cm" 2 CDW 5 150 135 -5 7 -.002 .02 .03 - .03 -1 220 260 13 17 .11 .11 .07 - .0018 2b 310 115 29 18 .16 .11 .14 - .0013 2a 320 80 43 18 .24 .10 .15 - .0019 3 340 115 30 32 .19 .26 .17 - .0013 6 . 570 120 60 32 .38 .22 .48 .46 .00095 4b 700 115 90 233 .45 .30 .76 - .00075 4a 800 ,115 154 184 .66 .36 .95 - .00093 (1) (2) (3) (4) (5) (6) (7) Notes: 1. A l l un i t s are c. g .s . 2. 6 i s wind d i r e c t i o n i n degrees (true) 3. 0~g and <T"r are the standard error of the means of E and f over the t o t a l number of data blocks analysed. 4. "7*w i s the mean momentum f l u x from wind to waves, computed from the f,^ c o r r e l a t i o n . 5. "f c i s the momentum f l u x from the a i r to the sea, computed from T c = ? a C D U 5 ' C D = - 0 0 1 2 -6. Xs i s the momentum f lux from the a i r to the sea, as measured by a sonic anemometer. ^Dy i s the dimensionless drag c o e f f i c i e n t computed from Cpy = ea U 5-125 0.85 T w ( f ) 6.6; that i s , 157o less than the calculated value. The l i m i t s of ~t Tf /2 are chosen since i n a p r a c t i c a l s i t u a t i o n the p r o b a b i l i t y of f i n d i n g wind-driven waves t r a v e l l i n g against the wind i s n e g l i g i b l e . Taking more sophisticated and accurate d i s t r i b u t i o n s i s not worthwhile i n view of the s i z e of the correction and the expected accuracy of "7J* (f) . A l l the *7j* w values except run 5 have been corrected for the assumed d i r e c t i o n a l spec-trum of the waves (Equation 6 .4) . The values of E and "J* were computed for each data block analysed (512 or 1024 samples, or time i n t e r v a l s of 10 or 20 seconds), and the <P columns for E and r7J*w give the standard error of the mean for these populations. 6.5.2b The Spectra of Energy and Momentum Flux The spectra of Energy and Momentum f l u x from wind to water are pre-sented i n Figures 40 to 47. They are i n units of dyne cm ""sec ''Hz - 2 -1 (E(f)) and dyne cm Hz ( T w ( f ) ) . Included on each figure i s the wave spectrum for the run. Also noted i s f^., the frequency where the wave phase v e l o c i t y at the peak of the spectrum equals U^, the wind speed at a distance V k = "X /2TT from the mean water surface. I t should be noted that, as on a l l spectra presented i n "Results" the l i n e s joining the spectral estimates are meant to be guides to aid the eye i n seeing the sp e c t r a l shapes; they are not meant to imply i n t e r p o l a t i o n between points. Thus E(f) and 7 T w ( f ) , since both are computed from exactly the same data, must cross the zero f l u x l i n e at the same frequency for a given run; the "guide l i n e s " r a r e l y do cross this l i n e at exactly the same frequency. Run 1, the only one taken i n westerly winds, shows low energy trans-f e r s . The wind speed was 220 cm s e c - 1 ; the l o c a l wave generation i s weak 126 and has a c o n t r i b u t i o n for unknown reasons at frequencies less than f^. I t i s considered u n l i k e l y that the apparent energy and momentum transfers at frequencies below 1.2 Hz are r e a l . Run 2a i s considered as roughly t y p i c a l of a fe tch l i m i t e d , e a s t e r l y wind s i t u a t i o n . Although a large wave peak i s present at 0.17 Hz, l i t t l e or no damping i s seen to occur. The peak of the wave spectrum i s at a frequency lower than f^; i n the l i g h t of r e s u l t s from other s i m i l a r runs t h i s may i n d i c a t e the presence of stronger winds upstream of the measure-ment s i t e . Energy transfer to lower frequencies through nonl inear i n t e r -a c t i o n s , such as those proposed by Benjamin and F e i r (1966), P h i l l i p s (1961), and Hasselman (1963) i s another p o s s i b i l i t y . The peaks i n E(f ) and (f) occur i n run 2a at frequencies higher than, but less than twice that of the peak of the wave spectrum. At higher frequencies the spectra approach zero, being governed p r i m a r i l y by the r a p i d decrease of the wave spectrum; the phase between p s + ^ Y £ and n^. i n th is run (see F igure 33) i s near - 9 0 ° at 1.4 Hz, while E(f) and f w ( f ) have dropped to 15-207o of t h e i r peak values by th i s frequency. The large low-frequency wave s p e c t r a l peak i s less we l l -de f ined i n run 2b than i t i s i n run 2a. Damping i s seen to occur at one f r e -quency, but th is cannot be considered as s t a t i s t i c a l l y s i g n i f i c a n t . The peaks i n E( f ) and f w ( f ) are i n th is part of the run s h i f t e d to lower frequencies than i n run 2a; they almost co inc ide with the peak of the wave spectrum. In fac t they peak c loser to the wave peak i n run 2b than i n any other . The wind speed being lower i n th is part of the run , E(f ) and 'fw(f) are somewhat lower than i n run 2a, the larges t d i f ferences occurr ing between 0.7 and 1.2 Hz. The two subs id iary peaks i n the spectra , at 1.04 and 1.62 Hz, are remarkably c lose to the f i r s t and second harmonics of the frequency at the peak of the wave spectrum. Because of the large v a r i a b i l i t y i n the i n d i v i d u a l s p e c t r a l estimates E(f ) and f ( f ) , the two peaks are not considered s t a t i s t i c a l l y s i g n i f i c a n t ; i f they appeared i n the spectra of every run they would have to be taken s e r i o u s l y , but they do not . In run 3, some wave damping i s seen to occur at the low-frequency (0.2 Hz) swel l peak i n the wave spectrum. Not enough information i s present , however, to compute the amount of damping. I t has n e g l i g i b l e e f f ec t on E and T W , the mean t o t a l energy and momentum f luxes . The wind-generated wave peak occurs at 0.62 Hz, c lose to f^, i n d i c a t i n g that the spectrum was s t i l l growing. The M i l e s ' c r i t i c a l height (Table 6.2) was 9.7 cm for th is run and 100 cm for run 2; th i s and the f a c t that the frequency of the peak of the wave spectrum i s higher than i n run 2 a lso suggest that i t i s a "younger" spectrum. E(f ) at i t s peak i s less than i n run 2a but greater than i n 2b. Presumably because of the presence of many spikes i n the pressure r e c o r d , the spectra converge to zero more s lowly at high frequencies than i s the case for run 2. I t may be th i s e f f e c t which masks any evidence of the harmonic peaks so evident i n run 2b. On the other hand, the pressure s i g n a l of that run i s jus t as h e a v i l y contaminated with spikes as i s that of run 3. Runs 4a and 4b, the two high wind speed runs (U^ = 800 and 700 cm/sec r e s p e c t i v e l y ) , show the l arges t energy and momentum transfers (note the larger scales i n Figures 44 and 45). Neither show s i g n i f i c a n t damping at low frequencies , i n sp i t e of a small swel l peak at 0.2 Hz i n run 4a. In both runs the peaks of E(f) and "U w(f) occur at frequencies 128 above that of the peak of the wave spec tra . As i n run 2, they appear to occur on the higher-frequency "rear face" of the wave spectrum; they occur w e l l below the frequency of the f i r s t harmonics. In both, however, the spectra l e v e l o f f i n the reg ion of the frequency of the f i r s t har -monics. The high-frequency region of run 4a i s marred by going negative; th i s i s not considered to be r e a l , and i s associated with the blowup of the phase spectra discussed on p. 121. The comparative behaviour of E(f ) and f w ( f ) i n runs 4a and 4b shows some s i m i l a r i t i e s with those of runs 2a and 2b. In both run 2 and i n run 4 the wind speed dropped between the two sect ions of the runs , with the r e s u l t that the spectra show the l arges t d i f ferences at frequencies jus t above t h e i r peaks; i n runs 4a and 4b th is i s from 0.8 to 1.2 Hz. On the other hand the behaviour of the run 4 spectra at frequencies below t h e i r peaks i s s t r i k i n g l y d i f f e r e n t from that observed i n run 2; th i s d i s p a r i t y can be put down to the f a c t that the waves i n run 2 were saturated , while i n run 4 the wave spectrum was growing r a p i d l y , the peak advancing from 0.6 Hz i n run 4a to 0.5 Hz i n 4b. E(f ) and TTw(f) advance with i t , E(f ) showing the most spectacular change. Run 5 was taken as a "noise run"; the wind speed was so low that the water surface was unruf f l ed for two th irds of the run , and the i n t e n t i o n was to see what E(f ) and ^ ( f ) would be when phys i ca l condit ions precluded the p o s s i b i l i t y of wave generat ion. I t i s p l o t t e d on the same sca le as the others (with the exception of runs 4a and 4b) and gives a good q u a l i t a t i v e p i c t u r e of the r e s o l u t i o n of the system. For almost the f u l l frequency band where waves ex i s ted damping i s i n d i c a t e d . The 129 "tightness" of E( f ) and * C w ( f ) at high frequencies , although at f i r s t s u r p r i s i n g having seen the phase spectrum of the run (Figure 38), i s explained by not ing the r e l a t i v e l y small s i z e of the pressure and wave power spectra (Figure 30) compared with those from, for instance , run 6 (Figure 31). The small p o s i t i v e values of E(f ) and t w ( f ) occurr ing above 0.8 Hz are considered to be no ise . Note i n Table 6.5 that the s tress during run 5 from the sonic anemometer agrees c l o s e l y with that measured by the buoy. The most s t r i k i n g feature of run 6 i s the small s i z e of the wave spectrum. E(f ) and T w ( f ) , on the other hand, are far from smal l . The f e tch for th i s run was short , and large energy and momentum fluxes might w e l l be associated with small waves i n such a s i t u a t i o n . Since the j u x t a p o s i t i o n of large E and " f w spectra with small waves i s i n th i s case reasonable, a d d i t i o n a l credence i s l en t to the accuracy of the c a l i b r a t i o n of the wave sensing system, which i n the case of run 6 has been c a r e f u l l y s c r u t i n i s e d because of the small observed wave spectrum. The s tress computed from the i n t e g r a l under the spectrum shown i s , as for runs 4a and b, somewhat lower (35% here) than that computed from Equation 6.1 or i n th is case from that measured by the sonic anemometer 1.75 m above the mean water l e v e l . The stresses computed from the i n t e g r a l under the t w ( f ) spectrum from runs 1, 2, and 3, on the other hand, exceed those computed from Equation 6.1. These runs d i f f e r most prominently from runs 4 and 6 i n the r e l a t i o n of the wave phase v e l o c i t y C p at the peak of the spectrum to the mean wind speed U5; U5 - c p i s near zero for runs 1, 2, and 3 and i s much greater than zero for runs 4 and 6. This evidence suggests that as the wind speed r e l a t i v e to the waves 130 increases an increas ing proport ion of the momentum input to the water goes not to the waves but to some other momentum s ink , such as currents . C e r t a i n l y from Table 6.5 the buoy measurements i n d i c a t e that a dominant f r a c t i o n of the t o t a l wind s tress over water goes i n i t i a l l y d i r e c t l y in to waves. The l a s t paragraph ind ica tes that the f r a c t i o n decreases as (U - c) increases . 6.5.3 The Spectra of T Miles (1957) defines £ as the negative damping r a t i o , or the f r a c t i o n a l growth rate i n mean wave energy per rad ian: T = - L = - | E 6 . 7 . CO E 9 t Since the wave spectra and E spectra are a v a i l a b l e , spectra of C are computed from f ( f ) = Q u r Z J £ ^ . 6.8 where use has been made of Equations 5.3 (p. 82) and 5.9 (p. 87). Two points must be made: f i r s t , the presence of spikes and of large d r i f t s i n the pressure s i g n a l a l l introduce low-frequency noise i n the p g quadrature spectrum, but there i s no corresponding noise i n ^ ^ ( f ) , the power spectrum of the wave s i g n a l . This causes large f luc tua t ions i n the low-frequency f spectra; these are not considered r e a l . Second, as for the other spec tra , the If spectra above 2 Hz (CJ > 13 rad/sec) are almost e n t i r e l y composed of no ise , and are not shown. Also inc luded on the graphs are f spectra ca l cu la ted from the formula (sc ~ ( u W c - 1) , - ^ / c  L) 6.9, Cw where c i s the wave phase v e l o c i t y and i s the mean wind speed at a height one wavelength above the water surface , extrapolated from a logar i thmic p r o f i l e assuming Cp = .0012. This formula i s an empir i ca l one proposed by Snyder and Cox (1967) which f i t s t h e i r wave growth data and those of Barnett and Wllkerson (1967) moderately w e l l . The spectra are presented, except for run 5, i n Figures 48 to 55. The f i r s t thing to note i s the s i m i l a r i t y of the present r e s u l t s with Snyder and Cox's empir i ca l curves i n the frequency range from the onset of ac t ive wave generation to u = 10 r a d / s e c . The curves, except near where they cross the l i n e of zero energy t r a n s f e r , are always w i t h i n a fac tor of two. They begin to diverge at higher frequencies , the present r e s u l t s f a l l i n g below those computed from the Snyder and Cox r e l a t i o n . A s i m i l a r tendency, although not near ly so marked, shows up i n the equi -va lent regions of the spectra shown by Snyder and Cox and Barnettand Wilkerson . In the case of the present data , the f a l l o f f i s probably explained as the e f f ec t on the pressure-waves coherence of in ter ference by the buoy with both s i g n a l s . In runs 3 and 4, the data f a l l below Snyder and Cox's p r e d i c t i o n at a l l frequencies . These runs , from t h e i r small c r i t i c a l heights ( a l l < 10 cm), were taken while the wave spectra were s t i l l growing; the generation process was not s ta t ionary or homogeneous. The f a c t that the data f a l l below those of Snyder and Cox, which were taken i n a more or less saturated wave f i e l d , ind icates the p o s s i b i l i t y that the genera-t i o n process becomes i n c r e a s i n g l y e f f i c i e n t (higher E for a given E) as the wave f i e l d approaches s a t u r a t i o n . Run 6 i s a s p e c i a l case; i t has a low c r i t i c a l height (1.2 cm), but 132 t f a l l s w e l l above ^ s c - In th is case the fe tch was much less (1.6 km as compared with 6.7 and 2.5 km; see F igure 2) than for the other runs. Also , water depths i n the d i r e c t i o n from which the wind was blow-ing shoal as the shore i s approached and some f a i r l y steep wooded c l i f f s 30 meters h igh she l ter the shore, fur ther reducing the e f f e c t i v e f e t c h . Thus, although E i s large s ince U - c i s l arge , E i s r e l a t i v e l y smal l , l eading to the high £ va lues . The other runs (1, 2) genera l ly show values of C i n the region, of ac t ive generation which exceed £ s c - This can be expected. Since i n these runs the wave spectrum i s saturated or even oversaturated (U,. < Cp), they are more c l o s e l y r e l a t e d to those taken by Snyder and Cox and Barnett and Wilkerson . These inves t iga tors measured not energy input at one po int as i n the present experiment, but wave growth with increas ing fe tch; t h e i r spectra therefore do not inc lude that por t ion of the energy input which passes from the waves to a l l other sources v i a breaking , e tc . whi le the spectrum i s being formed. SECTION 7: DISCUSSION OF RESULTS 7.1 In troduct ion 133 7.2 Recent Attempts to E x p l a i n Wave Generation 133 7.2.1 Mi les (1967) 133 7.2.2 Stewart (1967) 135 7.3 The Power Spectra 139 7.4 The Phase Spectra 141 7.5 The Energy and Momentum Flux Spectra 144 7.5.1 The Mean Energy Flux 145 7.5.2 The Wave-Supported Wind Stress 146 7 .5 .3 . The Energy and Momentum Flux Spectra . . . . 149 7.5.3a Observed and Pred ic ted T r a n s i t i o n s Fetches 150 7.5.3b Energy Transfer i n the Wave Spectrum . 152 7.6 The f Spectra 155 7.6.1 Mean Values of C 155 7 .6 .2 . The Spectra of f 156 7.6.3 A Dimensionless R e l a t i o n between { (f) and Wind Speed 159 SECTION 7: DISCUSSION OF RESULTS 7.1 In troduct ion The r e s u l t s presented i n the l a s t sec t ion are i n very sparse com-pany. In f a c t they are the only (two-dimensionally)' E u l e r i a n pressure measurements made on the surface of n a t u r a l l y generated sea waves; the only comparable measurements are those of Shendim and Hsu (1967), over mechanical ly generated waves i n a wind tunnel (described i n "Observa-t ions", p. 33 f f . ) . Whereas the wind tunnel work may i n some ways ( i . e . i n the use of the s ing l e s inuso id for a wave) more c l o s e l y approximate the condit ions set out as s i m p l i f i c a t i o n s i n p a r t i c u l a r i d e a l i z e d models, the present r e s u l t s r e l a t e d i r e c t l y to r e a l i t y , and thus i t i s hoped, w i l l enable theor i s t s to apply the necessary s i m p l i f i c a t i o n s to t h e i r models to bet ter e f f e c t . 7.2 Recent Attempts to E x p l a i n Wave Generation A f t e r the works of Snyder and Cox (1966) and Barnett and Wilkerson (1967) appeared and strong doubts i n the e f f i c a c y of M i l e s ' i n v i s c i d model were aroused, a l t e r n a t i v e mechanisms were considered for ge t t ing energy in to the waves. For the sake of completeness, those which appear to be re levant to the present r e s u l t s w i l l be discussed i n the fo l lowing paragraphs. 7.2.1 Mi les (1967) Mi les (1967) reviews the evidence for and against h i s i n v i s c i d model, and i s forced to reconsider the e f fects on the momentum transfer of wave-induced perturbat ions of the turbulent Reynolds s tress -f a uw. 133 134 He suggests that the r o l e of the Reynolds stresses may increase i n importance as the time scale of the turbulence increases r e l a t i v e to the time sca le of a given s p e c t r a l component i n the wave f i e l d . He therefore suggests the use of a dimensionless t ime-scale parameter where the slope of the mean v e l o c i t y p r o f i l e i s evaluated at the c r i t i c a l and a logar i thmic wind p r o f i l e i s assumed. Mi les then goes on to e luc idate the r o l e played by these e f fects of the turbulence. He reverts to h i s o r i g i n a l (1957) formulat ion of the equation for the momentum f l u x from a i r to waves, th i s time r e t a i n i n g the terms containing the turbulent Reynolds s tresses . A l l v a r i a b l e s are averaged i n the crosswind d i r e c t i o n . He obtains i n th i s way two terms i n the momentum equations; the f i r s t consis ts of the v e r t i c a l i n t e g r a l of the covariance between the v e r t i c a l v e l o c i t y and the v o r t i -c i t y along a flow streamline (corresponding to momentum trans fer v i a the i n v i s c i d laminar mechanism), and the second represents the wave-induced perturbat ions of the Reynolds s tress at the sea surface ( P h i l l i p s , 1966; Equation 4 .3 .23 , i s shown by Mi les to be equivalent to h i s formu-l a t i o n of the momentum equations, which i s the one under d i scuss ion here ) . The f i r s t of these two terms i n the momentum equations i s l inked to the second by an equation for the advect ion of the v o r t i c i t y by the per turbat ion Reynolds s tresses . Mi les remarks that the f i n d i n g of 1/kc 7.1, height z c f or waves of wavenumber k, K (en 0.4) i i s von Karman's constant, 135 P h i l l i p s (1966) that the second term can be neglected i s quest ionable , and suggests that further t h e o r e t i c a l progress i s barred u n t i l more experimental information becomes a v a i l a b l e on the Reynolds s tress d i s -t r i b u t i o n at the surface of the water. 7.2.2 Stewart (1967) Stewart (1967), i n a review paper, discusses several nonl inear mechanisms which had not up to that time been mentioned i n the l i t e r a t u r e . The f i r s t considers the e f fects of the turbulence on the momentum trans-fer v i a M i l e s ' i n v i s c i d laminar mechanism. The amount of momentum transfer by th i s mechanism i s propor t iona l both to the r a t i o of the curvature to the slope of the wind p r o f i l e , and to the rms v e r t i c a l v e l o -c i t y , a l l evaluated at the c r i t i c a l height z^ where the mean wind speed equals the wave speed. Since both the r a t i o and the v e r t i c a l v e l o c i t y are very nonl inear funct ions of z^ the turbulence, by causing l o c a l v a r i a t i o n s i n z^, can cause the increases i n momentum transfer during wind gusts to g r e a t l y exceed the decreases associated with l u l l s , thus grea t ly increas -ing the mean momentum transfer over that which would be computed by assum-ing a mean z^ which i s averaged over many wavelengths. He proposed that there may be d i f f e r e n t regimes of generat ion, depending on the s i ze of the r a t i o L of the c r i t i c a l height to the rms wave he ight . This idea i s connected with h i s arguments that the exponen-t i a l f a l l o f f of streamline amplitude with height used by Mi les as h i s model of the a i r f l ow i s probably u n r e a l i s t i c i n the l i g h t of observed spectra of turbulent v e l o c i t i e s above sea waves, which show no obvious 136 wave peaks. This leads him to propose that i n f a c t the streamline con-f i g u r a t i o n s over the waves may not be as shown i n L i g h t h i i l (1962) or P h i l l i p s (1966; p. 91) but ins tead , i n the coordinate system moving with the phase speed of the waves, the "cats-eye" associated with flow around the c r i t i c a l height may be s i tuated i n the troughs instead of over the c r e s t s . This conf igurat ion requires the presence not only of pressure gradients , but a lso of shear s tress gradients along the wave surface . The stresses must be arranged to produce the given streamline pa t t ern , and are thus low i n the troughs and high on the c r e s t s . This i n turn leads to a d i s t r i b u t i o n of low pressures to leeward and high pressures to windward of the wave c r e s t s , which i s exact ly the d i s t r i -but ion which generates waves. I f th i s mechanism i s indeed operat ive , i t would not be separable from M i l e s ' i n v i s c i d mechanism by measurement of normal pressures alone; the shear s tress d i s t r i b u t i o n would have to be measured as w e l l , i n regions very c lose to the water surface . In view of the great d i f f i c u l t i e s involved i n making such measurements, i t appears that the existence of th i s mechanism would be hard to detec t . The d i s t r i b u t i o n of the shearing s tress along the wave suggests another mechanism to Stewart. The e f fects of the s tress d i s t r i b u t i o n w i l l be f e l t by a th in surface layer i n the water. I f the momentum balance i n th i s layer i s considered, i t i s found that the divergence of the flow i n i t caused by the s tress d i s t r i b u t i o n induces v a r i a t i o n s i n i t s thickness A along the wave p r o f i l e . These v a r i a t i o n s i n A act on the wave as i f they were v a r i a t i o n s A i n the pressure on the wave. With the s tress d i s t r i b u t i o n described above th i s extra pressure lags the waves by 9 0 ° , and thus represents generat ion. Stewart ca lcu la tes 137 that the value for the momentum transport associated with the l a t t e r mechanism i s k a T o , where k i s wave number, a i s wave amplitude, and tT0 i s the amplitude of the shearing s tress ; i t has s ince been pointed out by Longuet-Higgins (1969a) that th i s should be ka t"o/2 . This represents a f r a c t i o n ka/4 (2: 0.05 for t y p i c a l sea waves) of the t o t a l momentum transfer from a i r to sea; for short wavelength waves with large s lopes , th i s f r a c t i o n might become considerably larger than the above-mentioned f i g u r e of 5%, which i s taken as that l i k e l y for normally observed slopes i n a wind-driven sea. Stewart then goes on to suggest a f i n a l p o s s i b i l i t y . Momentum can be added s e l e c t i v e l y to the crests of long waves v i a the breaking there of steep short-wavelength waves, which would presumably pass t h e i r momentum d i r e c t l y to the o r b i t a l v e l o c i t i e s of the long waves. He notes fur ther that ne i ther th i s mechanism nor the preceding one depends for i t s a c t i o n on normal pressures , and hence ne i ther would be observed by measurements of these pressures . Observations on wave growth which are considered to have been taken i n a regime i n some sense s i m i l a r to that obta in ing i n the present set of measurements have been reviewed e a r l i e r ("Observations", p. 17 f f ) . As was pointed out, they give c o n f l i c t i n g evidence; the f i e l d measure-ments genera l ly show larger discrepancies from M i l e s ' i n v i s c i d theory than those i n wind tunnels . M i l e s , as mentioned e a r l i e r (p. 134), has suggested that th i s may be due to the increased ef fects i n the f i e l d measurements of the turbulent Reynolds stresses on the momentum trans-f e r , as represented by the largeness of the parameter TV given by Equation 7.1. (Stewart (1967) has suggested i n turn that the type of 138 momentum transfer regime operat ing i n a given s i t u a t i o n may be governed by the s i ze of L = z c (V?)'k 7.2. P h i l l i p s (1966) describes the regimes of wave generation i n terms of the r a t i o c /u^, wave phase v e l o c i t y over f r i c t i o n v e l o c i t y . He con-cludes that three separate mechanisms are operat ive for d i f f e r e n t values of c /u, v ; for c / u * ^ 5 (although he does not s p e c i f i c a l l y say so) the mechanism must become c o n t r o l l e d by v i s c o s i t y (Miles ' v iscous laminar model); for 10 < c /u* ^ 20 the i n v i s c i d laminar model i s operat ive , while for c/u^y 20 the i n t e r a c t i o n of the turbulent Reynolds stresses with the a i r flow over the waves i s assumed to be dominant. A l l of the r a t i o s : J\. , L , and c /u^, are d isplayed for the present runs i n Table 7.1. A l l values are taken at the l oca l ly -genera ted peak of the re levant wave spectrum. From the tab le , i t i s c l ear that _A- and L are dominated by z c ; they are with the exception of run 6 near ly the same i n s i z e . They show moreover that the present measurements cover a large range of va lues , and thus should provide information on more than one wave generation regime. The range of values of c/u^. suggest that a l l three regimes: viscous laminar, i n v i s c i d laminar, and Reynolds stress-dominated, are l i k e l y to be present i n the runs. I t i s i n t e r e s t i n g that although the range of v a r i a t i o n of Cp/u^ (where Cp i s the wave phase v e l o c i t y at the peak of the wave spectrum) i s very much less than for .A. and L , the three parameters, which are a l l derived from wave c h a r a c t e r i s t i c s at the s p e c t r a l peak, are a l l arranged i n the same order . TABLE 7.1 Run No. Notes: u 5 cm/sec (1) CP / U * (On racl/ sec " c cm cm (2) (2) (2) A (3) (4) 5 150 5. ,2 332 (64) 2. .95 10 4 3. ,4 (>103) (>103) 1 220 7. .6 370 49 2. ,65 10 4 4. ,2 n o 3 n o 3 2b 310 10. .7 275 26 3. ,57 145 3. ,8 19 38 2a 320 11. .0 270 25 3. .63 83 4. ,3 11 19 3 340 11. .7 225 191 4. ,36 9.7 5. ,0 1.5 1.9 6 570 19. .6 270 13.8 3. .63 1.2 2. .4 9 x 10" 2 5 x 10" 1 4b 700 24. .1 305 12.6 3. ,21 0.7 4. ,6 3.7 x 1 0 - 2 1.5 x 10" 1 4a 800 27. .6 265 9.6 3. .70 0.2 4, ,8 1.1 x 1 0 - 2 4.2 x 10" 2 Notes: 1. 2. 3. 4. 5. u , ( C D U2 ) ^ i s ca l cu la ted as U 5 / 29 ( i . e . C D = .0012). Subscript "p" means 'evaluated at the peak of the wave spectrum'. A = 0.4 C J P Z c / u * . L = zc/p?)%. In run 5, the values for c /u* , A . , and L may not be r e a l i s t i c , s ince swel l comprises the major part of the wave energy present . 139 7.3 The Power Spectra The pressure power spectra are best character i sed as r i s i n g mono-t o n i c a l l y with decreasing frequency, with the p r i n c i p a l part of the 0.1 -5 Hz range having a wave-induced "hump". This hump i s superimposed on what w i l l be c a l l e d the "basic" pressure spectrum: that found over sand at Boundary Bay; see Figures 81 and 82. The Boundary Bay spectra were found to sca le as we l l as could be expected (see pp.199ff) to the best a v a i l a b l e f i e l d r e s u l t s (those of P r i e s t l e y 1965) and to the wind tunnel r e s u l t s of Wil lmarth and Wooldridge (1962). The p r i n c i p a l uncerta inty i n the s c a l i n g i s always the thickness of the atmospheric turbulent boundary layer ; while that of the wind tunnel boundary layers i s accurate ly known, there seems to be l i t t l e s a t i s f a c t o r y data besides those obtained from inferences on the thickness of the corresponding atmospheric l a y e r . The monotonic r i s e of the pressure spectra as the frequency decreases i s i n accord with r e s u l t s from many other inves t i ga t ions ; i t shows up very w e l l , for instance , i n the pressure spectra given by Gossard (1960). This f a c t makes i t hard to spec i fy a value for p 2 , the mean square pressure f l u c t u a t i o n s , as has been done i n the past by P h i l l i p s (1957), Longuet-Higgins et a l (1963), and Kolesnikov and Efimov (1962). The power spectra measured using the buoy are very noisy at low frequencies . At these frequencies they are high by factors of 2 - 10 when scaled to the r e s u l t s of P r i e s t l e y or Wil lmarth and Wooldridge. For th i s reason the d i s cuss ion of th i s por t ion of the pressure spectra i s not c a r r i e d beyond the above simple s c a l i n g . I t i s f e l t that the Boundary Bay r e s u l t s (see Appendix 2), although taken i n a d i f f e r e n t boundary l a y e r , are bet ter representat ions of the l i k e l y form of the 140 pressure spectrum at frequencies less than about 0.2 Hz. At higher frequencies , the power spectra are dominated by the e f fects of the waves, i n the sense that i t i s the wave-induced part of the spectrum which determines t h e i r s lope. Whereas the Boundary Bay spectra show a more or less continuous s lope, the present spectra d i s p l a y a "hump"; i t i s very tempting to t ry to integrate under i t , assuming the base spectrum to have the slope of a s t r a i g h t l i n e j o i n i n g the low-frequency p o r t i o n of the spectrum to the asymptote to the high-frequency " t a i l " . The seemingly high s p e c t r a l estimates at low frequencies , and the con-tamination of the pressure s igna l with spikes at high frequencies (des-cr ibed i n Appendix 1) i n d i c a t e that any such attempt would not prove f r u i t f u l . The p r i n c i p a l f i n d i n g pointed out by the p + faq^ spectra i s the large part of th i s pressure which i s coherent with the waves. This i s to be contrasted with the observations discussed by Stewart (1967), that even at r e l a t i v e l y small values of kz (where k i s the wave number of the waves and z the anemometer h e i g h t ) , spectra of turbulent v e l o c i t y f l u c t u -at ions show l i t t l e or no evidence of the e f f ec t of the waves. This might be expected to be the case; the rms v e l o c i t y f luc tuat ions as soc i -ated with the wave motion observed by a probe even at r e l a t i v e l y small values of kz are of order ka-v/Uu, compared w i t h V u 2 for the turbulent f l u c t u a t i o n s , the two being about equal i n t y p i c a l condi t ions . However the pressure f l u c t u a t i o n s due to the waves at the same value of kz should be of order f^^a (Uu) 2. whereas those due to the turbulence are of order 2 eu , which i s at l eas t an order of magnitude smal ler . Thus even i f the turbulent v e l o c i t y f luc tua t ions e f f e c t i v e l y mask those caused by the waves, the wave-induced pressure f luc tua t ions could s t i l l exceed those caused 141 by turbulence. One further f a c t should be mentioned. As f a r as can be determined from the spectra (which with the important exception of run 2a are con-taminated by the e f fects of spikes beyond 2 Hz) , they are modif ied by the e f fec t s of the wave generation process at a l l frequencies above the peak of the l oca l ly -genera ted wave spectrum. In the case of run 2a, for which the inf luence of spikes can be ignored, the inf luence of the wave generation process appears to extend to 2 Hz, which i s the l i m i t beyond which pressure measurements probably are gross ly contaminated with dynamic pressures (see p. 91 f f ) . 7.4 The Phase Spectra There are two phase spectra presented: that of pressure versus wave e l e v a t i o n , and that of pressure plus s t a t i c head pa3')2 versus wave e l e v a t i o n . For comparison with theory the obvious choice i s the former, s ince i t i s much less prone to error (see the Results sec t ion) ; at the same time the r e s u l t s of Shemdin and Hsu are presented i n terms of p + p^g i£ . For this reason both are d iscussed, bearing i n mind the larger errors i m p l i c i t i n the p + fjj'g phase r e s u l t s . Table 7.2 d i sp lays the comparison, for 0.6 Hz waves for a l l runs , between measured phases and those predic ted from M i l e s ' i n v i s c i d laminar model with appropriate assumptions for the wind p r o f i l e . The p,>^ phases are computed from Equation 4 .1 , which i s i d e n t i c a l with that used by Longuet-Higgins e_t _al (1963), and the (p + ^ 3 ^ ) ) ^ phases from Q t = t a n - 1 ft la.; <X and (5 are from Miles (1959a, Figures 4 and 5). I t i s immediately seen that the runs d iv ide themselves into three groups. For the f i r s t group TABLE 7.2 Comparison of Observed Phase Angle of Pressure Re la t ive to Waves at 0.6 Hz, with That Calculated from M i l e s ' I n v i s c i d Laminar Model u 5 Run No. _ I c / u * cm sec e t degrees 0 t 8 , degrees tes: (1) (2) (3) (4) (3) (4) 1 220 34 10-2 -180 mi -179.7 -2b 310 24 5 x 10" 3 -179.8 -164 -165 -152 2a 320 23 5 x 10-3 -179.5 -157 -149 -113 3 340 22 4 x 1 0 - 3 -179.3 -174 -145 -152 6 570 13.3 1 .5 x 10" 3 -174.5 -131 -156 -111 4b 700 10.8 10-3 -173.5 -156 -163 -146 4a 800 9.4 7 .5 x 10" 4 -172.5 -152 -166 -142 Notes: 1. c = 260 cm sec "1, the ph ase v e l o c i t y of a 0.6 Hz wave. 2. n = §z 0 / u i = 0 .565 /U2 , where a logar i thmic p r o f i l e i s assumed, U-^  = 2.5 U^, and Z Q = 3.6 x 10-3 cm (equivalent assuming C^ = .0012). 3. 6 t i s the phase lead of p r e l a t i v e to ->£ computed from e. tan" - P ( U l / c ) 2 1 - A ( U ] / c ) 2 _ see Longuet-Higgins et a l (1963), Equation 53. C*. and |3 are obtained from Miles (1959a), Figures 4, 5. For the phase of (p - ^ ^ ' t ^ r e l a t i v e to , 6 t = t a n - l P/ c 0 i s the corrected measured phase angle, and i s accurate to ~t 2 ° for the c o r r e l a t i o n (see "Data Ana lys i s and Inter -pre ta t ion" , p. 104). 142 c / u ,*> 30 ( i n r u n 1); for the second, c / u * 20 - 30, and for the t h i r d , c / u * 2J 10. Run 1 i s of p a r t i c u l a r i n t e r e s t i n comparing the present r e s u l t s with those of Longuet-Higgins et a l (1963). They show i n the ir Table 4 a comparison of measured versus t h e o r e t i c a l phase d i f ferences for various values of CO . The peak of t h e i r wave spectrum i s at ^ : 0.6 r a d / s e c , g i v i n g a c/u^ for a 960 cm/sec wind of 50.4. At a frequency of about 0.9 rad / sec , c /u > v for the i r data i s about equal to that of run 1 of the present data at CO = 3.8 rad/sec (f a* 0.6 Hz) . For run 1 at this frequency the measured phase i s about - 1 7 2 ° , and l i e s w i t h i n the expected error ( T l 0 ° ) of t h e i r reported value of 1 8 2 ° . The phase s h i f t predic ted by M i l e s ' theory i s less than 1 ° . The highest value of c /u^ at which the present r e s u l t s can be considered r e l i a b l e i s 10; the phase i n run 1 at the frequency (2 Hz) where c /u*2 i l0 i s - 1 2 6 ° , a s h i f t from - 1 8 0 ° of + 5 4 ° . At the corresponding frequency at which c/uV c.fiil0 i n t h e i r data, which i s 3 r a d / s e c , t h e i r phase s h i f t s are s t i l l e s s e n t i a l l y zero. Thus the two r e s u l t s are not incons i s t ent at low frequencies ( c / u ^ 20, say) but d i f f e r markedly for the higher frequencies . The d i f ferences must be put down to d i f ferences i n measurement technique or unaccounted-for e r r o r s . The presence of dynamic pressures i n antiphase with the waves i n the pressure s igna l from the ir buoy may account for a major part of the d i f f e r e n c e . For 20 < c/u^< 30 (runs 2 and 3 i n Table 7.2) i t appears that the i n v i s c i d laminar model pred ic t s phase s h i f t s i n p which are too small by angles of 5 to 2 0 ° . P h i l l i p s (1966) proposes that for c/uVi>20 the genera-t i o n process must be dominated by the i n t e r a c t i o n of the wavy a i r flow with the turbulent Reynolds s tresses , s ince at these values of c/ui( the 143 c r i t i c a l height:'is h igh , and so i s i n a region where the curvature of the mean wind p r o f i l e i s smal l . The a d d i t i o n a l phase s h i f t s may be caused e i t h e r by the turbulent Reynolds stresses or by some other mechanism such as that proposed by Stewart (1967), whereby the momentum transfer to the waves i s caused by the balance over a wavelength between h o r i z o n t a l turbulent shear stresses and gradients of normal pressure. The runs for which 10 <. c /u^ < 20 (4a, 4b, 6) are those i n which M i l e s ' i n v i s c i d mechanism should dominate. In these runs , the observed phase s h i f t s are about 20 to 40 degrees larger than p r e d i c t e d . Taking in to account the phase accuracy of the measurements i t i s c l ear that the i n v i s c i d laminar model i s not the only mechanism generating waves for th i s range of c /u^ e i t h e r . I t w i l l be seen l a t e r (p. 159) that observed rates of wave growth exceed those predic ted from the theory by fac tors of f i v e to e ight . Included i n Table 7.2 i s a comparison between the observed phases ° f Ps + 6a"J?2 with respect to ">£ and the corresponding phases predic ted from M i l e s ' i n v i s c i d laminar model. This comparison i s inser ted so that the r e s u l t s from the present experiment may be compared with those of Shemdin and Hsu (1966). (The e a r l i e r manuscript report of Shemdin and Hsu i s used for th is comparison s ince some r e s u l t s presented i n i t but not r e a d i l y a v a i l a b l e elsewhere have s ince been re f erred to i n the l i t e r a t u r e . ) Shemdin and Hsu make a comparison of c/u-^ = 0.4 c /u^ versus measured (p + PJ^^ ),•*£ phase and that computed from M i l e s ' (1957) theory. The values of c /u^ for t h e i r study range from 3.0 to 13.5. For c /u Ca 10 t h e i r measured phase s h i f t s exceed the t h e o r e t i c a l l y predic ted ones by 10 - 2 0 ° i n t h e i r "best" run (the r e s u l t s from which 144 are shown i n Shemdin and Hsu (1967) i n the ir Table 2) and by 3 0 ° i n the only other run presented. The phase discrepancies from runs 4a, 4b, and 6 range from 1 7 ° to 4 5 ° , and are thus not incons i s tent with those observed by Shemdin and Hsu i n t h e i r wind-wave tunnel . The wind-wave tunnel r e s u l t s for lower values of c /u^ can be com-pared with the present data at higher wave frequencies . At 1.5 Hz c /u^ ranges from 3.8 to 5.3 i n runs 4a, 4b, and 6. The discrepancies between observed and t h e o r e t i c a l phase s h i f t s are 8 ° , 2 3 ° , and 1 7 ° for the three runs (see Figures 36, 37, and 39); Shemdin and Hsu f i n d discrepancies ranging from 7 ° i n t h e i r "best" run to 4 7 ° i n the other run at c /u^ = 4 .5 . Because of the large scat ter i n the present phases and the apparently even larger scat ter i n those of Shemdin and Hsu, i t i s not worthwhile pursuing the comparison f u r t h e r . I t i s evident that i n sp i t e of the great d i f ferences i n the condit ions of measurement of the wind-wave tunnel work and the present f i e l d work, the measured phases are not i n -cons i s tent , both e x h i b i t i n g s i g n i f i c a n t l y larger s h i f t s from - 1 8 0 ° (they average 2 0 ° i n Table 7.2) than those pred ic ted by M i l e s ' (1957, 1959a) theory. 7.5 The Energy and Momentum Flux Spectra These spec tra , shown i n Figures 40 to 47, represent one of the major contr ibut ions of th i s thes i s . In the fo l lowing sec t ion they w i l l be discussed and compared where poss ib le with other measurements. F i r s t to be discussed w i l l be the i n t e g r a l s under the spectra; this w i l l be followed by a cons iderat ion of the spectra themselves. 145 7.5.1 Mean Energy Flux The values of E and r e s p e c t i v e l y the integrated energy f lux to the waves and i t s standard error of the mean over the run , are shown as funct ions of mean wind speed i n Table 7.3. Also inserted are values obtained by Kolesnikov and Efimov (1962) from t h e i r f r e e - f l o a t i n g buoy i n condit ions of ac t ive wave generat ion. These are marked with a KE i n the "Run Number" column. I t has already been pointed out (p. 22) that these r e s u l t s are suspect; however they are to the w r i t e r ' s knowledge the only energy f l u x measurements comparable with those presented here, and so they are included because of t h e i r uniqueness. From the table i t can be seen that both sets of r e s u l t s show consider-able s ca t ter when compared with wind speed alone. Nonetheless those of Kolesnikov and Efimov are w i th in an order of magnitude of those found i n th i s experiment. Lack of reported wave data i n the Russian p u b l i c a t i o n precludes any more objec t ive comparison. The standard errors of the mean ( (J*]| ) for each run are a lso given i n Table 7.3; they represent the expected v a r i a b i l i t y over a given run of the values of E obtained for each data block i n the run , and are com-puted from Equation 5.5. I t w i l l be immediately seen that the values of <T~^ , are large; i n run 4b <T"|. i s more than double the value of E , while i n general i t ranges from one-half to about one times E . I t should be noted that two runs (4 and 5, for which the r a t i o s (J-g/E are larges t ) showed p o s i t i v e evidence of a lack of s t a t i o n a r i t y ( i n the sense that a l l runs are f e t c h - l i m i t e d they a l l can be considered as s p a t i a l l y inhomogeneous, with the poss ib le exception of run 1). Run 4 146 was taken i n two sect ions and a l l spectra for the two runs are s i g n i f i -cant ly d i f f e r e n t , i n d i c a t i n g a development of the wave and wind f i e l d s with time. The large scat ter from block to block i n the energy and momentum f l u x spectra appears to hide any trends w i th in runs 4a and 4b. Run 5 d e f i n i t e l y shows such a trend; the wind speed was observed to increase from s t a r t to f i n i s h , and to change d i r e c t i o n by 20 to 3 0 ° i n the course of the run. The d i s t r i b u t i o n s of the E values about t h e i r means have been p lo t ted for each run, and compared using the Chi-squared test against normal d i s -t r i b u t i o n s with the same means and standard deviat ions; no s t a t i s t i c a l l y s i g n i f i c a n t deviat ions from the normal d i s t r i b u t i o n s were found. No attempt was made to compute h igher-order moments, although a v i s u a l in spec t ion of the E d i s t r i b u t i o n s showed no obvious skewness of kur tos i s i n any of them. The above f indings are taken to mean that the wave generation pro-cess, although i t causes on the average a p o s i t i v e energy input to the waves, i s b a s i c a l l y a random process with large s c a t t e r . Fur ther , there appears from this data to be no evidence of strong "intermittency" i n the sense that a l l of the energy input occurs at r e l a t i v e l y infrequent i n t e r v a l s during which large transfers take place; the generation pro-cess should rather be considered as one which i s ac t ive a l l the time but which at a given time i s e i ther enhanced or diminished by random v a r i a t i o n s i n the r e l a t i v e phase between the pressure and the waves. 7.5.2 The Wave-Supported Wind Stress One of the most important questions which must be answered about TABLE 7.3 Comparison of Observed Values of E with Those Obtained by Kolesnikov and Efimov (1962) i No. U 5 cm sec -"-E erg cm-^sec - 1 1 _ o dyne cm (1) (2) (2) (2) (2) 5 150 -5 7 -.002 .02 1 220 13 17 . 11 .11 2b 310 29 18 .16 .11 2a 320 43 18 .24 .10 3 340 30 32 .19 .26 KE 500 10 KE 510 50 KE 540 60 6 570 60 32 .38 .22 4b 700 90 233 .45 .30 KE 710 90 KE 790 130 4a 800 154 184 .66 .36 KE 1050 350 KE 1060 310 Values of U5 for Kolesnikov and Efimov (KE i n "Run No.") are extrapolated assuming a logar i thmic p r o f i l e with Cp = .0012 from given speeds at 2.5 m. E and Tvw are the mean in tegra l s under the E(f ) and TTv/f) spectra for each run; ^"~"E and 6% are standard errors of the mean over the number of data blocks done (always ^ 20). Note: 1. 2. 147 the wave generation process i s , "What f r a c t i o n of the Reynolds s tress - £> f t Uur exerted by the wind on the water surface goes d i r e c t l y into waves?". Stewart (1961) considered th is quest ion, and by examining wave data measured at known fetches and wind speeds came to the conclus ion that a lower l i m i t to the f r a c t i o n '^-nj/t, where 7TW i s the mean wave-supported s tress and = - p^U-xw , i s about 0.2. He considered th i s to be a lower l i m i t because the parameter he estimated i n determining T w was wave momentum present at a given f e tch , which because of wave d i s s i p a t i o n mechanisms i s i n f a c t only a part of the t o t a l momentum f l u x to the waves. His conclus ion was that i t would be poss ib le for T T w / t r to be near ly one. P h i l l i p s (1966) estimated T w / ' r f or the waves for which c > 5u,. by computing "cX* from h i s ver s ion of the M i l e s - P h i l l i p s theory (which includes the e f fec t s of the i n t e r a c t i o n of the turbulent Reynolds stresses i n the a i r with the flow over the waves) and by comparing th i s computed value with ^ u ^ . He found that for these longer waves tw/'t* 4z 0 .1. Since the waves considered by Stewart i n h i s analys i s were a l l such that c > 5u*, P h i l l i p s ' t h e o r e t i c a l f i n d i n g contradicts Stewarts ' , which was based on observat ions . In view of the observations by Snyder and Cox (1966) and Barnett and Wilkerson (1967) which show that observed wave growth rates exceed those predic ted by the M i l e s - P h i l l i p s theory by about one order of magnitude, P h i l l i p s ' r e s u l t , which was ca lcu la ted from the pred ic t ions of th is theory, must be regarded as doubt fu l . The average momentum input to the waves has been computed for a l l the runs. These values for "cXw are shown i n Table 6.5; they are r e d i s -2 played i n Table 7.4 as f rac t ions of P ^ Q U ^ , where C^ = .0012. Also 148 shown are values of cp/u y . , and a "wave-drag coe f f i c i en t" = f-^/ ^ a u 5 -The mean C D y for the runs i s .0014, and a l l values shown f a l l w i th in the combined standard deviat ions of the mean values of quoted by Smith (1967): .0010 + .0003, and Weiler and B u r l i n g (1967): .0015 ± .0004, which were obtained from d i r e c t measurements of - ^ U l t / 1 at the same s i t e . Thus, although the observed scat ter i n appears large at f i r s t s i gh t , i t i s not large when compared with other observat ions . In runs 5 and 6 - ^ U.ur was-, measured with a sonic anemometer, and the observed agreement (see Table 6.5) of the stress measured by the buoy with the sonic anemometer measurements i s p a r t i c u l a r l y heartening . I t ind icates that the values for the s tress given by the buoy measurements give as accurate a measure of 1£* as does the formula = .0012 ^ U 2 ' . This means that the most d i r e c t evidence from which the value of T w / f can be i n f e r r e d i s that of run 6, which ind icates that the r a t i o i s about 0.8. The large experimental uncer ta in t i e s i n the buoy measurements, p a r t i c u l a r l y i n the pressure sensor c a l i b r a t i o n , i n d i c a t e an expected error of ~J"207o i n th i s f i g u r e . I t thus becomes f a i r l y c e r t a i n that Stewart's estimate that a large f r a c t i o n of the wind s tress over water i s supported d i r e c t l y by the waves i s s u b s t a n t i a l l y correc t ; the value ind icated for the f r a c t i o n i s 0.8. Since th i s f i n d i n g disagrees with that of P h i l l i p s , i t s trongly suggests that h i s t h e o r e t i c a l estimate for i s too low by about one order of magnitude, i n agreement with the f indings of the wave growth measurements of Snyder and Cox, and Barnett and Wilkerson. The r e s u l t obtained here f o r * ? ^ , / f , 0.8, i s larger than that (0.1 - 0.4) observed i n a recent study by Wu (1968). His measurements TABLE 7.4 F r a c t i o n of the Wind Stress Supported by the Waves Run No. tes: y u 5 ( i ) c „ / u p * (2) (3) UDW (4) 1 1.7 2b 0.9 2a 0.9 3 0.7 6 0.5 4b 0.4 4a 0.3 Mean Values Standard E r r o r of Means 49 26 25 19 13.8 12.6 9.6 1.6 1.1 1.6 1.1 0.8 0.6 0.7 1.1 0.4 .0018 .0013 .0019 .0013 .00095 .00075 .00083 .0014 .0005 Notes: 1. Cp i s the wave phase v e l o c i t y at the l oca l ly -genera ted peak of the wave spectrum. 2. — u 5/29 ( C D = .0012); U 5 i s mean a i r speed at a height of 5 meters. 3. , s r ; w i s the mean s tress measured by the buoy system; " f c i s ca l cu la ted from *7J"C = P^CnU^* except i n run 6, where T*C i s replaced with the s tress "~£ s measured by the sonic anemometer. % = ' V e a u 5 2 . 149 were made i n a wind-water tunnel over natura l ly -generated waves, and Cp/u,v i n his tunnel never exceeded 7.5, a value somewhat less than the lowest Cp/u,v observed i n the present study. Thus i t can be assumed that h i s r e s u l t s apply to a d i f f e r e n t generation regime. His measured values of "Cw were obtained i n two ways, ne i ther of which can give r e s u l t s s t r i c t l y comparable with those presented here. He f i r s t took the d i f f e r -ence between drag c o e f f i c i e n t s measured over waves and over s o l i d surfaces with the same measured roughness length; he also observed the increase i n wave momentum with f e t c h . The l a t t e r observat ion was then used to confirm the v a l i d i t y of the former. Since the present r e s u l t s are i n terms of momentum f lux to the waves and not observed momentum increase , and s ince the c /u^ values are so d i f f e r e n t , the discrepancy between the two experimental values of T*W/ ""/*• i s not considered to be an important one. I t i s i n t e r e s t i n g to note the apparent decrease of Z^/^c with C p / U ^ i n Table 7.4. :<<>If th i s were r e a l i t would i n d i c a t e that the wave genera-t i o n was becoming more e f f i c i e n t at absorbing the wind s tress with increas ing Cp/U^; Tj"^  , however, i s computed from C^, which i s not known with s u f f i c i e n t accuracy; i t may indeed be that v a r i a t i o n s i n cause a l l of the observed v a r i a t i o n i n XVI/'XC '. 7.5.3 The Energy and Momentum F l u x Spectra Three frequencies associated with the energy f lux spectra w i l l be s ing led out for comment. These are the frequencies f p , of the l o c a l l y -generated maxima i n wave spectra; f g , at the maxima of energy f lux spectra; and f^, the frequency at which the wind speed at a height 1/k 150 = A/2TT equals the wave phase speed. The choice of *X/2Tf for the height used to determine f^ i s pure ly a r b i t r a r y ; i t i s genera l ly small enough i n these measurements so that i t i s w i th in the height range over which anemometer wind speed measurements are a v a i l a b l e but not so small that the curvature of the p r o f i l e i s l arge . In th i s way inaccurac ies i n the ex trapo la t ion of the wind p r o f i l e by assuming i t to be logar i thmi are minimized. The value of f^ =^/2tT i s r e l a t e d to u* i f a logar i thmic wind pro-f i l e i s assumed ( f k i s defined as the frequency at which the phase speed of the wave equals the mean wind speed at a height 1/k =X/21T) U a " U k = U a " S / « k  = 2 * l n ( ^ ) = l n ( E £ k ) 2 > K U k / K V § / where K i s Von Karman's constant and the subscr ip t a denotes "at anemometer height", i n th i s case 5 meters. This can be put i n the trans cendental form f k 2 e x p (gK/2lT f k u*) = (g/4f l 2 z a ) exp (K/ V c p ) = constant where = u^/U^2 i s the dimensionless drag c o e f f i c i e n t , assumed to be 0.0012. I f u can as a f i r s t approximation be taken as constant for the f u l l f e tch over which wave generation i s o c c u r r i n g , then f^ w i l l be con-s tant . Since f p can be expected to decrease with increas ing f e t ch , the r a t i o f^ / fp can be expected to increase with f e t c h . I t should also be noted that f k / f p i s r e l a t e d d i r e c t l y to c^/u^ through the d i s p e r s i o n r e l a t i o n for the waves and Equation 7.3 above. 7 .5 .3a Observed and Pred ic ted T r a n s i t i o n Fetches The question of the value of the f e tch at which the growth with 151 time of a given wave component s h i f t s from a l i n e a r to an exponential rate of increase i s s t i l l not s a t i s f a c t o r i l y answered. There are now many measurements of the frequency of the steep f ront face of the wave spectrum for given fetches and wind condit ions (see, for instance , Barnett and Wilkerson,1967); these show a great deal of s c a t t e r . Almost W II i n v a r i a b l y , however, the observed t r a n s i t i o n fetches are less than those predic ted by M i l e s ' i n v i s c i d laminar model. P h i l l i p s (1966, F igure 4.6) gives a p l o t of N^, the t h e o r e t i c a l l y predic ted number of wave periods to t r a n s i t i o n , versus c / u , ; i f i s r e l a t e d to the t r a n s i t i o n fe tch through Frj , = c^N^/f^, where c^ i s the group v e l o c i t y of the waves and f f the frequency of the f ront face of the wave spectrum, then the fetches at which the observed spectra have reached t r a n s i t i o n can be compared with t h e o r e t i c a l l y predic ted fetches; th i s comparison i s shown i n Table 7.5. The observed t r a n s i t i o n fetches are less than predic ted except i n two cases: run 1 and run 3. Run 1 has a long fe tch (40 km), and was taken i n l i g h t winds. Since no information on condit ions upwind of the sensors i s a v a i l a b l e i t i s not known how homogeneous the wind f i e l d r e a l l y was; for th i s reason the r e s u l t i s not in terpre ted as showing s i g n i f i c a n t disagreement with theory. Run 3, i n which the wind d i r e c t i o n was 8 0 ° (True) , has a much larger f e tch than runs 2 or 4 i n which the wind was from 1 1 5 ° . A large change i n f e tch with azimuth occurs at about 1 0 5 ° ; see F igure 2. This f a c t alone makes the comparison with theory d i f f i c u l t ; unaccounted-for errors of ^10° i n wind d i r e c t i o n or unnoticed s h i f t s just before the run began could b r i n g the observed and pred ic ted t r a n s i t i o n fetches into good TABLE 7.5 Observed and Predic ted T r a n s i t i o n Fetches f f C f F T F o Run No. Hz cm/sec cm/sec C$/u, v N T (km) (km) Notes: (1) (2) (3) (4) 1 .3 520 7.6 68.5 6 x 10 2 5.2 40. 2b .49 329 10.7 30.8 1 x 10 3 3.2 2.4 2a .48 325 11. 29.6 1 x 10 3 3.4 2.4 3 .6 260 11.7 22.2 2 x 10 3 4.6 6.7 6 .50 312 19.6 15.9 3 x 10 3 9.4 1.6 4b .44 354 24.1 14.7 2 x 10 3 8.9 2.4 4a .48 325 27.6 11.8 1 x 10 3 3.4 2.4 Notes: 1. The subscr ip t f means "evaluated at the low-frequency face of the l oca l ly -genera ted wave spectrum". 2. Nrp i s the number of wave periods to t r a n s i t i o n . I t i s obtained from P h i l l i p s (1966, F igure 4 .6 ) . 3. F^ i s the t h e o r e t i c a l l y predic ted t r a n s i t i o n f e t c h , and i s given by F^ = (N^Cg), where Cg i s the group v e l o c i t y of the f f waves at the low-frequency face of the wave spectrum. 4. F 0 i s the observed fe tch for the given wind d i r e c t i o n ; see Table 6 .1 . 152 agreement or throw them into greater disagreement. Furthermore a l l wind-drivenwave f i e l d s have a f a i r l y broad d i r e c t i o n a l d i s t r i b u t i o n , so that the wave spectra from runs 2 and 4 may be p a r t l y made up of energy from waves t r a v e l l i n g i n d i r e c t i o n s as much as 3 0 ° o f f that of the wind, and hence over fetches as large as 7 km. These waves may produce a large f r a c t i o n of the t o t a l energy at low frequencies , near the l o c a l l y -generated s p e c t r a l peak. This means that the downwind fe tch may not be the e f f e c t i v e one, and i n the present circumstances may be too small i n runs 2 and 4. This error would increase the observed t r a n s i t i o n fetches i n runs 2 and 4, b r i n g i n g them into c loser agreement with theory; i t s e f f e c t i s , however, at l ea s t p a r t i a l l y o f f s e t by larger t h e o r e t i c a l N^'s i n th i s case. By the same token the predic ted t r a n s i t i o n fe tch for run 3 may be too small i f the waves at frequencies near the s p e c t r a l peak were t r a v e l l i n g at an angle to the wind. I t i s considered that the r e s u l t s from run 3 i n d i c a t e approximate agreement between pred ic ted and observed fetches , while r e s u l t s from run 2 are equ ivoca l , and those from run 4 show s i g n i f i c a n t disagreement. 7.5.3b Energy Transfer i n the Wave Spectrum A glance at the E(f) spectra (Figures 40 to 47) shows that i n a l l cases the frequency fp of the peak of the wind-generated wave spectrum occurs at or s u b s t a n t i a l l y below fg and fx, the frequency of the peaks of the energy and momentum f l u x spectra r e s p e c t i v e l y . Two fac tors besides that of growth v i a normal pressures can contr ibute to t h i s . F i r s t , i f wind speeds had r e c e n t l y dropped or i f swel l from another source had entered the reg ion , damping of swel l at frequencies jus t below the wave 153 peak could cause a s h i f t of f £ and f T to higher frequencies; th is e f f ec t i s thought to be smal l , s ince the swel l peaks i n the wave spectra are genera l ly at frequencies less than f p / 2 . The second f a c t o r i s the presence of some mechanism whereby energy can be t rans ferred to the waves at the peak of the spectrum other than by normal pressures . A number of such mechanisms are a v a i l a b l e . The f i r s t and most obvious i s energy transfer from higher frequencies v i a nonl inear interact ions among wave components. One such transfer process , which i s probably the most important wave-wave i n t e r a c t i o n near the peak of the spectrum of a wind-generated wave f i e l d , i s that of sideband feeding (Benjamin and F e i r , 1967), whereby a " c l a s s i c a l Stokes wave", being unstable , generates sidebands at lower and higher frequencies . The low-frequency sidebands generate sidebands of t h e i r own at s t i l l lower frequencies , and thus move the f ront face of the spectrum to lower frequencies; th i s process i s completely independent of the wind. As has been pointed out by Stewart (1967), i t i s a lso poss ib le for the wind to generate waves without the a id of normal pressures . He suggests two such mechanisms (see pp. 136, 7); both of these have been fur ther considered and enlarged upon r e c e n t l y by Longuet-Higgins (1969a, 1969b). The f i r s t mechanism, which i s caused by a v a r i a t i o n along the wave propagation d i r e c t i o n of the Reynolds s tress on the water surface , has a lready been discussed on p. 136. The second i s o u t l i n e d i n Longuet-Higgins (1969b). In essence i t involves the "sweeping up" by long waves of momentum from shorter waves; because the ac t ion of the mechanism i s s i m i l a r to that of a maser, Longuet-Higgins re fers to i t as a "maser mechanism". The short waves are presumed to grow v i a normal wind pressures; 154 these become steep and may break at the convergences on the f ront faces of the longer waves. Whether they break or not momentum i s t rans ferred to the longer waves, but the mechanism i s most e f f e c t i v e i f the short waves a c t u a l l y do break; presumably most energy i s t ransferred to f r e -quencies near the peak of the wave spectrum. An energy f l u x spectrum measured by c o r r e l a t i n g normal pressure with wave v e r t i c a l v e l o c i t y would not show this energy transfer at the peak of the wave spectrum, but would have a maximum at a higher frequency. Therefore the integrated energy f l u x measured i n this way would inc lude the energy transfer associated with the "maser" mechanism, provided that the maximum i n the energy f l u x spectrum was not at such a high frequency that i t was outs ide the frequency range of the sensors. In order for the maser mechanism to be e f f e c t i v e , the energy loss of the short waves to the long ones v i a breaking must exceed t h e i r energy gain through the working of the Reynolds s tresses of the long waves. For th i s to occur, Longuet-Higgins shows that the v e l o c i t y r a t i o c s / c £ must be much less than 1, where c g and c e are the phase v e l o c i t i e s of the short and long waves. Since c = g/co , t h e i r frequency r a t i o equals c g / c g and so must be l arge . The observed r a t i o f g / f p i s always one or greater , but never exceeds two. In fac t the present measurements cease to be accurate above about 4-fp, and so i t might be expected that the energy input i n the maser process i s from waves with frequencies higher than any measured by th i s experiment. Two p o s s i b i l i t i e s are open; e i ther the present measurements are a gross underestimate of the energy f l u x becuase they cannot measure f l u x e s ' t o high-frequency waves, or the maser mechanism 155 i s r e l a t i v e l y i n e f f i c i e n t , and accounts for only a small f r a c t i o n of the energy f l u x to the waves.. The l a t t e r p o s s i b i l i t y seems more l i k e l y , i n view of the e f f i c i e n c y of the normal pressures exh ib i ted by the observed large phase s h i f t s of the pressure from - 1 8 0 ° towards - 9 0 ° near fp , and the f a c t that the energy and momentum f l u x spectra do f a l l to low values at 3-4 times f £ . 7.6 The i* Spectra A d i scuss ion of the spectra of £ , the f r a c t i o n a l increase i n wave energy per r a d i a n , w i l l be put of f i n order to begin the sec t ion with a comparison of the mean values of £ with those predic ted by theory. This comparison i s given i n Table 7 .6 . 7.6.1 Mean Values of t £ , the measured mean values of C, are given as E / o E ; i n the present case s ince the measured E and E = f^g ig5" are integrated values for the energy f l u x and wave spectra for a given run, the rad ian f r e -quency cjp of the peak of the l oca l ly -genera ted wave spectrum i s chosen for cj . The t h e o r e t i c a l pred ic t ions £ t corresponding to ^ m are com-puted from Miles (1957, Equation 2.10): the values of |3 t are obtained from Mi les (1959a, h i s F igure 4) . Thus whereas E and 1£ 1 are in tegrat ions under the corresponding spectra , the t h e o r e t i c a l pred ic t ions are for a s ing l e s inuso id at the observed f r e -quency of the peak of the wave spectrum. The comparison may therefore TABLE 7.6 Comparison of Observed and Predic ted Values of { = E/wE ; E <^p y^2 Erg Run rad c m - 2 CI P t cm 2 sec" 1 J m J t No. sec-1 C p / U ! Notes: (1),(2) (3) (3) 1 2. 6 19 .6 1 X 10-2 <10" 2 18 13 2.8 X 10" 4 2. 5 X 1 0 - 8 2b 3. 6 10 .4 5 X 10" 3 0. 25 14 ::29 5.9 x: i l O " 4 2. 8 X 10" 6 2a 3. 6 10 .0 5 X 1 0 - 3 0. 3 18 43 6.8 X 10" 4 3. 0 X 10-6 3 4. 4 7 .6 4 X 10" 3 1. 9 25 30 2.8 X 10" 4 4. 1 X 10-5 6 3. 6 5 .5 1. .5 X 10" 3 3. 2 6 60 2.8 X l O " 3 1. 3 X 10" 4 4b 3. 2 5 .0 1 X 10-3 3. 5 21 90 1.4 X 10-3 1. 7 X 10" 4 4a 3. 7 3 .8 7. .5 X 10" 4 3. 6 23 154 1.8 X 1 0 - 3 3. 1 X 10" 4 Notes: 1. The subscr ip t "t" re fers to values ca l cu la ted from M i l e s ' i n v i s c i d laminar theory; the subscr ip t "m" re fers to measured values . 2. ^ t i s obtained from Miles (1959a, F igure 4) . 3 - Cm = E / ^ g c o p T 2 ; r t ~ejtvfnypt. 156 be u n f a i r ; i t i s made p r i n c i p a l l y because the E spectra are sharply peaked quite near <jp (the choice of <Jp instead of = 2 It fg i s a r b i t r a r y ; the value of i s not changed s i g n i f i c a n t l y by using one instead of the other) ; they can i n some sense be thought of as represent-ing the spectra from a s ing l e s i n u s o i d . The values of J * m and j'j. and of Cp/U^, given for a l l the runs i n Table 7.6, show that the measured integrated values ( exceed the ° •> m t h e o r e t i c a l ones for s ing le s inusoids by fac tors of 8 - 200 (the average r a t i o i s about 100) except for run 1, for which the fac tor i s 10^ and which must be considered an exception. The large value of Cp/U^ for run 1 and the f a c t that wind condit ions were unknown over the long (40 km) fe tch i n d i c a t e f i r s t , that any l o c a l generation was d e f i n i t e l y not caused by M i l e s ' i n v i s c i d mechanism and second, that i t i s probable that the waves present were generated by stronger winds upstream from the measurement s i t e . This second i n d i c a t i o n i s considered h i g h l y probable . Two fac ts shown by Table 7.6 i n d i c a t e the presence of a mechanism d i f f e r e n t from the i n v i s c i d laminar model. The f i r s t i s the large d i s -crepancies between measured and predicted values of f , and the second i s the much smaller range-.and percentage v a r i a t i o n between runs i n the measured values as: compared with the t h e o r e t i c a l ones. The l a t t e r phenomenon i s as s t r i k i n g as the former; whereas f var i e s over the runs by a fac tor of 100, C var ie s by only a fac tor of 5. There are m not enough values to judge the form of the v a r i a t i o n of £ with c/U-^; i t does not appear to be s i m i l a r to that of £ with c/U-^. 7.6.2 The T Spectra The !i spectra themselves, shown i n Figures 48-55, d i s p l a y a remark-157 able resemblance to curves drawn according to Equation 6.9 (p. 130). This i s an empir i ca l r e l a t i o n suggested by Snyder and Cox (SC), which f i t s t h e i r data and those of Barnett and Wilkerson (BW) quite w e l l . The frequency where the SC curve crosses the ^ = 0 axis i s that at which the wave phase v e l o c i t y equals the wind v e l o c i t y at a height of one wavelength; i t appears not to be as good an approximation to the "cross-over frequency" of the observed ^ curve as i s UJy., although ne i ther i s p a r t i c u l a r l y good. The only part of the observed ^ curves which d i f f e r s from the SC r e l a t i o n i s the high-frequency reg ion , where the observed values f a l l below the SC p r e d i c t i o n s . As was mentioned i n "Results", there i s some evidence of s i m i l a r high frequency behaviour i n the SC and BW data . The f a l l o f f i n the present data , however, occurs at values of c/u& always less than 10, while the lowest c /u* i n the SC and BW data i s greater than 10. Since t h e i r measurements are of wave growth, i t could be expected that the high-frequency wave components i n t h e i r spectra were approaching e q u i l i b r i u m and had therefore ceased to grow; th i s would exp la in the f a l l o f f of t h e i r r e s u l t s at high frequencies . The present measurements, on the other hand, are of energy . f lux . This means that what should be observed (by an instrument with s u f f i c i e n t r e s o l u t i o n at short wavelengths) i s the t o t a l energy input to the system due to normal pressures; th is input w i l l presumably come from d i f f e r e n t sources i n d i f f e r e n t frequency bands ( d i f f e r e n t ranges of c/uV (.); i f i t i s assumed to vary i n v e r s e l y as c / u * i t should increase with frequency up to 13.4 Hz, the frequency at which the waves reach t h e i r minimum phase v e l o c i t y . I f the high-frequency f a l l o f f i n ^ i s to be be l ieved i t must be in terpre ted i n some other way. Stewart (personal communication) suggests 158 that the behaviour of £ at h igh frequencies can be explained i n terms of a "she l ter ing" of short-wavelength waves by the longer ones near the peak of the wave spectrum, so that over most of a long wave cycle l i t t l e energy can be added to the short waves by the wind. The term "she l ter -ing" immediately brings J e f f r e y s ' theory (1925) to mind; the "she l ter ing" being considered here i s not, however, of the type considered by J e f f r e y s . I t i s concerned instead with flow r e v e r s a l i n the coordinate system mov-ing with the wave phase v e l o c i t y c at the c r i t i c a l height where U = c. This r e v e r s a l can be considered as a type of separat ion , and i f the "separat ion bubble" formed by the flow i n th i s coordinate system l i e s i n the trough of the waves as suggested by Stewart (1967) rather than near the cres t as suggested by P h i l l i p s (1966; h i s F igure 4 .3 ) , then the short-wavelength waves i n the troughs would be i n an "eddy" which keeps pace with the long waves for most of the time, and thus could indeed spend a large proport ion of t h e i r time i n regions of the a i r flow where l i t t l e energy could be added to them. Another piece of i n f e r e n t i a l evidence ex i s t s which suggests that the f a l l o f f i n the observed f spectra at high frequencies i s r e a l and not an a r t i f a c t of the measurement system. From the d i scuss ion on p. 148, i t i s c l ear that the momentum transfer to the waves measured by the buoy i s a large f r a c t i o n of the t o t a l wind s tress on the water surface . Since the momentum transfer to the waves i s ca l cu la ted from the same information as i s f , then i f the observed f a l l o f f i n £ were not r e a l the computed momentum transfers to the waves would be much larger than those given i n Table 7.4, and the values of computed from these large momentum transfers would become much greater than those observed by 159 Smith (1967) and Weiler and B u r l i n g (1967) . Since these workers measured the t o t a l transfer from the a i r to the sea, i t would be d i f f i c u l t to see how the measured momentum input to the waves v i a normal pressure could be l a r g e r . Since the observed t spectra f a l l below the empir i ca l curve suggested by Snyder and Cox at high frequencies , i t may be that t h e i r empir i ca l curve overestimates the energy (and momentum) transfer at h igh f r e -quencies . Snyder and Cox integrated under t h e i r curve to compute the momentum transfer to the waves i n t h e i r experiment; i t may therefore be that t h e i r anomalously high wave drag c o e f f i c i e n t (C T T — .007) i s DW accounted for by th is overest imation of the energy f l u x . The i * spectra predic ted by M i l e s ' (1957) i n v i s c i d laminar model are a lso shown i n Figures 48-55. They c l e a r l y l i e f ar below the observed /* spectra and those ca lcu la ted from Snyder and Cox's empir i ca l formula, except at the higher frequencies ( — 10 rad/sec or f 1.6 Hz); at the lower frequencies the observed values of exceed those pre-: d ie ted from M i l e s ' i n v i s c i d laminar mechanism by factors of 5 to 8. The conclus ion to be reached i s that at a l l values of c /u^ (at the peak of the re levant wave spectra the c^/ui( range covered by the data i s 10 < Cp/u^<50) the i n v i s c i d laminar model of Mi les (1957) i s i n s u f f i c i e n t to account for the observed f luxes of energy and momentum to the waves. 7.6.3 A Dimensionless R e l a t i o n between (f) and Wind Speed At the suggestion of Dr. A. E . G i l l a dimensionless p l o t of ^(f) versus U^/c has been prepared, and i s shown i n F igure 55. I t can immediately be seen that the spectra are f a i r l y well-grouped i n the 160 reg ion 0.5 ^ U^/c ^ 5; the l i n e drawn through the points i s a v i s u a l estimate of the "best f i t " which has been forced to pass through the value of fa/P w , the densi ty r a t i o of a i r to water, at U^/c = 0 . I t passes through zero at U^/c at 1.1, and i f a logar i thmic p r o f i l e with a drag c o e f f i c i e n t of .0012 i s assumed, i t i s found that U^/Ug ~ 1.0 at a height z a 2 meters. This suggests that a reasonable formula for computing T(f) might be f ( f ) - ? a / ew ( U 2 / c - 1) 0 . 5 < U 5 / c < 6 * 7.5, where i s the mean wind speed at a height of 2 meters. The equation i s not a p p l i c a b l e above U^/c= 6; £ (f) should probably be set to zero at higher values of U^/c to avoid the p o s s i b i l i t y of computing excessive energy and momentum transfers at high frequencies . I t w i l l be not iced that th i s empir i ca l r e l a t i o n i s s i m i l a r to that of Snyder and Cox (1966). In order to compare the two r e l a t i o n s , the ca l cu la ted Snyder and Cox curves for the runs were put on the dimension-less p l o t with the measured values of f ( f ) ; i n general i t was found that the Snyder and Cox curves f i t t e d the data as we l l as r e l a t i o n 7.5 d i d ; d i f ferences were smal l . In fac t the only advantage claimed for Equation 7.5 over the Snyder and Cox formula i s that i t i s s l i g h t l y s impler , s ince w i t h i n i t s range of v a l i d i t y ^*(f) can be ca l cu la ted from i t without the need of matching a logar i thmic p r o f i l e to a computed wavelength. *The range 0.5 < U^/c < 6 trans la tes to 50 < c /u , v <. 6 i f a drag co-e f f i c i e n t of .0012 i s assumed. SECTION 8: CONCLUSIONS 8 .1 In troduct ion 161 8 . 2 The Power Spectra 162 8 . 3 The Cross-Spectra 162 8 .4 The Energy and Momentum Flux Spectra . 163 8 . 5 Mean F lux of Energy to the Waves 163 8 .6 Mean F lux of Momentum to the Waves 163 8 .7 £ , the F r a c t i o n a l Energy Increase of the Waves per Radian 164 8 .8 The Boundary Bay Experiment 165 8 .9 A i r Flow over Waves Moving in to the Wind 166 SECTION 8: CONCLUSIONS .8.1 Introduct ion A measurement system cons i s t ing of a wave sensor and a buoy i n which was mounted a pressure sensor has been designed, b u i l t , thoroughly tested, and used to obta in simultaneous recordings of wave height and normal pressure at the water surface i n the presence of wind-driven sea waves. Spec ia l ana lys i s techniques have been developed for dea l ing with large spurious spikes i n the pressure s i g n a l , and i t has been shown that these spikes do not m a t e r i a l l y a f f e c t the r e s u l t s i n the frequency range of i n t e r e s t , which i s between 0.1 and 2 Hz. The recordings obtained with the measurement system are the f i r s t simultaneous f i e l d measurements of pressure and waves i n which the sensors have been constrained to move only v e r t i c a l l y ( q u a s i - E u l e r i a n measurement), and they are the most comprehensive of a l l the e x i s t i n g measurements. They should thus provide the necessary basis for future pred ic t i ons of the magnitude and phase ( r e l a t i v e to the waves) of the normal pressures which appear to be the p r i n c i p a l source of the energy transfer from the wind to the waves. Power spectra of the pressure and wave height s ignals and the cros s - spec tra between them have been presented for s i x separate runs, which cover a wide range of wind and wave speeds. Energy and momentum f l u x spectra and the in t egra l s E and t w under these spectra have been computed and are presented for a l l the runs. The frequency d i s t r i b u t i o n s of £ , the f r a c t i o n a l increase i n wave energy per rad ian , and a dimension-161 162 less p l o t of f ( f ) versus U5/c have been ca lcu lated and are a lso pre-sented . 8.2 The Power Spectra The wave power spectra behave as expected for the s i t e , e x h i b i t i n g high-frequency slopes between -4 .5 and -5 . In some runs a considerable amount of long-wavelength swel l was present which was presumably generated by marine t r a f f i c i n the area. The pressure power spectra appear to be made up of two p a r t s , con-s i s t i n g of a "basic" spectrum s i m i l a r to those observed over land, on which i s superimposed a wave-induced "hump". A d d i t i o n of Pj]^ to the pressure s i g n a l has the e f f e c t of removing the superimposed hump at low frequencies whereas i t has l i t t l e e f f e c t at high frequencies , i n d i c a t i n g that at high frequencies most of the pressure s igna l which i s coherent with the waves i s i n quadrature with them. 8.3 The Cross-Spectra The phase between the pressure and the waves has been compared with the phase pred ic ted by M i l e s ' (1957) " i n v i s c i d laminar" model of wave generat ion. At frequencies near the loca l ly -genera ted peaks i n the wave spec tra , observed phases are sh i f t ed from -180 (that i s , pressure high over wave troughs) by an amount which exceeds predic ted values by 2 0 ° ; the estimated error i n th is measurement i s less than ~t 5 ° . This discrepancy occurs over a range 9.5 £ Cp/u* 4 35, where Cp i s the wave phase v e l o c i t y at the peak of the wave spectrum and u^ i s the f r i c -t i o n v e l o c i t y . 163 8.4 The Energy and Momentum F l u x Spectra Spectra of the f luxes of energy and momentum from the wind to the waves are presented here for the f i r s t time. They both are peaked sharply at a frequency at or (usual ly) s l i g h t l y higher than f p , the frequency of the peak of the wave spectrum. The sharply-peaked *^(f) spectrum i s contrasted with the rather broad - f^EW spectrum measured at r e l a t i v e l y short distances above the waves (Smith, 1967; Weiler and B u r l i n g , 1967). The f a c t that the peaks of the E(f ) and 7j*w(f) spectra are at f r e -quencies higher than fp suggests e i ther that wave-generation mechanisms are ac t ive which transfer energy by means other than normal pressures , or that energy i s being t rans ferred from higher to lower frequencies i n the wave f i e l d i t s e l f . 8.5 Mean F l u x of Energy to the Waves The i n t e g r a l s under the E(f) spectra have been determined at 20-second i n t e r v a l s throughout each run; the E values presented are the means of these averages. The i n d i v i d u a l i n t e g r a l s under the E(f) spectra are found with a Chi-squared tes t to be approximately normally d i s t r i -buted about t h e i r means, and are found to have large standard dev iat ions; these fac t s i n d i c a t e that the wave generation process i s random i n nature and var i e s considerably i n time (or space) . 8.6 The Mean F l u x of Momentum to . the Waves The mean values of f , the wave-supported momentum f l u x , have been computed for each run and found NOT to be s i g n i f i c a n t l y d i f f e r e n t from 2 the mean f luxes computed from ^ Op U 5 , where Cp, the drag c o e f f i c i e n t , 164 i s taken to be .0012. A "wave-drag c o e f f i c i e n t " has been computed from the T"' . and i s .0014 with standard error .0005. In one measure-i» w ment the t o t a l momentum input to the a i r - s e a boundary f = - £ a u w was measured; the r a t i o ^ w / ^ for th is case was 0.8. On the basis of these r e s u l t s , i t i s f e l t that the a s ser t ion made by P h i l l i p s (1966) that <, 0.1 i s probably i n c o r r e c t . .8.7 , the F r a c t i o n a l Energy Increase of the Waves per Radian Frequency d i s t r i b u t i o n s of measured values of !> have been prepared and compared with the pred ic t ions of M i l e s ' i n v i s c i d laminar model and with an empir i ca l curve suggested by Snyder and Cox (1966). The measured values of £ are greater than those predic ted by M i l e s ' theory by fac tors of between 5 and 8, and are about the same s i ze as those pred ic ted by Snyder and Cox's empir i ca l r e l a t i o n , over most of the f r e -quency range of i n t e r e s t . Thus the present data support the f i n d i n g of Snyder and Cox that M i l e s ' (1957) i n v i s c i d laminar mechanism i s inadequate to exp la in observed rates of wave growth. At high frequencies the measured values of C f a l l below the Snyder and Cox curve. This ind icates that the slow-moving short-wavelength waves i n th i s frequency range are not generated as e f f i c i e n t l y as those nearer the peak of the wave spectrum; the p o s s i b i l i t y therefore ex is t s that shorter waves may be "sheltered" for a large p o r t i o n of the time by the long waves at the peak of the spectrum. This .could happen i f , as suggested by Stewart (1967), the flow d i s t r i b u t i o n over the waves i s such that the "cats-eye"- flow d i s t r i b u t i o n ( i n the coordinate system moving with the wave phase speed) which was f i r s t described by L i g h t h i l l 165 (1962) i s s i tuated i n the wave trough. A dimensionless p l o t of £ (f) versus U5/Cp has been presented; the observed values of ^ ( f ) , although somewhat scat tered for U 5 / c > 4 , are approximated quite we l l by f(f) = ? a / ? w ( U 2 / c - 1) , where U2 i s the wind speed at a height of two meters. This simple r e l a t i o n f i t s the data as w e l l as the Snyder and Cox curve does over the range 0.5 < U^/c ^ 6; for U^/c ^ 6 i t does not apply, s ince at these values of U^/c the measured values of ^ ( f ) may f a l l r a p i d l y towards zero. The f a l l o f f i n ,T(f) at high frequencies , i f i t i s r e a l , must be taken in to account i n any computation of integrated f luxes of energy or momentum to the waves. 8.8 The Boundary Bay Experiment This experiment was f i r s t conceived as a f i e l d tes t of the buoy sensor and a sensor developed by J . A. E l l i o t t of the I n s t i t u t e , against a standard sensor. The t e s t , which was made on a g e n t l y - s l o p i n g sand f l a t , ind ica ted that a l l three instruments produced r e s u l t s which were e s sent ia l l y , i n d i s t i n g u i s h a b l e over the frequency range of i n t e r e s t i n the pressure-waves experiment (0.1 to 2 Hz) . Besides th i s r e s u l t the experiment provided some i n t e r e s t i n g information on the s t ruc ture of the pressure f i e l d i n the turbulent atmospheric boundary l a y e r . These r e s u l t s are l i s t e d below. 1. The pressure power spectra obtained show reasonably good agree-ment with other measurements, both i n the f i e l d ( P r i e s t l e y , 1965) and 166 i n a wind tunnel (Willmarth and Wooldridge, 1962). 2. The pressure power spectra have been found to group w e l l when 2 2 non-dimensionalized with ( €a u*) for radian wave-numbers greater than 10"" 2 cm"l; th is ind icates the existence of a r e l a t i o n s h i p between the 2 pressure and the Reynolds s tress C = £ u* i n the boundary layer . 3. The v e r t i c a l "scale s ize" of the pressure-generat ing eddies i s about one-hal f , and the crosswind scale s i z e i s about one-tenth, of t h e i r downwind scale s i z e . The l a t t e r r e s u l t agrees roughly with the more d e t a i l e d f indings of P r i e s t l e y (1965). 4. The broadband advection v e l o c i t y of the pressure-generat ing eddies i s about 0.6 U,*, } where U,,, ± s taken to be 30 u, v . 5. At h igh frequencies (2-5 Hz) the pressure i n the a i r shows a large phase lead (90 to 1 8 0 ° ) over the downwind a i r speed at the same e l e v a t i o n . 6. There i s some evidence that low-frequency (0.05 - 2 Hz) v e r t i c a l acce lerat ions ex i s t i n the shear flow near the ground; this i s based on the apparent presence of a v e r t i c a l gradient of the part of the pressure coherent with the downwind a i r speed. 8.9 A i r Flow over Waves Moving in to the Wind A f a i r l y d e t a i l e d study has been made of the phase r e l a t i o n s between wave e l e v a t i o n , surface pressure, and downwind and v e r t i c a l a i r v e l o c i t y over a group of waves of four-second period which were t r a v e l -l i n g against a l i g h t wind. The a i r flow over the wave group was found to be approximately given by p o t e n t i a l flow theory; the observed d i f f e r -167 ences, although smal l , are important. The phase of the pressure was + 1 6 5 ° r e l a t i v e to the waves instead of 1 8 0 ° , and the phase of the wind v e l o c i t y along the waves was + 1 2 ° instead of 0 ° ; the phase of the v e r t i c a l v e l o c i t y was 9 0 ° , as pred ic ted . The energy f l u x from the waves 9 1 to the wind was computed and found to be 20 erg cm - z- sec , which represents a Q for the waves of about 4000. The momentum f l u x from - 9 waves to wind has a lso been computed and found to be 0.04 dyne cm , the t o t a l momentum f l u x from water to a i r was s imultaneously measured at a height of 1.75 meters with a sonic anemometer, and was 0.002 dyne _ o cm . This may i n d i c a t e the presence of a v e r t i c a l s tress gradient of magnitude -2 x 10 dyne cm /cm. The observed perturbations from p o t e n t i a l flow are much smaller than those which would be extrapolated from the pred ic t ions of P h i l l i p s (1966); however i t i s considered that the flow regime over the waves i n th is case was quite d i f f e r e n t from the case considered by P h i l l i p s . APPENDIX 1: SPIKE REMOVAL A l . l In troduct ion 168 A1.2 The Spike Removal Process 169 A1.3 D e f i n i t i o n s 171 A1.4 The E f f e c t s of a Regular Spike Window 175 A1.5 The E f f e c t of the Spikes on the Power Spectra 177 A1.6 The E f f e c t of the Spikes on the Phase Spectra 178 A1.7 The E f f e c t of the Spikes on the Energy F l u x Spectra . . 180 A1.8 Conclusions 182 APPENDIX 1: SPIKE REMOVAL A l . l In troduct ion This appendix deals with the analys i s problems introduced by the presence i n the pressure s igna l of l arge , spurious "spikes". These spikes occur i n the data whenever water washes over the pressure-sensing diaphragm at the surface of the buoy. They are character i sed by genera l ly large amplitudes and a r i s e - t i m e much f a s t e r . t h a n that of the normal turbulent pressure f l u c t u a t i o n s ; the i r amplitude i s often large enough to saturate the recording system. The ir e f fects on the s i g n a l l a s t for periods of 0.1 - 2 seconds. Their occurrence i n time cannot be sa id to be f u l l y random, since they are caused by the motion of the buoy on the waves; they occur most often at or near the crests of the large waves and also on some of the smal l , steep waves. Thus the pressure s i g n a l which i s analysed must be a modif ied one; i n the ac tua l ana lys i s i t s mean value i s set to zero, so wherever a spike occurs i n the o r i g i n a l pressure s igna l and i s removed, the d i g i t i z e d data values are replaced with zeros for the durat ion of the sp ike . This modified pressure s igna l i s w r i t t e n where p g ( t ) i s the observed pressure s igna l from which the spikes have been removed, p(t ) i s the s igna l which would be observed i n the absence p s ( t ) = p(t) S(t) A l . l , of sp ikes , and S(t) i s a "Spike funct ion" defined by S(t) = 0 during a spike A1.2 . = 1 at a l l other times 168 169 Equation A l . l cannot be inverted to recover p ( t ) ; that i s to say the pressure information l o s t during a spike i s i r r e t r i e v a b l e . This represents a fundamental l i m i t a t i o n i n the i n t e r p r e t a t i o n of the pressure data . This sec t ion therefore w i l l be devoted to explor ing the e f fects of the spikes , i n the hope of gaining an understanding of the spike- induced d i s t o r t i o n s of the observed pressure spec tra . The f i r s t s ec t ion of th is appendix w i l l deal with a f a i r l y d e t a i l e d account of the inner workings of the spike removal process . This w i l l be fol lowed by a s ec t ion g iv ing some d e f i n i t i o n s and d e r i v a t i o n s . Three d i f f e r e n t analyses follow-;; the f i r s t i s a c r o s s - s p e c t r a l analys i s of computer-generated sine waves i n which a r t i f i c i a l spikes have been i n s e r t e d . The next i s a comparison of the power spectra of wave s ignals obtained i n the f i e l d with power spectra of the same s ignals i n which holes have been inserted at times when spikes occur i n the pressure s ignals which were measured at the same time. L a s t l y , phase and quad-rature spectra between modified pressure and modif ied wave s ignals from complete runs made i n the f i e l d are compared with phase and quadrature spectra between unmodified pressure and unmodified wave s ignals from scattered data blocks i n the same runs, i n which no pressure spikes occurred . A1.2 The Spike Removal Process A b r i e f o u t l i n e of the funct ions of the spike removal subroutine SPKSKP has been given i n "Data Ana lys i s and In terpre ta t ion" , p. 86. This o u t l i n e w i l l be enlarged upon i n the fo l lowing paragraphs. The detect ion of spurious data by the rout ine i s on the basis of 170 slope; i f the absolute magnitude of the pressure d i f ference between successive data points i s l arger than a threshold value (DIFMAX) which must be determined separate ly for each run, then SPKSKP w i l l automat ica l ly zero a preset number (NSKIP) of the data points succeeding the point at which the spike i s f i r s t detected. Before these points are zeroed they are checked against the maximum pressure value (VALMAX) which i s l i k e l y to occur; at the same time the d i f ference between each point to be zeroed and i t s succeeding point i s checked against DIFMAX. I f e i ther l i m i t i s exceeded, SPKSKP r e s t a r t s the zeroing procedure and skips a further NSKIP points beyond the newly detected spike . I f NSKIP successive points beyond a "spike" pass both c r i t e r i a they are a l l zeroed; the (NSKIP + l ) s t and fo l lowing points pass unmodified unless a new spike i s detected. The checking continues u n t i l the end of a given data block i s reached. SPKSKP i s c a l l e d once for every data block analysed, and accepts two data arrays : p Q ( t ) , the observed pressure data , and S ( t ) , an array i n which a l l data values are set i n i t i a l l y to one. I t sets both p Q ( t ) and S(t) to zero for at l eas t (NSKIP + 1) successive values during spikes . SPKSKP returns p 0 ( t ) , the pressure with spikes removed, and S ( t ) , the "Spike funct ion" of Equation A1 .2 . The computer program (FORTRAN IV) for SPKSKP i s reproduced i n Appendix l a . In the F o r t r a n program there i s included an opt ion not mentioned here because i t i s not used i n the f i n a l ana lys i s whereby the spikes , instead of being replaced by zeroes, can be replaced by a s t r a i g h t l i n e in terpo la ted between t h e i r beginning and end data po in t s . SPKSKP returns to the main program v i a one of two routes . I f dur-ing the zeroing procedure the number of data points zeroed i n one block 171 exceeds a preset"- number of points (JFRAC; u s u a l l y set to one-half the number of data points i n the b l o c k ) , the main program i s ins t ruc ted to ignore the block being analysed and s t a r t on the next one. During the normal r e t u r n process SPKSKP p r i n t s out the t o t a l number of data points skipped and then d e l i v e r s p s ( t ) and S(t) to the d i g i t a l precondi t ion ing program FIDDLE. There the pressure s igna l has any r e s i d u a l mean (over each block) removed from i t and i s returned to ther.main program for F o u r i e r a n a l y s i s . A1.3 D e f i n i t i o n s In order to make the fo l lowing accounts c l earer some bas i c d e f i n i -t ions are given f i r s t . I t i s hoped that the i r i n c l u s i o n w i l l e l iminate uncer ta in t i e s as to the l o c a t i o n of factors of Tt , 2, e tc . We wish to perform a spectral ana lys i s on a s i g n a l v ( t ) , which begins at time t = 0 and extends to t = T. This s i g n a l i s sampled at f i xed i n t e r v a l s of time t j = j A t , j = 0, 1, 2, I / A t A1 .3 , where A t i s the sampling i n t e r v a l , and i s the r e c i p r o c a l of the sampling frequency f g ( v ( t ) i s arranged to contain no s i g n i f i c a n t energy at f r e -quencies exceeding f s / 2 ) . The r e s u l t i n g c o l l e c t i o n of data points v ( t j ) i s then d iv ided in to equal time i n t e r v a l s T , where N A t = T « T A1.4 . These sets of N A t data points are hereafter c a l l e d "blocks". The s p e c t r a l ana lys i s technique used i n the present work i s one developed by Cooley and Tukey (1965), and commonly r e f e r r e d to as the 172 "Fast F o u r i e r Transform" (the name of the ac tua l computer program is PKFORT). This program i s used to produce for each block of data points the set of c o e f f i c i e n t s of the d i s c r e t e F o u r i e r ser ies which exact ly describes the sampled (d i screte ) s i g n a l w i t h i n that b lock . These co-e f f i c i e n t s are complex, and occur at frequencies separated by. i n t e r v a l s The r e a l part of the F o u r i e r c o e f f i c i e n t at f = 0 i s the mean value of the data points i n the b lock. From these F o u r i e r c o e f f i c i e n t s power or c r o s s - s p e c t r a l estimates at each frequency are determined. This ana lys i s i s repeated for each complete block of data points i n the run; the f i n a l power or c r o s s - s p e c t r a l estimates for a complete run are the averages over the number of blocks i n the run. The Fast F o u r i e r Transform program thus accepts an array v ( t j ) of data points and replaces them with an array of complex F o u r i e r c o e f f i c i e n t s A ( f k ) where f k = k A f i s the frequency of the k ^ harmonic of the s igna l w i th in the b lock . A f = ! / - £ • A1.5 and extending from f = 0 to f s / 2 , the f o l d i n g or Nyquist frequency. N - l v ( t O exp (-2TT i j k / N ) A1.6; where N i s the number of data points i n the block and i = V - 1 . Put t ing k/2TI = k / N A t and using A1.3 and A1 .4 , A ( f k ) = I - i w . t j A t A1 .7 . A ( f k ) = B ( f k ) + i C ( f k ) A1.8 173 i s complex; i t contains amplitude and phase information on the Four i er harmonic at the frequency f k . In the s p e c t r a l ana lys i s program SCOR (see "Data Ana lys i s and I n t e r p r e t a t i o n " , p. 82) estimates of (power) s p e c t r a l dens i t i e s (t>e(fk) = k<£kf = X . U f k ) 2 + C ( f k ) 2 l A1.9 2 A f 2 1 J are computed at each frequency for each b lock. The f i n a l estimate of the power spectrum for a run cons is ts of a s ing le set of ^ ) e ( f k ) > ^ n which the estimate at each frequency i s averaged over the t o t a l number of complete blocks i n the run. Since the block and run durations are f i n i t e the^^ e ( f k ) are estimates of some kind of "ensemble" averages ( J ) ( f k ) . The ( J ) (fk) would a r i s e from the average over several r e a l i z a t i o n s of v ( t ) which derive from the same set of p h y s i c a l condi t ions; that i s to say i f the v ( t ) measurement were begun on several occasions at i d e n t i c a l time i n t e r v a l s fo l lowing the on-set of the ( i d e n t i c a l ) causal processes, then c e r t a i n parameters such as the var iance of v ( t ) , i t s estimated power spectrum ^ e ( f k ) (and v ( t ) i t s e l f ) should each be s t a t i s t i c a l l y d i s t r i b u t e d i n the same fashion for each r e a l i z a t i o n of v ( t ) . We seek to f i n d "good estimates" of these para-meters or of t h e i r d i s t r i b u t i o n s . The ana lys i s of v ( t j ) i n a ser ies of blocks i s equivalent to m u l t i p l y -ing v ( t ) by the "data window" D(t) = 1 0 - t ^ T A L I O = 0 otherwise Lving0 £ This i s equivalent to c o n v o l v i n g U X ( f ) with the "spectra l window" 174 ( J ) D ( f ) = T s i n 2 ( 2 1 T f ^ / 2 ) A l . l l ; ( 2 T T f T / 2 ) 2 th i s i s exact ly the "rectangular lag window" discussed i n Blackman and Tukey (1959; pp. 68-70). The modif ied (spike-contaminated) pressure s i g n a l p s ( t ) = p ( t )S ( t ) i s the only one a v a i l a b l e with which to form a cross-spectrum with the wave s i g n a l 7£ (t) . The question a r i s e s : should i t be * £ ( t ) or * £ s ( t ) = ? £ ( t ) S ( t ) which i s used to obta in the cross-spectrum? The F o u r i e r transforms F T p s ( f ) and F T * £ s ( f ) of p s ( t ) a n d ^ s ( t ) are convolut ions: F T p s ( f ) = F T p ( f ) * F T s ( f ) and A1.12, F T , s ( f ) = F T ^ ( f ) * F T s ( f ) where F T g ( f ) i s the "Fourier transform window" of S ( f ) . At a given f r e -quency, the phase spectrum Q p g 7 £ s ( f ) i s given by the d i f ference of the arc tangents of the r a t i o s of imaginary to r e a l parts of F T p g ( f ) and F T ^ s ( f ) . Because FTp(f) (which i s i n p r a c t i c e not ava i lab le ) and F T t j ( f ) have both been convolved with the same transform window, the r a t i o s of imaginary to r e a l parts of F T p ( f ) and FT^(f) are both modified i n the same way by the convolut ion and therefore ^ P g ^ g C f ) i - s tb^a f i r s t approxi-mation the same as ^ ^ ( f ) . I f , on the other hand, p s ( t ) i s corre la ted with ^ ( t ) , the r a t i o of the imaginary to the r e a l part of the p s ( t ) F o u r i e r transform i s modif ied by convolut ion with the spike window while the ^ ( t ) transform r a t i o i s not. For th is reason the p s ( t ) ^ s ( t ) cross-spectra are used e x c l u s i v e l y for the presentat ion of the r e s u l t s . 175 A1 . 4 E f f ec t s of a Regular Spike Window In order to assess the inf luence of "holes" i n the pressure s i g n a l r e s u l t i n g from the removal of sp ikes , hypothe t i ca l pressure and wave s ignals have been generated on the computer and analysed with the same procedures used on the f i e l d data. Both s igna l s cons i s t of a s ing le s inuso id with zero mean; they have the same amplitude and frequency but d i f f e r i n phase by 3 0 ° . The ir frequency i s arranged so a data block does not contain an i n t e g r a l number of cyc le s . I f de s i red , e i ther s i g n a l may be zeroed for a f ixed length of time once every L cyc l e s , L being any p o s i t i v e integer; both L and the durat ion of the "holes" pro-duced by the zeroing procedure can be preset for a given run . These hypothe t i ca l pressure and wave s ignals have been analysed exact ly as the f i e l d s ignals are analysed; power and cross spectra of the pressure and wave "signals" are obtained. A number of d i f f e r e n t combinations of s ine waves, hole durat ions , and values of L have been analysed. For the sake of b r e v i t y the cross - spec tra from only one com-b i n a t i o n i s presented, for which the "pressure" and "wave" s ignals both cons i s t of a s ing le s inuso id , and the spike funct ion repeats once per cycle of this s i n u s o i d . This combination was chosen because i t c l e a r l y shows the s p e c t r a l d i s t o r t i o n s introduced by the r e p e t i t i v e spike funct ion used and because the contamination introduced by removing data from every cycle of the s igna l i s more severe than the contamination which occurred i n any of the runs analysed. The p a r t i c u l a r "signals" used were _2 p ' ( t ) = 5.0 s i n ( 2 1 T N f 0 / f s ) dyne cm , 176 •^'(t) = 5.0 s i n ( 2 1 T N f o / f s + TT/6)cm, and A1.13 S 1 (t) = 0 (Mf / f + 30) 4 N L. (Mf It + 50) s o — s o = 1 otherwise. In Equation A1.13 N i s the sample number and f the sampling frequency s (chosen here to be 50 Hz) , f i s chosen to be 0.4 Hz, and M = 0, 1, 2, . . . o i s the cyc le number. Thus from Equation A1.13 the per iod 6"fthe s i n u -soids? was 125 samples, and a "hole" occurred once per cyc le at the phase 2 If (M + 30/125) and ended 20 samples, or (2TT x 20/125) rad ians , l a t e r . Ten blocks of 1024 samples each were analysed for two cros s - spec tra : p ' ( t ) S ' ( t ) versus ^ ' ( t ) , and p ' ( t ) S ' ( t ) versus ( t ) S ' ( t ) . The r e s u l t s of these analyses are d isp layed i n Figures 56 to 63. Figures 56 and 59 show r e s p e c t i v e l y the power spectra of p 1 ( t ) S ' ( t ) = P g ( t ) and * £ ' ( t ) ; F igure 59 c l e a r l y shows the envelope of the block s p e c t r a l window given by Equation A l . l l . The v e r t i c a l bars on the power s p e c t r a l estimates are standard errors about the mean over the 10 blocks for each spectral estimate; the h o r i z o n t a l bars give the band-, width. The pressure power spectrum i n F igure 56 shows that the e f f ec t of m u l t i p l y i n g the pressure s i g n a l with the "Spike funct ion" (which repeats once per cyc le s t a r t i n g at 2 IT (1.24) and ending at 2TT (1.4) radians) i s the generation of harmonics of the fundamental; r e p l i c a s of the block s p e c t r a l window appear at 0.8, 1.2, and 1.6 Hz. The coherence spectrum i s shown i n F igure 57 and the phase spectrum i n Figure 58. I t w i l l be remembered from Equations A1.13 that the phase of the pressure s igna l s lags the wave s i g n a l by 3 0 ° . Although the phase spectrum i s 177 almost constant with frequency near 0.4 Hz the measured phase lag at 0.4 Hz i s 3 8 ° . Turning now to Figures 60 to 63,which d i s p l a y cross - spec tra between p s (t) and s ( t ) , the phase at 0.4 Hz i n Figure 62 i s seen to be w i t h i n 0.1 of 3 0 ° , lending credence to the arguments made on p.174 that the Ps s c o r r e l a t i o n should be used for f i n d i n g the phase spectrum rather than the fo s /^ c o r r e l a t i o n . Note the large coherences at harmonics of 0.4 Hz i n F igure 61 as compared with the rather small coherences at the harmonics i n F igure 57; the high-frequency coherences of the P s °7 s c o r r e l a t i o n s from the f i e l d data are also found to be larger than those of the p s ^ c o r r e l a t i o n s . This ind icates that at high frequencies i n the p s ^ s s P e c t r a from the f i e l d data the observed phases are probably e n t i r e l y determined by the inf luence of the spikes . A1.5 The E f f e c t of the Spikes on the Power Spectra Because the information from the regions of the pressure s i g n a l i n which spikes are present i s i r r e t r i e v a b l y l o s t , only a q u a l i t a t i v e estimate can be made of the e f fec t of the spikes on the pressure power spectra . This estimate has been made by m u l t i p l y i n g an observed wave s i g n a l ^ ( t ) by the spike funct ion S(t) derived from the simultaneously observed pressure s i g n a l , and observing the e f fec t of the spike window on the wave power spectrum. The spectra of ^ ( t ) and of * £ ( t ) S ( t ) = ^ s ^ a r e s n o w n f o r t n e f i e l d runs i n which appreciable spike contamination occurred (runs 1, 2b, 3, and 6) i n Figures 64 to 67. The general e f fec t of the spike contamina-t i o n i s seen to be a smearing out of the uncontaminated spec tra , with 178 energy being removed from the peak and added at the h igh frequencies; energy i s a l so added at frequencies below the wind-generated s p e c t r a l peak i n runs 1 and 6, while i t i s removed from these regions i n runs 2a and 3. I t appears that i n the four runs shown energy i s added to the frequencies below the wind-generated peak i f no swel l peak appears i n the spectrum and removed i f the swel l peak i s present . This apparent c o r r e l a t i o n between the ac t ion of the spikes on the spectra and the presence of the swel l peak may be r e l a t e d to the t o t a l number of spike occurrences , assuming that the presence of swel l i n the wave f i e l d would increase the l i k e l i h o o d of large numbers of spikes . The a c t i o n of the spikes on the wave spectra at high frequencies i s to decrease t h e i r f a l l o f f r a t e . The contaminated spectra cross the uncontaminated spectra near 1.5 Hz, the contaminated spectra being larger at higher frequencies . Above 2.5 Hz the spectra diverge r a p i d l y ; s ince ne i ther the pressure nor the waves are well-measured at these frequencies , these large discrepancies are not discussed f u r t h e r . A1.6 The E f f e c t of the Spikes on the Phase Spectra The recorded pressure s i g n a l consisted genera l ly of c l ear s i g n a l interspersed with regions of spike a c t i v i t y . As a check on the e f fec t the spikes have on the phase spec tra , the spike detect ion and removal program SPKSKP was a l t ered s l i g h t l y so that the detec t ion of a spike anywhere i n a data block caused the e n t i r e block to be ignored. This a l t e r a t i o n i n SPKSKP allowed the analys i s of blocks of c l ear 179 data; these were,however, scattered at random time i n t e r v a l s through-out the run . In p r a c t i c e , to get a s u f f i c i e n t number of blocks for good s t a t i s t i c s the block length had to be shortened from 20 seconds for the "spikey" data to 5 seconds for the c l ear data. Therefore the lowest frequency which could be resolved was r a i s e d from 0.05 Hz i n the case of the "spikey" data to 0.2 Hz for the c lear data. Phase spectra between pressure and waves for runs 1, 2b, 3, and 6 (see "Results" for d e t a i l e d descr ip t ions of these runs) are shown i n Figures 68 to >71. In each f igure two curves are drawn corresponding to two ways of precondi t ion ing the data; the f u l l curve shows the phases between p& and ^ f r o m a l l blocks i n each run for data subjected to the spike removal process , the dotted curve shows the phases between p and from the blocks during which no detectable spikes occurred. The dotted and f u l l v e r t i c a l l ines at high and low frequencies are the f r e -quency l i m i t s beyond which the re levant phases become u n r e l i a b l e ; that i s , the frequencies where the coherence f a l l s below about 0.3 (see the d i scuss ion of this coherence l i m i t i n "Results", p. 115). The phase spectra computed i n the two d i f f e r e n t ways show sub-s t a n t i a l agreement. The larges t d iscrepancies are about 2 0 ° , but i n most cases the phases are w i t h i n 1 0 ° of each other at a given f r e -quency. This comparison of c lean and spike-contaminated data provides 180 d i r e c t evidence that the e f fec t of the spikes on the phase spectra i s quite smal l . A1.7 The E f f e c t of the Spikes on the Energy Flux Spectra The p o s s i b i l i t y s t i l l ex i s t s that s i g n i f i c a n t spike- induced d i s -tor t ions can occur i n the quadrature spectrum Qu (f) , i n sp i t e of the genera l ly small d i s t o r t i o n s i n the phase spectra; for th is reason the energy f lux spectra E(f) (which are derived from the Qxif^(f) through r e l a t i o n 5.9) computed from the "clear" and "spikey" pressure data are a lso compared, i n Figures 72 to 75". I t w i l l be seen that although the two E(f) spectra agreesquite c l o s e l y i n runs 2b and 6, there occur what appear to be large d i s c r e -pancies between the two spectra i n the other two runs (1 and 3) . The s i g n i f i c a n c e of these d iscrepancies cannot be assessed s t a t i s t i -c a l l y s ince the sca t ter over the data blocks analysed i s i n v a r i a b l y large - - so large i n f a c t that a l l of the "large" discrepancies noted between the two E(f) spectra i n runs 1 and 3 l i e w i t h i n one standard error of each other. The poss ib le phys i ca l s i g n i f i c a n c e of this large scat ter i n the E(f) spectra i s taken up i n "Discuss ion of Results" on p. 146. 181 The observed discrepancies i n runs 1 and 3 between the E(f ) spectra are hence w i t h i n the observed sca t ter i n the r e s u l t s . The discrepancy i n run 1 occurs at one frequency only , 0.37 Hz; th is i s the frequency of the peak of the wave spectrum. I t w i l l be noted that the analys i s of the "clean" data indicates stronger wave generation (has a larger negative E( f ) spectrum) at this frequency than does the "spikey" data a n a l y s i s . Since the wind speed for th i s run was lower (see Table 6.1) than the wave speed at 0.37 Hz there i s l i t t l e l i k e l i h o o d that generation was a c t u a l l y occurr ing; therefore the r e s u l t from the "spikey" data i s probably c loser to being r e a l i s t i c . I t i s i n t e r e s t i n g that i n sp i t e of d i screpancies i n the E( f ) spectra from the two sets of data the in t egra l s under these spectra are almost i d e n t i c a l ; the "spikey" data gives E = — 9 1 9 1 16.6 erg cm sec" and the "clean" data gives E = 16.3 erg cm s e c - . This agreement i s the r e s u l t of the weighting of the higher-frequency s p e c t r a l estimates (they have larger bandwidths) i n the i n t e g r a t i o n pro-cess, which minimizes low-frequency d i screpanc ies . The good agreement between the two values of E makes i t unnecessary to pursue further the reasons for the d i f ferences i n the two E(f) spectra . Run 3 (Figure 74) i s somewhat d i f f e r e n t . The discrepancy noted at the higher frequencies (1.1 to 2 Hz) i n the phase spectra for the run (Figure 70) ind ica te s that the average phase angles of the "clear" data are d i f f e r e n t from those of the spikey data . The integrated energy f l u x — - 2 - 1 E from the c l ear data i s 62 erg cm sec , while the value determined - 2 - 1 the spikey data i s 30 erg cm sec , 507, lower. Both of the E(f) spectra from run 3 i n d i c a t e wave damping at 0.2 Hz, at which frequency a large amount of swel l energy was present i n the wave power spectrum. Therefore there i s i n th i s case no objec t ive c r i t e r i o n by which to judge which of 18>2 the two analyses i s more near ly c o r r e c t . The "spikey" data might be expected to underestimate E( f ) s ince i t has had energy removed from i t v i a the spike removal process; on the other hand i f i t i s assumed that energy input was larges t to the largest -ampl i tude waves, then s ince the l arges t waves caused spikes more of ten , the c l ear data would miss th i s energy t rans fer and hence might be expected to underestimate E ( f ) . The only choice remaining i s to assume that E for run 3 i s not known to bet ter than -50%. A1.8 Conclusions The object of th is appendix has been to study the e f fects of the removal of spikes from the pressure s i g n a l on the power and cross spectra of pressure and wave s ignals which form the most important r e s u l t s of th is thes i s . The conclusions r e s u l t i n g from this study are l i s t e d below. 1. By computing the power and cross spectra of computer-generated s ine waves of the same frequency but d i f f e r e n t phase, one or both of which was modif ied to be replaced by zeros once every cycle for a durat ion of one-s ix th of a cyc le (a "hole 1 ' ) , two conclusions have been reached. The f i r s t i s that the power spectrum of the "hole-contaminated" sine wave i s d i s t o r t e d so that energy i s removed from the s p e c t r a l peak at the fundamental frequency and inser ted at the harmonics of the s i g n a l . The second conclus ion i s that the phase d i f f erence between the two sine waves compared i s accurate ly reproduced by the c r o s s - s p e c t r a l ana lys i s only i f the cross spectrum between two s igna l s contaminated i n the same way by the presence of holes i s used to determine the phase. 2. The e f f ec t of the spike removal on the power spectra of the f i e l d data i s a r e d i s t r i b u t i o n of energy, as i n conclus ion (1) . There 183 i s however no evidence that the excess energy i s s h i f t e d in to harmonics of the frequency of the peak of the wind-generated wave spectrum. 3. A comparison has been made between phase spectra between pressure and waves for two d i f f e r e n t methods of treatment of the pressure data. In one case the data were analysed normally i n the sense that when a spike occurred i t was replaced with zeroes; i n the other case only data blocks i n which no spikes appeared were analysed. The comparison shows f a i r l y good agreement, suggesting that the phase spectra are af fected l i t t l e by the energy r e d i s t r i b u t i o n noted i n conclus ion (2) . 4. A comparison has a lso been made of energy f l u x spectra between pressure and waves for the aforementioned two methods of t r e a t i n g the pressure data . Of the four runs with appreciable numbers of sp ikes , the agreement between the energy f l u x spectra computed from the data analysed i n the two ways was very good i n two cases (runs 2b and 6); agreement i n the other two cases (run 1 and 3) was poor. The integrated energy f luxes E were computed for the two runs where agreement was poor; the E values computed from the clean and spikey data agreed c l o s e l y for run 1, while the E computed from spikey data underestimated that computed from clean data by 507o i n run 3. As a r e s u l t the run 1 discrepancies are considered to be unimportant while those i n run 3 i n d i c a t e that serious spike contamination i s present i n th i s run and that the accuracy of the computed r e s u l t s for the run are suspect. IC I F S P I K E D E T E C T E D , S T A R T L C O U N T I N G , S T A R T C O R R E C T I O N - ' R O U T I N E )C "'-(' O R , I F IFSKP I S 1; R E T U R N T O G A L L I N G P R O G R A M W I T H I E R S P =•• 2 - .). | l *9 IF( I F S K P . G E . l ) G O T O 4 1 0 [50 1 = 1 ! • IF( M . G T . J F R A C ) G O T O 3 1 0 C C O R R E C T I O N ROUTINE ALSO C H E C K S A H E A D . I F A N O T H E R S P I K E O C C U R S D U R I M G - N S - K - I F •C T H E D O W I L L C O R R E C T A L L . P O I N T S T O N S K I P B E Y O N D IT . K T O T A L I Z E S C O R R E C T E D lC • - P O I N T S . M S T A R T = I + K N E N D = l+K + N S K I P - 1 D O 2 0 0 J = M S T A R T , M E N D C C H E C K F O R E N D O F A R R ( N P T S ) I F ( J . G E . N P T S ) G O T O 3 0 0 •C T E S T F O R . S A T U R A T I O N - O F A T O D . . . . . . . . I F ( A B S ( A R R ( J ) ) . G E . V A L M A X ) G O T O 2 5 0 C C H E C K A H E A D F O R N E X T + G O I N G S P I K E DIFJ = A R R C J + l ) - A R R (,J) I F ( D I F J . G T . D I F M A X ) G O T O 2 5 0 1 A R R ( J ) = 0 . • - •• SARR(J-) = - 0 . • - - - . . . K = K + 1 M = M + 1 2 0 0 C O N T I N U E \ G O T O 3 0 0 v. 2 5 0 A R R ( J ) =0. ) S A R R ( , ! ) - = • 0 . - - • • ? K = K + 1 ; M = M + 1 G O T O 5 0 3 0 0 C O N T I N U E G O T O 3 2 0 •310-- NSTOP= I + K • - ... . ... . ' W R I T E ( 6 , 3 1 5 ) J F R A C , N S T O P 3 1 5 FORMAT ( 1 H 0 , 2 X , 2 9 H D A T A Q U E S T I 0 N A B L B M O R E T H A N hl\\ V A L U E S S K I P 1 P E D . S P I K E R E M O V A L S T O P P E D A T , 14, 9 H T H V A L U E . ) , . C I E R S P I S R E T U R N E D A S 2 I F F R A C * I B L I S E X C E E D E D * I E R S P = 2 — G O T O - 3 3 0 - - - — • 4 1 0 W R I T E C R , 4 1 5 ) I 415 F O R M A T ( 2 H , 3 2 H S P I K E E N C O U N T E R E D A T S A M P L E M O . , I 4 , 1 7 H . B L O C K . I G N l O R E H ) I F R S P = ? R E T U R N 3 2 0 - - I F L I N E = I F L I N • +- 1- -W R I T E ( 6 , 3 2 5 ) A I F L I N C I F L I N E ) , M 3 2 5 FORMAT( 1H , 2 X , 5 4 H S P ! K E R E M O V A L C O M P L E T E D . SKIPPED V A L U E S R E P L A C E D . • 1 W I T H , A 8 , 1 H . , 1 X , I 5 , 2 1 H D A T A V A L U E S S K I P P E D . ) -' ' C IERSP - 0 MEANS N O R M A L R E T U R N I E R S P = 0 3 3 0 - • R E T U R N - - • - • - . . . . . E N D , SUBROUTINE S P K S K P C ARR, S A R R , NPOW, I E R S P / I F U N " " ) fC S P I K E R E M O V A L S U B R O U T I N E . R E M O V E S SPI.KES F R O M P R E S S U R E D A T A . , A M D A L S O p F R O M A T E S T A R R A Y WHICH I S I N I T I A L L Y . ' A L L 1 . '"; N P T S = 2**NPOW P T S = N P T S • ~ • . . . . . . . - , ' D I M E N S I O N A R R ( l ) , S A R R ( l ) , A I F L I N ( 2 ) ' C R E A D I N R E Q U I R E D D A T A . N S K I P I S BASIC N U M B E R OF D A T A P O I N T S CORRECTED C AFTER A- S P I K E . . D I F M A X IS M A X I M U M ALLOWED D I F F E R E N C E B E T W E E N S U C C E S S I V E C P O I N T S . V A L M A X IS MAXIMUM ALLOWED V A L U E OF D A T A P O I N T S . C FRAC IS M U L T I P L I E R TO SET M A X . ALLOWED NO. OF P O I N T S TO B E S K I P P E D . C I F S K P S E T S M O D E O F O P E R A T I O N I F 0 , S P K S K P S K I P S S P I K E S O N E N C O U N T E R C I F 1 , S P K S K P S K I P S WHOLE B L O C K S ON E N C O U N T E R I N G A S P I K E . D A T A A I F L I N ( l ) , A I F L I N C 2 ) / 6H Z E R O S , 6H L I N E S / - F ( - I E R S P . • H E . 0 ) 0 0 - T O . 2 0 . . . _ . . . . . . R E A D ( 5 , 1 6 ) F R A C , D I F M A X , V A L M A X , N S K I P , I F L I M , I F S K P 1 6 F O R M A T ( U X , F 6 . 5 , 2 ( i * X , F 6 . 1 ) , 7 X , 1 3 , 2 ( 9 X , I D ) J F R A C = P T S * F R A C W R I T E ( 6 , 1 7 ) J F R A C , D I F M A X , V A L M A X , N S K I P 17. FORMAT ( 1 H 0 , 2 X , I ^ H M A X I M U M ALLOWED N U M B E R OF SKIPPED V A L U E S I S , • 1 | i f , - l H . / - 3 X , 8 H D I F M A X - = , F 7 . 3 , 2 X , 8 H V A L M A X = , F 8 . 3 , 2 X , 7 H N S . K I P . . = , . 1 . 3 ) C S P I K E REMOVAL ROUT I M E . I N I T I A L I Z E COUNTERS. 2 0 K = 0 L = 0 M = 0 M E N D = 0 | C START -CH-ECK I HG D A T A B Y - L O O K I N O - F O R D I F F - E R - E N C E S B E T W E E N S U C C E S S I V E DATA. P Q.I . N T ! }C GREATER THAN M A X D I F . P R O G R A M CORRECTS DATA POINTS BY S E T T I N G T H E M TO ZERO O R |'C BY R E P L A C I N G T H E M W I T H A S T R A I G H T L I M E F R O M S T A R T T O E N D O F S P I K E . I T | C CORRECTS N S K I P POI NTS A F T E R A S P I K E , AND W I L L , I F A NEW S P I K E OCCURS- IN T H E x C R A N G E OF N S K I P , C O R R E C T N S K I P P O I N T S B E Y O N D THE NEW S P I K E . K C O U N T S ' C T O T A L P O I N T S C O R R E C T E D , L I S U S E D T O S H O R T O U T T H E S P I K E T E S T R O U T I N E W H I L E •G I I S B E I N G I N C R E M E N T E D TO T H E E N D OF THE S E R I E S OF C O R R E C T E D P O I N T S . D O 3 0 0 1 = 2 , N P T S I F ( L . E O . O ) G O T O 2 5 L = L + 1 IF( I F L I N . E O . 0 ) G O T O 2h I F ( L . N E . 2 ) G O T O 2 2 I N I T •=- N E N D - K - — - • . . . J E N D = N E N D + 1 R K = K + 1 A D D = ( A R R ( J E N D ) - A R R ( I N I T ) ) / R K 2 2 A R R ( I - 1 ) = A R R ( I - 2 ) + A D D C T E S T C O U N T E R T O S E E I F L A S T C O R R E C T E D P O I N T R E A C H E D -2h I F - C L . - G T . C K + 1 ) ) G O T O - 2 5 - - •• - - - • G O T O 3 0 0 C C H E C K F O R S P I K E S 2 5 D l F = A R R ( I ) - A R R ( 1 - 1 ) jC R E S E T K , M A F T E R M U L T I P L E S P I K E R E M O V A L C O M P L E T E K = 0 .. - ; . ^ ? v .-• •C T E S T FOR- P R E S E N C E O F - I N I T I A L - S P I K E - ( + Q R - - G O l - N G ) - - ' -I F C A B S C D I F ) . G T . D I F M A X ) G O T O If 9 C I F D A T A O K , R E S E T L , R E T U R N T O D A T A C H E C K I N G L O O P L = 0 • ' G O T O 3 0 0 ' APPENDIX l a : FORTRAN IV PROGRAM FOR SPKSKP APPENDIX 2: THE BOUNDARY BAY EXPERIMENT A2.1 In troduct ion 184 A2.2 D e s c r i p t i o n of S i t e 185 A2.3 Equipment 186 A2.3 .1 The Buoy Sensor 186 A2.3 .2 The Reference Barocel 187 A2.3 .3 The E l l i o t t Probe 188 A2.3 .4 A u x i l i a r y Equipment 188 A2.4 C a l i b r a t i o n 190 A2.5 Data Ana lys i s 191 A2.6 Results 192 A2 .6 .1 Summary of Relevant Information 192 A2 .6 .2 The Hot-Wire Data 192 A2.6 .3 The Pressure Power Spectra 195 A2.6 .4 Non-Dimensional Pressure Spectra 198 A2 .6 .5 Comparison of Power Spectra with Other Measurements , 199 A2.6 .6 The Pressure Cross-Spectra 202 A2 .6 .7 P r e s s u r e - V e l o c i t y Corre la t ions 208 A2v6.7a Comparison with Favre et a l (1957) . . . 210 A2.7 Conclusions 215 A2 .7 .1 Sensor Tests 215 A2.7 .2 The Structure of the Pressure F i e l d 216 APPENDIX 2: THE BOUNDARY BAY EXPERIMENT A2.1 In troduct ion In the summer of 1968 the opportunity arose of comparing the pressure sensor used i n the buoy with others being developed and used by Mr. J . A . E l l i o t t , a fe l low student at the I n s t i t u t e of Oceanography. Two sensors were used i n the comparison with the buoy sensor. One was an unmodified "Barocel" transducer, a commercial device made by Datametrics Inc . of Waltham, Mass . , U . S . A . , which was used to measure pressure f l u c t u a t i o n s on a f a i r l y l e v e l sand surface . The other was a probe developed by E l l i o t t which used a Barocel as i t s transducer and with which pressure measurements were made i n the body of the turbulent a i r flow above the sand surface . The p r i n c i p a l objec t ives of the experiment were to obta in pressure measurements with each of the three sensors, to compare the r e s u l t i n g spec tra , and thus to e s t a b l i s h whether or not the buoy sensor and the E l l i o t t probe were measuring true pressures . The p o s s i b i l i t y ex i s ted that pressures measured by the two might be contaminated by dynamic pressures o r i g i n a t i n g from the disturbance to the flow caused by t h e i r shapes. The unmodified Barocel was taken as the standard for the t e s t s , i t s pressure-sensing port being arranged i n a standard manner for surface measurements. A secondary objec t ive was to make some pre l iminary inves t iga t ions of the pressure f i e l d i n the a i r , at various heights and at various down-184 185 stream and cross-steam separat ions . A2.2 D e s c r i p t i o n of S i t e A s u i t a b l e , e a s i l y access ib le land s i t e was at hand, thanks to the e f f o r t s of another I n s t i t u t e student, Mr. N. E . J . Boston, who was conducting measurements of smal l - sca le temperature s t ruc ture i n the atmospheric boundary layer over a sand f l a t at Boundary Bay, about ten miles by road from the I n s t i t u t e . F igure 76 shows a map of the immediate area of the experiment. The sand f l a t , completely covered at high t i d e , i s bare for about 2.5 km seaward of the dike at low t i d e . I t s surface of f ine sand i s near ly l e v e l on the average with a seaward slope of less than 1 : 2,500. There are some shallow pools of r e s i d u a l water (averaging 1 - 10 m i n diameter) at low t i d e . Growing on the sand are numerous clumps of a l ow- ly ing p lant ; these clumps are about 60 cm i n diameter and about 10 cm h igh . Thus the sand, although smooth on the average, has roughness elements present which have h o r i z o n t a l scales of 10 - 100 cm. The h igh t ide shore l ine (about 10 meters from the base of the dike) i s strewn with logs , some as much as 20 m long and 1.5 m i n diameter. The dike i s about 3 m high and 10 m i n width. There i s a drainage d i t c h on the shoreward s ide; beyond i t f l a t farmlands (part of the d e l t a of the Fraser River) s t r e t c h in land for 5 - 1 0 km. The three hunting shacks on the shore l ine a l l have e l e c t r i c power (115 VAC, 60 Hz); Mr. H. R. Hipwe l l , the owner of one, k i n d l y consented to al low us the use of i t . Cables were constructed (Boston) which allowed the experiment to be set up about 100 m from the shore l ine . 186 A2.3 Equipment The three pressure sensors and the ways they were deployed i n the experiment w i l l be described f i r s t . A2 .3 .1 The Buoy Sensor The transducer, e l e c t r o n i c s , and buoy have been descr ibed e a r l i e r (see "Experiment"). During the Boundary Bay experiment the apparatus was arranged i n two ways. At f i r s t "case 1" the buoy was not used. The microphone and i t s o s c i l l a t o r were f i r m l y clamped into a thermally in su la ted (5 cm thickness of styrofoam) wooden box along with the r e f e r -ence sensor. The aluminum backup volume used i n the buoy was replaced with a glass jar of twice the usual volume which was i t s e l f in su la ted with 3 cm of styrofoam. The microphone was connected to a 0.8 mm diameter pressure port which was f l u s h with the top of the box and which was a c a r e f u l l y d r i l l e d and po l i shed hole i n a brass plug (3 cm i n d i a -meter) . The pressure connection to the sensor was made with a 16 cm length of 2 mm I . D . s t e e l tubing. Long-term (periods of 100 seconds or more) pressure v a r i a t i o n s were f i l t e r e d out by a pneumatic high-pass f i l t e r cons i s t ing of the b u i l t - i n leak around the diaphragm of the micro-phone and the backup volume. To make measurements with th i s arrangement the e n t i r e box was buried f l u s h with the sand. Some care was taken to insure that roughness elements w i t h i n 2 m upstream of the pressure tap were kept small ( scale s izes less than 1 cm). For the second set of measurements ("case 2") the buoy microphone and o s c i l l a t o r were placed i n a separate box, and the buoy was f i xed to the top of the box. The buoy surface was made as near ly as poss ib le 187. i d e n t i c a l with that used during tests on waves, the only d i f f erence being that the rubber diaphragm used i n the sea experiments was replaced by a c a r e f u l l y d r i l l e d 0.8 mm diameter pressure tap i n the perspex sheet which normally covered the e l e c t r o n i c s package (see Experiment, F igure 5) . The e l e c t r o n i c s , microphone, and backup volume were i n the thermally insu la ted box below. The tap was connected to the microphone by 20 cm of 2 mm I . D . s t ee l tubing. The experiments were conducted with the buoy and the box below i t buried i n sand to the depth at which i t would normally f l o a t i n calm water. No attempts were made to see the e f f ec t of vary ing i t s angle of t i l t with respect to the sand surface . A2 .3 .2 The Reference Barocel The Datametries "Barocel" transducer i s of the capacitance type. I t has a r e s o l u t i o n of less than 0.1 dyne cm d i f f e r e n t i a l pressure on i t s most s e n s i t i v e range and a frequency response i n the arrangement used that was f l a t from 0 to 10 Hz (with some phase error above 7 Hz; see F igure 78). Stated accuracy and l i n e a r i t y are 0.25% of f u l l s ca le . I t has 9 pressure ranges covering pressures from 0 - 1 3 m i l l i b a r s (1 mb = 10 3 dyne c m - 2 - 1 0 - 3 atmos.) . For the f i r s t ser ies of experiments the reference Barocel was placed i n the same box as the buoy sensor; i t measured pressure at a pressure tap i d e n t i c a l with that used for the buoy sensor and s i tuated 5 cm crosswind from i t . One s ide of the transducer was connected to the tap with 13.5 cm of the same (2 mm I . D . ) tubing as that used for the buoy sensor; the other s ide was connected to a small vacuum f l a s k which served as a backup volume. Long-term pressure changes were equal ized 188 with a 4 cm length of 0.15 mm I . D . hypodermic needle, which was connected between the two pressure ports of the transducer and acted as a pneumatic high-pass f i l t e r . This arrangement and the thermal box was designed and b u i l t by E l l i o t t . For the second set of measurements the reference Barocel pressure tap was i n the box, which was bur ied as for case 1; the buoy and i t s box were bur ied at d i f f e r e n t times, crosswind and downwind from the r e f e r -ence B a r o c e l . A2 .3 .3 The E l l i o t t Probe The d e t a i l s of th is probe w i l l be discussed i n a forthcoming p u b l i -c a t i o n . I t i s mounted on the end of a 50 cm "st ing" along which pressures sensed by the probe are transmitted by tubes to a Barocel transducer with a backup volume arrangement s i m i l a r to that used for the reference sensor. The transducer and backup volume are contained i n a 12.5 cm O . D . , 35 cm long aluminum c y l i n d e r with a con ica l nose; cables carry the transducer output to the s i g n a l condi t ion ing e l e c t r o n i c s which can be located as f a r as 80 m from the probe (the same i s true for the reference B a r o c e l ) . A2 .3 .4 A u x i l i a r y Equipment Wind speed was measured with two "Casel la" cup anemometers; these returned e l e c t r i c a l pulses to counters at the recording area which counted once per cup r e v o l u t i o n ; t o t a l counts were recorded every f i v e minutes with a one-minute break to wr i t e the numbers down. Wind speed was obtained from c a l i b r a t i o n curves provided by the manufacturer. Wind 189 speed was obtained from c a l i b r a t i o n curves provided by the manufacturer. Wind d i r e c t i o n was measured with a small vane which turned the wiper of a potentiometer,and readout on a meter c a l i b r a t e d at f ive-degree i n t e r v a l s . Downwind f luc tua t ions of turbulent wind speed were measured for a l l of the runs with a "Disa" constant temperature hot-wire anemometer; the wire was placed v e r t i c a l l y i n the a i r flow at various heights above the sand. The hot-wire s i g n a l , the three pressure s i g n a l s , and a vo ice log were recorded at 1\ ips tape speed on a 14-channel "Ampex FR-1300" portable instrumentat ion tape recorder i n the FM mode (voltages are con-verted to frequency-modulated audio s igna l s and recorded; they are demodulated on playback. Frequency responsesis f l a t from 0 - 2 kHz.) For some runs a mercury thermometer was used to measure a i r temperature and sand temperature. F igure 77 shows a t y p i c a l setup. A portable mast 4.5 meters h igh i s erected on the sand at low t ide about 150 meters down from the high t ide l i n e . S igna l cables are l a i d from the sensors on and near the mast back to the panel truck 40 meters d i s tan t which contains the tape recorder and a l l s i g n a l condi t ion ing equipment. Power i s obtained from two cables l a i d from the hunting shack on shore^to the truck. On the mast are three cup anemometers at heights of 4 .5 , 2.0, and 0.5 meters (the middle anemometer did not funct ion r e l i a b l y and r e s u l t s from i t were not used). Also on the mast, at a height of 4 meters for the p a r t i c u l a r setup show, i s the probe of the hot-wire anemometer. The 190 three pressure instruments are buried w i t h i n 2'meters of the base of the mast and or iented in to the wind. The r o c k e t - l i k e object on the sand i s the c y l i n d e r containing the e l e c t r o n i c s for the E l l i o t t probe. The wind d i r e c t i o n vane i s located halfway between the truck and the mast (the experiments d id not depend for t h e i r v a l i d i t y on an accurate measurement of wind d i r e c t i o n ) . A2.4 C a l i b r a t i o n A l l three pressure sensors were c a l i b r a t e d i n the laboratory with the equipment described i n the "Experiment" sec t ion (p. 58 f f ) . They were i n every case c a l i b r a t e d with the same pressure tubing conf igurat ions used i n the f i e l d . The amplitude and phase c a l i b r a t i o n for the buoy sensor at frequencies from 0.05 to 10 Hz are d isplayed i n Figure 78. These were obtained by s i n u s o i d a l l y vary ing the pressure i n the c a l i b r a -t i o n drum at various frequencies . The buoy sensor s i g n a l was compared with that from a Barocel which had one port at drum pressure and the other at atmospheric pressure . The amplitude and phase of the Barocel s i g n a l began to d i f f e r a measurable amount from the buoy sensor at frequencies above 7 Hz; th is i s assumed to be caused by the Barocel and not the buoy, s ince a reson-ance i s known to occur i n the drum at 45 Hz and there was no reason to suspect a r o l l o f f i n the buoy sensor below 100 Hz (the resonant f r e -quency of i t s forevolume i s 300 Hz) . No c a l i b r a t i o n s were made above 7 Hz. Assuming the buoy sensor to be correc t at high frequencies and the Barocel to be correc t at low frequencies , the frequency response of the drum-driver system i s shown i n F igure 79. 191 A2.5 Data Ana lys i s The data was analysed using the same general methods described i n "Data Analys i s and In terpre ta t ion" . The analog tapes were f i r s t played back in to a s ix -channel chart recorder; sect ions s u i t a b l e for ana lys i s were chosen using as c r i t e r i a the length of time during which a l l s ignals functioned simultaneously and the general steadiness of the "eyeballed" var iance of the s igna l s (as an i n d i c a t i o n of s t a t i o n a r i t y ) . Promising sect ions were marked. These sect ions were then played back at e ight times r e a l speed, f i l t e r e d with c a r e f u l l y phase-matched low-pass f i l t e r s (to prevent a l i a s i n g ) , and d i g i t i z e d at a sampling rate of 500 samples per second on the I n s t i t u t e A/D converter (62.5 Hz "real - t ime" data sampling r a t e ) . The r e s u l t i n g d i g i t a l tapes were checked for p a r i t y errors and other l i k e l y d i g i t a l f a i l u r e s . I f they proved s u i t a b l e they were then analysed with the Fast F o u r i e r Transform package developed at the I n s t i t u t e . The computer output from this package gave power spectra and cross spectra i n c l u d i n g Co, Qu, Phase and Coherence for a l l s ignals analysed (see "Data Ana lys i s and In terpre ta t ion") . The data was F o u r i e r transformed i n blocks of 2048 points each (32.7 seconds of r e a l t ime). The f i n a l s p e c t r a l estimates were averages over the t o t a l number of data blocks i n a given run. Also pr in ted out were standard error of the mean and average trend for each s p e c t r a l estimate and c r o s s - s p e c t r a l estimate at each frequency. For th i s ana lys i s s p e c t r a l dens i t i e s at i n d i v i d u a l frequencies were averaged over roughly logar i thmic (approxi -mately \ octave) bandwidths. 192 A2.6 Results A2.6 .1 Summary of Relevant Information A summary of information re levant to the f i e l d runs i s presented i n Table A 2 . 1 . The r e s u l t s from the data of these runs w i l l be presented i n three sec t ions . In the f i r s t the determination of the " f r i c t i o n v e l o c i t y " u^ from the hot-wire data i s descr ibed; th is permits normal iza-t i o n of pressure spectra into a form su i tab le for comparison with other observat ions . In the second sec t ion power spectra of the s igna l s from the three pressure sensors are compared. These power spectra are a lso compared with those obtained by other i n v e s t i g a t o r s . In the t h i r d sec t ion the cros s - spec tra of the buoy sensor data with the data from the two pressure sensors and from the hot-wire anemometer are presented. A2 .6 .2 The Hot-Wire Data For a l l of the runs simultaneous hot-wire anemometer data were a v a i l a b l e . Power spectra were computed for the data taken, and i n some cases cross spectra with the pressure data were also computed. The power spectra are discussed f i r s t . downwind speed f luc tua t ions u , i s p lo t ted versus log (kz) , where z i s a hot-wire he ight , for one of the runs analysed. I t i s considered to be t y p i c a l of a l l the v e l o c i t y spectra obtained. The l i n e f i t t e d to the points i s of slope -5 /3; as for a l l the v e l o c i t y spectra this l i n e f i t s the points very w e l l over one decade of frequency. These spectra have been used to obta in a value for u , the " f r i c t i o n In F igure 80 log the log of the spectrum of turbulent 193 TABLE A2.1 Information Summary for Boundary Bay Runs:September, 1968 Wind Speed Date Time at 4m Wind H o r i z o n t a l Cloud (Sept. of (cm D i r e c t - Height Separation Cover tun 1968) day sec - 1) ion Sensor (cm) (cm) (tenths) totes: (1) (2) (3) l a 4 1530 390 NW Pa 30 0 0.0 Pb (ground) 0 5 cw u 1 400 200 cw 5a 27? 1430 310 NW Pa 30 60 cw 0.2 Pb (buoy) 2 60 cw u' 30 50 cw 5b 27 1500 340 NW Pa 30 60 cw 0.2 Pb (buoy) 2 60 cw u 1 30 50 cw 5c 27 1600 400 NW Pa 30 60 dw 0.2 Pb (buoy) 2 60 dw u' 30 60 dw, 10 cw 5d 27 1700 420 NW Pa 30 60 dw 0.2 Pb (buoy) 2 60 dw u' 30 60 dw, 10 cw 5e 27 1730 450 NW Pa 100 60 dw 0.2 Pb (buoy) 2 60 dw u' 100 60 dw, 10 cw Notes: 1. Mean wind speed measured with cup anemometers at heights of 50 and 400 cm. 2. p a = E l l i o t t a i r pressure sensor; pb = buoy pressure sensor (on ground or i n buoy on ground); u' = hot-wire anemometer. 3. cw = crosswind from reference pressure sensor; dw = downwind from reference pressure sensor. 194 v e l o c i t y " (see, for instance , Lumley and Panofsky (1964) ) from the value of CJ)uu.(k) , i n the k - ^ ^ 3 region of the spectrum. I t was shown by Taylor i n 1960 (see, for example, Weiler and B u r l i n g (1967) ) that u 2 - (mz)2/3 . k^/3 d ) u a ( k ) > v u ; ~i - 1.13 ^ ) 2 / 3 f 5 / 3 0 u a < f ) . where K 2i 0.4 i s Von Karman's constant, z i s the height of the anemometer and U i s the mean wind speed at that height; = 0.48 i s Kolmogorov's u n i v e r s a l constant for a i r i n the Universa l I n e r t i a l Subrange of turbu-lence scales (the value of i s that given i n Pond, Stewart, and B u r l i n g (1963) ); k = 2 T T = 21T f X U i s the wavenumber and X the scale s i z e of the turbulent eddies . The l a s t statement i s the "Frozen Turbulence" assumption, or T a y l o r ' s hypo-thes i s . o In the arguments leading to the expression for u, v i s inc luded the assumption that i n the I n e r t i a l Subrange under d i s cuss ion the rate of product ion of mechanical energy i s equal to the rate of energy d i s s i p a t i o n . For the Boundary Bay runs no record of the mean temperature p r o f i l e or of the mean humidity p r o f i l e was kept. Both of these quant i t i e s a f f ec t the buoyancy of the a i r and hence the ra te of product ion of energy by convect ion. The above assumption excludes convective energy; therefore the ca l cu la ted value u^ may be i n e r r o r . From the sparse temperature measurements taken i t was concluded that buoyancy ef fects were small 195 (see Lumley and Panofsky (1964), p. 104) compared with other errors,, for instance those due to d r i f t s i n the hot-wire c a l i b r a t i o n . The hot-wire wind speed c a l i b r a t i o n for these experiments i s poor; i t i s probably only known to w i t h i n T307o of i t s assumed value for the determination of The r e s u l t s of the analys i s jus t described are presented i n Table A2 .2 . They have been used i n the non-dimensional spectra of pressure versus wave number discussed below,(p.198 f f ) . A2 .6 .3 The Pressure Power Spectra Pressure spectra for runs l a and 5a (see Table A 2 . 1 ) , are presented i n Figures 81 and 82. On each f i gure are superimposed a set of three pressure spec tra , one for each of the three sensors being compared. The f i r s t set i s t y p i c a l of spectra found with the buoy sensor i n the "case 1" conf igurat ion (pressure tap f l u s h with sand); the second i s t y p i c a l for the "case 2" conf igurat ion (pressure tap on buoy, " f loa t ing" on the sand). The two f igures show the comparison of the three sensors on the two separate days on which runs were taken which are used i n th i s Appendix. The spectra i n F igure 81 are from the e a r l i e r of the two days, for which a l l gains and c a l i b r a t i o n s were well-known. The buoy sensor spectrum i n F igure 82 i s the f i r s t of f i v e spectra taken on the second of the two days; for a l l of these f i v e runs the p a r t i c u l a r gain s e t t i n g of one amplifier-.was not noted c o r r e c t l y . Upon analys i s the buoy sensor s p e c t r a l dens i t i e s for a l l f i v e runs were found to be higher by a fac tor of two than those of the other two sensors. For th i s reason the buoy 196 Run TABLE A2.2 Summary of Results of u, v A n a l y s i s on Hot-wire Anemometer Spectra Wind. Speed at hot-Time wire Wind Hot-wire H - W - ^ u ^ z ^ of height D i r e c t - height (<mzsec"z) day (cm s e c _ l ) ion (cm) Hz (cm sec--'-) (cm) (x 10~ 3) Notes: (1) (2) l a 1530 375 NW 400: 9.5 30.5 2.4 6.1 5a 1430 255 NW 30 8.1 18.0 0.1 4.9 5b 1500 280 NW 30 8.1 17.8 0.05 4.0 5c 1600 340 NW 30 13.8 22.3 0.07 4.3 5d 1700 350 NW 30 18. 23.9 0.09 4.7 5e 1730 370 NW 100 8.9 25.2 0.07 4.7 Notes: 1. zQ = z exp | -KU ^ i s the "roughness length", and 2 2. = u* i s the "drag coe f f i c i en t" for the boundary layer . U2" 197 sensor s p e c t r a l dens i t i e s for a l l f i v e runs taken on the second day have been halved to b r i n g them in to accord with those of the other sensors. The curves, a f t er the above-mentioned adjustments have been made to the buoy sensor spectrum i n F igure 82, f a l l w i t h i n the l i m i t s of c a l i b r a t i o n errors (see F igure 78) with the exception i n the "case 2" conf igurat ion of the estimates at frequencies above 2 Hz for the buoy sensor. T y p i c a l l y the buoy sensor showed higher s p e c t r a l values at frequencies less than 0.05 Hz i n a l l runs. The cause of the former discrepancy i s not understood. The p o s s i -b i l i t y ex i s t s that the extra energy may be assoc iated with a low-q eddy shedding from the s ides of the buoy. I f the buoy were assumed to be a v e r t i c a l cy l inder of i n f i n i t e he ight , then an eddy-shedding f r e -quency can be ca l cu la ted from f = ^ H , where S, the Strouhal number, i s D about 0.2 for such a c y l i n d e r , D i s the cy l inder diameter and U the wind speed. I f the log p r o f i l e for run 5a i s extrapolated to obta in a wind speed at 5 cm and D i s taken as the buoy diameter (23 cm), then f * 0.2 x 100 / 23 * 1 Hz. Another p o s s i b i l i t y i s that the buoy s i g n a l i s contaminated with some f r a c t i o n of the dynamic pressures associated with the a i r flow over the buoy. This would occur i f the prov i s ions made for r e j e c t i n g the dynamic pressures were not c o r r e c t l y set up. Their e f f ec t would be "broadband"; that i s they would be larges t where u was l a r g e s t , and would push the slope of the buoy sensor power spectrum compared with the 198 spectra from the other sensors towards -~'/3, which i s expected for the v e l o c i t y spectrum. This i s not observed; therefore dynamic pressures do not appear to be the cause of the high-frequency "hump" being d i s -cussed. They may be the cause of some other d i screpanc ies , which w i l l be mentioned i n the d i scuss ion of the buoy sensor - hot wire c o r r e l a t i o n s (p.209). The "extra" energy at low frequencies i s d i r e c t l y a t t r i b u t a b l e to super ior thermal i n s u l a t i o n of the backup volume of the E l l i o t t - d e s i g n e d reference sensor (the thermal noise of the E l l i o t t a i r probe also appears to be somewhat higher than that of the reference sensor) . The noise i s almost c e r t a i n l y caused by temperature-induced pressure f l u c t u a t i o n s i n the backup volume of the buoy sensor; such temperature s e n s i t i v i t y had been observed before and during the experiment. I t s p r i n c i p a l e f f ec t i s d r i f t , which because of the "block" s p e c t r a l window (see Appendix 1) i s spread over the low frequencies . Neither of these two discrepancies i s l i k e l y to have caused s i g n i f i c a n t errors i n the r e s u l t s obtained with the buoy over the water, s ince they are i n parts of the spectrum where r e l a t i v e l y l i t t l e energy was present i n any of the wave spectra . The low-frequency "noise" would be incoherent with the waves, and would be kept low by the much smaller temperature v a r i a t i o n s encountered by the buoy on water. Perhaps the most s i g n i f i c a n t error w i l l be i n the observed values of p 2 , which w i l l be high i f the "thermal noise" i s high ( th i s amounts to a fac tor of about 1.5 over the reference sensor i n the Boundary Bay runs) . A2 .6 .4 Non-Dimensional Pressure Spectra As was mentioned i n the sec t ion on v e l o c i t y spectra (p. 192), they 199 were used to obta in estimates of the f r i c t i o n v e l o c i t y u which i n turn were used to produce non-dimensional p lots of the pressure spectra . Because of uncer ta in t i e s i n the hot-wire c a l i b r a t i o n s , th i s was done only for runs 5a - 5e, which were r a i l taken on the same day. The assumption therefore i s that the c a l i b r a t i o n of the hot-wire d id not change during, the three-hour period over which the runs were taken. A p l o t of k ^)pp(k) / versus log (k) i s presented i n Figure 83; the spectra shown are a l l from the buoy sensor. I t should be noted that the "frozen turbulence" hypothesis i s invoked i n determing k, and that the v e l o c i t y at 50 cm height has been used i n this determination as a s c a l i n g f a c t o r . Since the "frozen turbulence" hypothesis i s probably not correc t for any sca l e , a bet ter non-dimensional p l o t i s probably not worth seeking unless the "advection" v e l o c i t y U a ( j of the pressure f l u c t u a t i o n s at each frequency i s known, so that a "k" may be ca l cu la ted for each "f" from k = 2 t r f / u a d to put into the non-dimensional formula. A2 .6 .5 Comparison of Power Spectra with Other Measurements Gossard (1960) reports spectra of pressure f luc tua t ions i n the f r e -quency range 10 - 6 - 10 Hz. He reports a -2 slope (on a l o g - l o g p lo t ) i n the frequency range from 0.5 - 5 Hz. The slopes i n th is range i n the Boundary Bay r e s u l t s are roughly -4/3 and are thus at variance with Gossard's r e s u l t s . His high-frequency spec tra , as f ar as can be i n f e r r e d from h i s paper, were obtained with a microbarograph i n the shape of a cy l inder one foot i n diameter, located i n a well-exposed l o c a t i o n on a 200 tower. With th i s conf igurat ion i t i s hard to see how the instrument was not measuring a large percentage of the t o t a l head f luc tuat ions ^ f<t (see p. 42 ) , which would lead one to expect a slope of -5/3 for h i s r e s u l t s . This i s not too far from h i s reported -6/3 slope over one decade. P r i e s t l e y (1965) c o l l e c t e d many pressure spectra on the ground beneath the atmospheric boundary l a y e r . He also measured l o n g i t u d i n a l and l a t e r a l space-time c o r r e l a t i o n s i n the pressure f i e l d . His power spectra were i n most cases quite l i n e a r on a l o g - l o g p l o t , with slopes vary ing from -1.6 to -2.4 at frequencies above 0.06 Hz. The slopes appeared to be weakly r e l a t e d to so lar heat ing . Below these frequencies , the curves sloped less s teeply than the slopes just quoted. P r i e s t l e y s h i f t e d a l l h i s spectra v e r t i c a l l y , matching them i n the frequency range .03 - .35 Hz; he then used the amount of v e r t i c a l s h i f t as an "absolute power densi ty s c a l i n g fac tor" , by which each spectrum must be m u l t i p l i e d to give the ac tua l value of p . These s c a l i n g fac tors are p lo t ted by Snyder and Cox (1966) against wind speed at some f ixed height , and are found to fo l low a U 4 power law f a i r l y c l o s e l y . The same technique was appl ied to the present data, f i t t e d to P r i e s t l e y ' s matched spectra . The r e s u l t i n g s c a l i n g fac tors fol lowed a U 4 l i n e as w e l l , but the s c a l i n g fac tors were larger than P r i e s t l e y ' s by a fac tor of about 1.5. This d i f ference might be explained by the f a c t that P r i e s t l e y ' s mean U was measured at a height of 10 meters while U was measured at 4 meters at Boundary Bay. The pressure spectra observed i n wind tunnel boundary layers are not e a s i l y compared with the present data. Wil lmarth and Wooldridge 201 (1962), for ins tance , provide dimensionless pressure power spectra for f * which the s c a l i n g parameters are o , the displacement thickness of the 9 2 boundary l a y e r , U«» , the free-stream v e l o c i t y , and q = % f aU«, , the f ree -stream dynamic pressure . None of these i s known for the atmospheric boundary layer under study. I f guesses for them are made, the shape of the spectra could be compared. Reasonable values for Boundary Bay might be S* * 0 . 1 5 2s 0.1 x 2 x 10 4 cm = 2 x 10 3 cm, U « ~ 30u* 900 cm/sec, and q 2 = \ ^ J J 2 ~ 0.5 x 1.23 x 10" 3 x 81 x 10 4 = 500 dyne cm Here § , the thickness of the atmospheric boundary l a y e r , i s taken to be 200 m. I f the Boundary Bay runs are normalized i n this way, and i f Corcos' (1963) correct ions are appl ied to the Wil lmarth and Wooldridge spec tra , t h e i r slope i s found to be - 1 , as compared with the - 4 / 3 slope commonly occurr ing i n the present data. Some recent r e s u l t s published by Gorshkov (1967) are d i f f i c u l t to f i t into any pa t t ern . When scaled to P r i e s t l e y ' s curves the r e s u l t i n g s c a l i n g factors do not have a -4 slope when p lo t t ed against wind speed (on a l o g - l o g p l o t ) . The spectra i n some cases are s t r a i g h t l ine s (slopes vary ing from -1 .8 to -2 .3 ) ; i n other cases they cons i s t of two l i n e s , the higher-frequency part having the higher negative slope; and i n other r a r e r cases they contain "jogs" s i m i l a r to those seen i n some of the Boundary Bay spectra (see Figure 82 at 0.1 Hz) . Gorshkov's spectra are cons i s t en t ly higher at low frequencies than any other r e s u l t s . The best that can be sa id i s that a few of the spectra roughly agree with 202 some of the Boundary Bay spectra . A2 .6 .6 The Pressure Cross -Spectra Information w i l l be presented from three sets of cross - spec tra : between buoy sensor pressure and E l l i o t t a i r sensor pressure , between buoy sensor pressure and reference sensor pressure; and between buoy sensor pressure and hot-wire v e l o c i t y . The r e s u l t s w i l l be given i n terms of coherence and phase. Coherences for a l l runs from the cross-spectrum between the buoy sensor and the a i r sensor are shown i n F igure 84. In a l l cases the a i r sensor was d i r e c t l y above the buoy sensor; i t was at a height of 30 cm i n a l l runs except 5e, when i t was at 100 cm. The genera l ly lower coherences for run 5 are p o s s i b l y due to the presence of larger s i g n a l d r i f t s than those encountered i n run l a , which i n the case of run 5 prevented the f u l l use being made of the dynamic range of the tape recorder; however, the feature might be associated with the greater drag c o e f f i c i e n t and roughness apparently p r e v a i l i n g during run l a (see Table A2 .2 ) . This would be p a r t i c u l a r l y l i k e l y i f condit ions were more unstable during th i s run. The e f f ec t on the coherence of the larger height of the a i r sensor i n run 5e i s shown i n s t r i k i n g fash ion . This f a l l o f f i n coherence i s r e l a t e d to the lowest scale s i ze of the turbulent eddies near one height which generate pressures at the ground. By making some rather broad assumptions a v e r t i c a l scale s i ze of these eddies can be c a l c u l a t e d . The c a l c u l a t i o n w i l l involve f i n d i n g the frequency ( f | ) at which 203 the coherence i n F igure 84 drops to one-half of i t s low-frequency value for each of the two he ights , 30 and 100 cm, and r e l a t i n g th i s to a sca le s i ze ( ^ t h r o u g h the "frozen turbulence" hypothesis . This i s a rather dubious c a l c u l a t i o n at best; however i t i s expected to y i e l d reasonable q u a l i t a t i v e answers. Scale s izes found th is way can be compared with those found by assuming that at the "half-coherence" frequency the height of the a i r sensor i s one-half the wavelength ( A m = 2z^) of the pressure-generating eddy above the ground sensor. When th is i s done the r e s u l t s are shown i n Table A 2 . 3 . I t seems that the v e r t i c a l scale s i z e of the pressure-generat ing eddies i s between one-half and one times as large as that predic ted from the frozen turbu-lence hypothesis and the assumption concerning the "half-coherence" wave-length . The phase r e l a t i o n between the buoy sensor and the a i r sensor ( d i r e c t l y above i t ) i s presented i n F igure 85 for a l l runs. The phase agreement i s considered good, the small d iscrepancies at frequencies above 2 Hz probably being associated with inaccurac ies i n the c a l i b r a t i o n system (see F igure 78) and the genera l ly low coherences at these f r e -quencies (Figure 84). The coherences between the buoy sensor and the reference sensor are shown i n F igure 86; the r e l a t i v e pos i t ions of the sensors for these runs are given i n Table A 2 . 1 . The coherences show a d e f i n i t e tendency to f a l l o f f at low frequencies . This would be expected because of the greater low-frequency "thermal noise" of the buoy sensor. The high coherences for run l a are p a r t l y due to the c loser proximity 3- 'he 204 TABLE A2.3 Advect ion V e l o c i t y of Pressure-Generat ing Eddies Ik-Run Z A u 4 u A U M £% U a d Am U H A / f ^ ites: (1) (2) (3) (4) (5) (6) (7) (8) l a 30 390 30 200 3.5 - 60 55 5a 30 310 18 190 1.8 - 60 100 5c 30 400 22 260 2.6 310 60 100 5e 100 450 25 360 1.2 340 200 400 Notes: 1. Z A = height of a i r sensor (cm). 2. U 4 = mean wind speed at 4 m (cm sec ^). 3. u * = f r i c t i o n v e l o c i t y . 4. U HA = mean wind speed at a i r sensor (cm sec ^; extropolated from log p r o f i l e ) . 5. frequency at which coherence drops to \ i t s low f r e -quency va lue . 6. ' U a d = measured advection v e l o c i t y (see p. 207). 7. ^m 2Z. ; see text . A 8. ^ ^ = wavelength obtained from "frozen turbulence" hypothesis at f, . 205 of the sensors i n run l a than i n run 5, and p a r t l y due to the much larger d r i f t s encountered i n run 5. The e f f ec t of separat ing the sensors by various amounts i n the crosswind and downwind d i r e c t i o n s i s shown d r a m a t i c a l l y . The p r i n c i p a l inference to be made from this i s that the l a t e r a l s p a t i a l c o r r e l a t i o n c o e f f i c i e n t f a l l s o f f more r a p i d l y than does the l o n g i t u d i n a l one. This has been observed prev ious ly both i n wind tunnels (Willmarth and Wooldridge, 1962) and i n f i e l d studies (Pr i e s t l ey ,1965) . P r i e s t l e y (compare h i s F igure 8) finds that i f C and D are r e s p e c t i v e l y crosswind and downwind sca le s izes , then C 21 0.8 D ^ - ^ . For a frequency of 0.5 Hz the "frozen turbulence" hypothesis gives a downwind scale s i ze of 700 cm; the above formula then gives 80 cm for the crosswind "scale size" ( scat ter i n P r i e s t l e y ' s data could make th i s as low as 55 cm, however). In F igure 11B coherences for 60 cm crosswind separations have dropped to about one-half of the coherences for 5 cm crosswind separations at 0.5 Hz; th i s ind icates that the crosswind scale s i ze i s about 60 cm. The r e s u l t s , then, are i n rough agreement with P r i e s t l e y ' s conclusions on th is po int . The corrected phase lead of the buoy sensor over the reference sensor is shown i n F igure 87. At low frequencies the buoy i s seen to lag the reference sensor by an amount which decreases with increas ing frequency. The phase of the a i r sensor (not shown) shows the same tendency. This cannot be explained by the hypothesis that the i n s t r u -ments were misal igned with respect to the wind, s ince i n th i s case the phase lag would increase with frequency. 206 The only p o s s i b i l i t i e s that remain are e i ther (a) that the h igh-pass pneumatic f i l t e r i n the reference sensor (see p. 188) changed i t s c h a r a c t e r i s t i c s a f ter c a l i b r a t i o n , or (b) that the c a l i b r a t i o n s were inaccurate . C a l i b r a t i o n s of the reference sensor were performed both before the experiment and af ter i t s completion, and the phase character-i s t i c of the reference sensor f i l t e r was i n fac t observed to have changed between the two c a l i b r a t i o n s . The correct ions introduced by the second c a l i b r a t i o n , were such that they brought the phase between the buoy sensor and the a i r sensor in to c loser agreement, but caused larger d i s -crepancies i n the buoy sensor-reference sensor phase. Thus case (a) i s not thought to be the l i k e l y cause of the discrepancy. The l i k e l i h o o d of case (b) i s best seen by examining the standard errors i n the phases from c a l i b r a t i o n s and comparing these with the sca t ter i n the r e s u l t s . The s tandard .error i n the determination of phase by the c a l i b r a t i o n i s + 1 0 ° ; th is includes instrumental errors as w e l l as observed sca t t er of the points about the c a l i b r a t i o n curve. The standard d e v i a t i o n of the s p e c t r a l dens i t i e s about the mean phase curve i n F igure 12B i s " i l 0 ° . Thus measurements from the buoy and reference sensors l i e w i th in each others' standard errors except at the three lowest frequencies . The high-frequency phase behaviour of the two instruments i s i n t e r e s t -ing . The phase d i f ferences remain around zero for those runs where the instruments were crosswind from each other , but i n the runs where the buoy sensor was downwind from the reference sensor buoy pressures lag the reference sensor pressures by increas ing amounts as the frequency increases . The d i f ferences between the three curves for downwind separa-207 t ions i s not s t a t i s t i c a l l y s i g n i f i c a n t , but the consistency of the curves suggests that the phase i s smaller at high wind speeds than at low speeds. For the low wind-speed case (run 5c) , the phase changes near ly l i n e a r l y at a mean rate of about - 7 0 ° per Hz. For a downwind separat ion of 60 cm th i s gives a broadband "advection v e l o c i t y " (U a ( j ^ 3 6 0 ° - ( A x ) / ( d Q / d f ) , where A x i s downwind separat ion and 98 /5 f i s the ra te of change of phase with frequency) of the pressure f l u c t u a -t ions of 360 x 60 210 cm s e c - ! ; the high wind speed case gives 6 3 ° 70 per Hz, or anMadvection v e l o c i t y of 360 x 60 /v, 340 cm sec"-*" (run 5e) . 63 To see i f these values are cons is tent with those quoted by Wil lmarth and Wooldridge (1962; p. 206) i t i s necessary to sca le the present measure-ments, s ince they quote t h e i r r e s u l t s as applying over ranges of the dimensionless f requency C J S * / U O O , where <$* and Uoo are defined as before (p.201). For these cases, Cs 2 x 103 cm (as be fore ) , and Uoo — 700 cm sec"-*- from the logar i thmic p r o f i l e ; the angular frequency to ranges from 2.5 to 12.5. This g ives , for the present cond i t ions , 7 ^^^/Uoo 4 36. For dimensionless frequencies between 0.4 and 0.95 Wil lmarth and Wooldridge obta in 0.83 U,p for an advect ion v e l o c i t y ; they get 0.69 U w for 4 . 0 i « 6 / U ^ 7.0. For the present two runs (U,*, « 700 and 740 cm sec"^ for runs 5c and 5e r e s p e c t i v e l y ) , the r a t i o s are 0.45 and 0.46. Considering the large expected error i n the phase measurements, the broad assumptions made, and the f a c t that the dimensionless frequencies here are higher than those of Wi l lmarth and Wooldridge,who obta in a value of 0.69 for the r a t i o , the agreement i s good; c e r t a i n l y the comparison with the wind tunnel work 208 should not be taken as any more than q u a l i t a t i v e . Unfortunate ly , the r e s u l t s presented here are not d i r e c t l y comparable with those of P r i e s t l e y (1965); h i s advect ion v e l o c i t i e s are r e l a t e d to measured wavelengths (his wind speed measurements were minimal) , while those presented here are broadband and are re la t ed to mean wind speed. A2 .6 .7 Pressure - V e l o c i t y Corre la t ions The coherence and phase of the buoy sensor s igna l with that from the hot-wire located e i ther 30 or 100 cm above i t are shown i n Figures 88 and 89. These r e s u l t s show marked d i f ferences from the coherence and phase (shown i n Figures 90 and 91) between the a i r sensor and the hot-wire ("these were both the same d i s tance , 30for 100 cm, above the buoy sensor; the hot-wire was 10 cm crosswind from the a i r sensor) . The coherence between the reference sensor (on the ground, 60 cm upwind of the buoy sensor) and the wind i s shown i n Figure 92. At frequencies below 1 Hz the buoy sensor shows coherences higher on the average than those of the a i r sensor. This d i f f erence increases with decreasing frequency, poss ib ly to a maximum of 0.2 at 0.05 Hz; however the low points at the lowest frequency may be af fected by "thermal noise" i n the buoy sensor. At low frequencies the phase between the two pressure instruments i s zero (Figure 85) but whereas the phase of the buoy sensor remains at 0 ° r e l a t i v e to the wind f luc tua t ions u n t i l 1 Hz, the phase lead of the a i r sensor r e l a t i v e to the (nearby) wind sensor r i s e s to 9 0 ° at 1 Hz. At frequencies above 2 Hz the phases of both pressure instruments r e l a t i v e to the wind are roughly the same, but both are scat tered and the coherence i s smal l . 209 Because of the phys i ca l separat ion of the three sensors (buoy pressure, a i r pressure, h o t - w i r e ) , i t i s only poss ib le to make i n f e r -ences about the d i f ferences between these corre la t ions of each pressure sensor with the wind at. low frequenc ies - - say less than 0.5 Hz. The most s t r i k i n g d i f ference i s that the low-frequency coherence between the a i r sensor and the nearby wind sensor i s 0.1 - 0 .3 , while that of the buoy sensor with the wind sensor i s 0.35 - 0.6. This would seem to i n d i c a t e that 50 - 1007„ of the l a t t e r coherence ar i ses through dynamic pressures ( i . e . noise) at the buoy. Since the s ignals from both sensors are i n phase with the hot-wire at low frequencies any such "noise" must have been i n phase with the hot -wire . For the Boundary Bay runs, a dummy wave probe was inadver tent ly not placed i n the buoy as i t was for the wind tunnel tests of the design i.and c a l i b r a t i o n of the buoy; i t i s known that i t s absence would cause a disturbance at the pressure port of -1.5% of the stagnation pressure at the bow of the buoy i f p o t e n t i a l flow i s assumed. This would cause the phase of the r e s u l t -ant s i g n a l observed by the buoy to migrate towards 1 8 0 ° , but th i s i s not observed i n F igure 89. Therefore another cause must be sought for th is phenomenon. I t was also found that the pressure tap was mis-pos i t ioned on the buoy surface , being about 2.5 mm too close to the r i n g (see F igure 13). I f i t i s assumed that a 3 mm error was made th is would cause a contamination of about +3%,, th i s time i n phase with the wind speed. The r e s u l t a n t maximum expected dynamic pressure contamination i s 1.5 ~ 2%. The next step is to make an estimate of the r a t i o of dynamic to s t a t i c pressures i n the flow over the buoy. At 0.06 Hz i n run 5e, the rms s t a t i c pressure amplitude measured by the reference sensor (over 210 a 0.03 Hz bandwidth) was about 1.2 dyne cm" 2; the corresponding rms turbulent v e l o c i t y amplitude was 20 cm sec"^. Therefore the maximum expected dynamic pressure was \ P^ U ~J u 1 2 - 5 dyne c m - 2 (U at 4 meters height was 450 cm s e c - l ) ; th i s i s f i v e times the s t a t i c pressures observed. Thus a 2% contaminating dynamic pressure i s equivalent to a "noise" which i s roughly 10% of the measured s t a t i c pressures . This i s lower by at l eas t a f a c t o r of 5 than that required to produce the observed excess coherence between the buoy pressure and the wind 60 cm above i t . Further evidence of the r e a l i t y of the "high" coherences between the buoy pressure and the wind i s provided i n F igure 92, which shows the coherences between the reference sensor (on the ground) and the wind; these coherences are also higher than the coherences between the a i r sensor and the wind (although they are somewhat lower than the coherences of the buoy sensor pressure with the wind). i A2.6 .7a Comparison with Eavre et a l (1957) I t i s therefore necessary to consider the p o s s i b i l i t y that a v e r t i c a l pressure gradient e x i s t s , which may be r e l a t e d to v e r t i c a l acce lerat ions i n the flow between the sensors. Such v e r t i c a l acce lerat ions appear to have been observed i n boundary layer flows i n a wind tunnel by Eavre e_t al (1957). In t h e i r experiment they inserted two hot-wire anemometers into a boundary layer which was grown from n a t u r a l t r a n s i t i o n on a f l a t glass p l a t e , and measured c o r r e l a t i o n s between the downwind v e l o c i t y f luc tuat ions at various v e r t i c a l , l o n g i t u d i n a l , and l a t e r a l spacings. They found that for a given downstream separat ion A x and given time delay A t there i s 211 one v e r t i c a l separat ion A z for which the c o r r e l a t i o n i s a maximum; by vary ing both A t and A z there i s one combination of A t and A z for each A x at which the c o r r e l a t i o n i s a greatest maximum. A z for th i s greatest maximum was then p lo t ted against the corresponding A x ; the r e s u l t i n g "curve of maximum c o r r e l a t i o n " was convex to the boundary, reach-ing i t s lowest height at the l o c a t i o n of the f ixed sensor c lose to the boundary and curving upwards i n the up-and-downstream d i r e c t i o n s . This curvature may i n d i c a t e that i n the mean, v e r t i c a l acce lerat ions ex is ted i n t h e i r flow which caused downward-moving eddies to be decelerated and upward-moving ones to be acce lerated . This argument i s of course an over-s i m p l i f i c a t i o n ; t h e i r r e s u l t s are broadband, s ince the acce lerat ions involved larger and larger scales as the sensor separations increased. Nevertheless an attempt has been made by the present w r i t e r to estimate the s i ze of these a c c e l e r a t i o n s . The boundary layers i n the wind tunnel and at Boundary Bay were scaled by choosing r e s u l t s for the same range of the dimensionless frequency w S/U.O , which i n Favre, et a l ranges from 0 to 400. For the Boundary Bay experiment S was taken to be 2 x 10 4 cm; U n was 740 cm sec"-'- (run 5e) . These values of S and U,» correspond to a frequency range of 0 to 2 Hz i n the Boundary Bay spectra . The curvature of the "curve of maximum c o r r e l a t i o n " i n Favre e_t al was measured from t h e i r F igure 7 and a mean v e r t i c a l a c c e l e r a t i o n estimated from t h i s ; when scaled to the case of run 5e ( a i r sensor 100 cm d i r e c t l y above the buoy sensor on the ground, hot-wire at the same height and 10 cm crosswind) the v e r t i c a l a c c e l e r a t i o n amounted to -3 cm s e c - 2 . The a c c e l e r a t i o n had to be extrapolated from the r e s u l t s i n Favre e t ' .a l , s ince the lowest height at which they measured (that of t h e i r Figure 7) scaled to a 600 212 cm height at Boundary Bay. In order to compare the Boundary Bay r e s u l t s with the acce lerat ions estimated from Favre et a l the vector d i f ferences between the pressure, wind speed cross - spec tra at the two l eve l s were determined. The ampli -tudes >/co 2 + Q u 2 of the cross - spec tra between pressure and wind were drawn on a phasor diagram at angles to the r e a l axis given by t h e i r phase ( 8 = t a n ' l (Qu/Co) ) , and the vector d i f f erence taken between them at each frequency. Then the rms downwind v e l o c i t y f luc tua t ions (u) from the hot-wire spectra were d iv ided into the magnitudes A (lju) of the r e s u l t i n g d i f ferences to give A p as a funct ion of frequency; these were then summed from 0 - 2 Hz. The r e s u l t s of these c a l c u l a t i o n s are given i n Table A2.4 . - 2 The sum of the pressure d i f ferences i s 1.7 dyne cm" ... • Over a height d i f f erence of 100 cm this gives a v e r t i c a l pressure gradient of 1.7 x - 2 - 2 -1 10 dyne cm cm , which would produce a " t y p i c a l " v e r t i c a l a c c e l e r a t i o n ^ w = - I d p -14 cm sec" 2 A2 1 S t ? d z This f i g u r e , although larger than the -3 cm sec r e s u l t extrapolated from the wind tunnel work, i s not an order of magnitude l a r g e r . I t seems poss ib le therefore that the observed d i f f erence between the pressure, wind speed cross - spec tra at the two heights i s r e a l , and i s not associated with i n c o r r e c t sensing by one of the instruments. I t i s i n t e r e s t i n g to observe that the phase angles (Table A2 . 4 ) between the parts of the pressures coherent with u at the two heights are ne i ther at 1 8 0 ° or 9 0 ° , but are almost a l l confined to the region 213 between these two va lues . This f a i r l y t ight grouping i s rather remark-able i n view of the already small pressure, v e l o c i t y coherences; i f i t i s r e a l , then the phase r e s o l u t i o n of the instruments i s better than might be gathered from the scat ter i n the d i r e c t phase comparisons (see F igure Another method of comparing observed with expected r e s u l t s i s to estimate the v e r t i c a l component of the wind speed w from where the expression i s constant only i f the a i r i s n e u t r a l l y s tab le . Assuming for the sake of argument that neutra l s t a b i l i t y existed for run 5e allows w to be estimated. Panofsky e_t a l (1967) quote a value of 1.2 - 1.3 for the constant for measurements over land; measurements made at th i s I n s t i t u t e (Miyake e_t al, 1969) i n d i c a t e that the constant i s 1.4 for measurements over the sea. For run 5e u, v i s 25 cm/sec; using 1.25 for the constant gives — 30 cm s e c ' l . The "observed" v e r t i c a l a c c e l e r a t i o n of 14 cm s e c - z would reduce th i s to zero i n roughly 2.1 seconds; i n th is time an "eddy" at 100 cm height would t r a v e l h o r i z o n t a l l y about 10 meters. A l l of these values are appropriate for the flow. Both of these discuss ions i n d i c a t e that v e r t i c a l acce lerat ions near the ground can account for the observed di f ferences i n pH coherence at the two l e v e l s . The main f i n d i n g of th i s p o r t i o n of the i n v e s t i g a t i o n i s therefore that there probably ex i s t s a v e r t i c a l pressure gradient near the ground i n the atmospheric boundary l a y e r , which ex is t s i n response to n a t u r a l v e r t i c a l acce lerat ions i n the flow. The exact s i ze of the gradient and i t s v a r i a t i o n with sca le s i ze are unknown, and 85). / u * = cons tant 214 TABLE A2.4 Calcu la ted Pressure Differences Between A i r and Ground Sensors for Run 5e Frequency Bandwidth (Hz) (Hz) A p u Notes: (1) 0.0264 0.0305 216 0.0591 0.0305 186 0.0903 0.0305 135 0.121 0.0305 91 0.165 0.0610 35 0.227 0.0610 5 0.302 0.0916 12 0.407 0.122 6 0.543 0.153 4 0.723 0.214 5 0.965 0.275 5 1.28 0.366 2 1.71 0.488 0.5 0 Q)uu(f) A p 9 9 1 9 (degrees) (cnr* sec"^ Hz" ) (dyne cm" z) (2) (3) 156 1. .93 X 10 4 -0.25 137 1. .56 X 10 4 -0.25 147 7, .75 X 10 3 -0.25 112 7. ,6 X 10 3 -0.19 116 4. .4 X 10 3 -0.14 94 2. .4 X 10 3 -0.03 71 1. .8 X 10 3 -0.08 118 1. . 1 X 10 3 -0.06 58 8, .9 X 10 2 -0.05 204 5. .2 i : X.'. i o 2 -0.09 145 3. ,8 X 10 2 -0.13 148 2. .4 X 10 2 -0.08 74 1. .5 X 10 2 -0.03 T o t a l -1 .7 Notes: 1. Units of A p u , the vector d i f f erence between the a i r and 9 1 ground p , u cross spec tra , are (dyne c m ) • (cm sec" ) Hz 2. 0 i s the phase of A^Hv 3. A p = ( ApuA / ( t ) u u ) x vBandwidth . 215 deserve fur ther i n v e s t i g a t i o n . A2.7 Conclusions The general conclusions reached are re la t ed to the object ives of the experiment, which were f i r s t , to tes t the two sensors for dynamic pressure "noise" and second, to make inves t iga t ions of the turbulent pressure f i e l d with a view to designing future experiments. A2 .7 .1 Sensor Tests The f i r s t objec t ive has been met, with reservat ions . The two instruments produce power spectra which over the mid-frequency range (0.1 - 1 Hz) are i d e n t i c a l w i t h i n expected e r r o r . The higher s p e c t r a l estimates i n the low-frequency pressure power spectra of the buoy sensor are s a t i s f a c t o r i l y explained as greater "thermal noise" i n that sensor; th i s phenomenon should introduce l i t t l e or no error into the pressure-wave c o r r e l a t i o n experiments. The high-frequency "hump" i n the buoy sensor i s not wel l -understood, but i s at too high a frequency to have much e f f ec t on the pressure-wave c o r r e l a t i o n s . The c r o s s - c o r r e l a t i o n s between pressure sensors behave very w e l l . The only cause for concern l i e s i n the low-frequency phase d i f ferences between the buoy and a i r sensors and the reference sensor (both lag i t by about 2 0 ° ) ; i t i s considered that the d i f ferences are bare ly s i g n i f i -cant at the lowest frequencies ( ^ 0.1 Hz) only , and therefore need not be taken s e r i o u s l y for work over waves. The p r i n c i p a l cause for concern about whether the buoy sensor s igna i s contaminated by dynamic pressures i s i n the c o r r e l a t i o n s with the hot 216 wire v e l o c i t y s i g n a l . The coherences with v e l o c i t y are higher by a fac tor of two for the buoy sensor (30 cm below and 10 cm crosswind from the hot-wire) than they are for the a i r sensor (at the same height as, and 10 cm crosswind from, the ho t -wire ) . I t i s found that the observed coherence d i f ferences can be explained i f i t i s assumed that a v e r t i c a l pressure gradient ex is t s i n the flow between the pressure sensors; there i s some experimental evidence (Favre e_t a l 1957) that a pressure gradient does ex i s t and that i t s magnitude i s approximately that required to exp la in the observat ions . A2 .7 .2 The Structure of the Pressure F i e l d The second objec t ive of the experiment was to inves t iga te the s t ruc ture of the pressure f i e l d and thus lay the groundwork for a more d e f i n i t i v e future experiment. P r i e s t l e y (1965) , i n a ser ies of w e l l -c o n t r o l l e d experiments, has looked at the crosswind and downwind s tructure of the pressure f i e l d i n the atmosphere. The present r e s u l t s s u b s t a n t i a l l y agree with h i s , and the wind tunnel work of Wil lmarth and Wooldridge (1962), although i n the l a t t e r case the comparison i s d i f f i c u l t because the s c a l i n g parameters used i n the wind tunnel are not known for the f i e l d work. The v e r t i c a l s t ruc ture of the pressure f i e l d has never before been s tudied . I t should be noted here that the r e s u l t s quoted concern-ing v e r t i c a l s t ruc ture are from E l l i o t t ' s instrument, and are thus h i s work. They are presented here with h i s permission for completeness only , and the i n t e r p r e t a t i o n s are those of the w r i t e r . The conclusions reached were: 1. The power spectra of f i v e runs , a l l taken on the same day at 217 various wind speeds, are grouped f a i r l y we l l when p l o t t e d i n nondimension-2 a l form i f p u^, the ground shear s t r e s s , i s used as the parameter. This ind icates i t s importance i n determining the pressure f i e l d above the ground. 2. From the v a r i a t i o n i n coherence between a i r and ground sensors with v e r t i c a l separat ion , i t i s i n f e r r e d that the v e r t i c a l scale of the pressure-generat ing eddies i s s l i g h t l y compressed (by a fac tor of two) with respect to the h o r i z o n t a l sca les . 3. From the d i f ferences i n coherence between sensors on the ground depending on whether they are crosswind or downwind from each other , the r a t i o D/C of downwind to crosswind scales i s found to be about ten, which i s i n f a i r l y good agreement with P r i e s t l e y (1965), who found the two scales to be r e l a t e d by C ~ 0.8D ° ' 7 . 4. From the phase r e l a t i o n between sensors on the ground placed downwind from each other , two broadband ..advection v e l o c i t i e s for the pressure eddies were i n f e r r e d . They were i n the r a t i o s of 0.5 and 0.6 to anestimate of U,^ , the "free-stream ve loc i ty" i n the boundary layer over the ground, and showed rough agreement with the wind tunnel work (Corcos 1963). 5. Where the ground sensor and a i r sensor agreed, they showed large phase d i f ferences from the hot-wire s i g n a l at h igh frequencies; these amounted to about 1 8 0 ° at 5 Hz. Since , however, coherences i n both cases were less than 0.1 at high frequencies and hence not s t a t i s t i -c a l l y s i g n i f i c a n t , no inferences were attempted. 6. Coherences of the h o r i z o n t a l component of the turbulent wind 218 speed u with two pressure s i g n a l s , one from a sensor on the ground and one from a sensor i n the a i r 100 cm v e r t i c a l l y above the ground sensors, showed s i g n i f i c a n t d i f f erences . These can be explained as being caused by v e r t i c a l acce lerat ions i n the flow near the ground. The phase r e l a t i o n s among p, u, and w show some order i n the r e s u l t s presented; i t i s f e l t that an experiment c a r e f u l l y designed to map out these phase r e l a t i o n s i s required before the s tructure of the v e r t i c a l a c c e l e r a t i o n and pressure f i e l d s can be f u l l y understood. APPENDIX 3: THE INTERACTION OF SWELL WITH AN OPPOSING WIND A3.1 In troduct ion 219 A3.2 Theory 220 A3.3 The D i r e c t i o n of T r a v e l of the Waves 222 A3 .3 .1 Phase Information 223 A3.4 Phase Correct ions 224 A3.5 Comparison of Observed Amplitudes with P o t e n t i a l Theory 225 A3.6 The "Wave Driven Wind" 228 A3.7 Wave Damping 228 A3.8 Momentum F l u x 231 A3.9 Conclusions 232 APPENDIX 3:: THE INTERACTION OF SWELL WITH AN OPPOSING WIND A3.1 Introduct ion One of the data runs analysed was taken with very low wind speeds (November 17, 1967*: see "Results", Table 6 .1) . The wind speed was so low (100-200 cm/sec) that the water surface was not r u f f l e d . The buoy pressure sensor, the wave probe, and the sonic anemometer were operated throughout the run , g i v i n g simultaneous data on surface pressure and e l e v a t i o n and the three components of the t o t a l wind vec tor . The only water motions present were those caused by swe l l . In the course of the run, a we l l -de f ined group of waves which has an average per iod of four seconds passed through the measurement s i t e . They were presumably the wake from some large , f a i r l y f a s t f r e i g h t e r . : They were noted i n the log of the run , but t h e i r d i r e c t i o n of t r a v e l was unfortunate ly not mentioned and so th i s information was l o s t . When the run was played back l a t e r on a chart recorder the e f fects of th i s group of waves were c l e a r l y v i s i b l e , not only on the pressure and v e r t i c a l v e l o c i t y but a lso on the h o r i z o n t a l v e l o c i t y traces . The object of th is Appendix w i l l be to i n t e r p r e t these traces , and to see how w e l l they agree with f i r s t - o r d e r p o t e n t i a l flow theory (with a b r i e f excursion in to second-order theory, to c a l c u l a t e transport v e l o c i t i e s i n the a i r ) . Energy and momentum inputs to the waves w i l l be ca l cu la ted and from these the wave damping by the wind i s determined. I t 219 220 i s hoped too that excitement may be s t i r r e d i n the heart of a future student, who may then decide to i n v e s t i g a t e th is f a s c i n a t i n g (to me) s i d e l i g h t more thoroughly. A3.2 Theory The p o t e n t i a l theory for g r a v i t y waves on a densi ty d i s c o n t i n u i t y between two f l u i d s when there i s a uniform mean v e l o c i t y U D i n the upper f l u i d i s worked out i n many texts; the present b r i e f development i s from Lamb (1932, § 232). The f low i n both f l u i d s i s considered to be i r r o t a t i o n a l , incompress ible , and i n v i s c i d . The v e l o c i t y p o t e n t i a l 0 i s introduced so that u = - d<\>/ d x , w = - d<l)/dz A3.1 where u , x, w, and z are h o r i z o n t a l and v e r t i c a l a i r v e l o c i t i e s and distances r e s p e c t i v e l y , p o s i t i v e to the r i g h t and upwards. 1£ i s d i s -placement from the mean water surface; k = 2IT/X i s wave number and wavelength, c i s wave phase speed, and CO. = 2Tff the radian frequency. The condi t ion that the v e r t i c a l v e l o c i t y of the surface be given by the t o t a l time d e r i v a t i v e of t£ g ives , to f i r s t order i n f£ , d ^ / 9 t + U 0 3i£/<2x = - d<|>/ d z ] z = 0 A3.2 as the required surface boundary cond i t ion . The governing equation i s Laplace ' s equation: 32<f) / 3 x 2 + a 2 0 / 3 Y 2 = Q ( s e e L a m b j £ 227) A3.3. I f surface e l eva t ion i s taken to be a wave of amplitude a « % , p e r i o d i c i n x and t and t r a v e l l i n g i n the plus x d i r e c t i o n 221 ^ = a e i k ( c t - x ) 5 A 3 > 4 then the v e l o c i t y p o t e n t i a l for the a i r becomes to f i r s t order 6 = -U x + B e - k z e i k ( c t - x ) . A3.5 ~ o S u b s t i t u t i o n of A3.4 and A3.5 into A3.2 to obta in B gives . = - U D x = i a (U Q - c) e - k z e i k ( c t > x ) . A3.6 The a i r pressure i s obtained from B e r n o u l l i ' s equation for i r r o t a t i o n a l motion; d<b\ 2/ + constant, A3.7 = act) ? a t 2  ^ -gz - \ m+ m where p i s the dens i ty of the a i r and g the a c c e l e r a t i o n due to g r a v i t y . S u b s t i t u t i o n from A3.6 gives p = j^gz + k a e i k ( c t > x ) e " k z ( U 0 - c)2j + constant A3.8 where - \ ^ U G 2 has been absorbed i n the constant. At the water surface z = , P = { + k ^ ( U o " c ^ 2 | + c o n s t a n t A 3-9 where e~ k 7 Z i s replaced by ^ to th is approximation. I f one sub-tracts the mean from A3.9 i t can be put i n the form P - P = - f g ^ | 1 + (1 - U o ) 2 | J A3.10 th i s i s the f l u c t u a t i n g pressure which would be present at a buoy f l o a t -ing on the water surface i n the presence of a steady, v e r t i c a l l y uniform 222 wind U Q . I t i s i n t e r e s t i n g to note that i f U 0 i s zero A3.10 becomes (P " 7 ) u Q = 0 = " 2 £ § *L A3.11. Returning to examine A3.10, i t can be seen that i f U 0 i s negative c (waves t r a v e l l i n g against the wind) pressure f l u c t u a t i o n s of many times g ^ could be observed. Taking the r e a l part of A3 .4 , A 3 . 1 , and A3.10, where A3.6 i s used, gives as the f i r s t - o r d e r approximation for a s i n u s o i d a l wave observed at x = 0, u w p - p I t should be noted i n A3.13, 14 and 15 that the quanti ty (U Q -c) becomes (U ocos0 -c) for U Q t r a v e l l i n g at an angle 8 to c. A3.3 The D i r e c t i o n of T r a v e l of the Waves The chart recordings showing the s ignals as recorded on tape (ampl i f ied and inverted where necessary to preserve the phase r e l a t i o n s among the o r i g i n a l s ignals ) are shown i n Figure 93. The group of four-second waves i n the time i n t e r v a l from 420 to 500 seconds stand out c l e a r l y against the background of 1 - 1.5 second waves and turbulent f l u c t u a t i o n s . The three bottom traces are from the sonic anemometer, being the two h o r i z o n t a l wind components along the sonic paths A and B (shown i n F igure 94) and the v e r t i c a l v e l o c i t y w i n that order. The o r i e n t a t i o n of the anemometer, wave probe, and pressure sensor as they = a cos?, kc t U D + ka (U D -c) e" k z cos kct ka ( U 0 - c ) e " k z s i n kct - £ ga cos kc t ^1 + (1 - U Q ) 2 j A3.12 A3.13 A3.14 A3.15 223 were during the run are shown i n F igure 94; the sonic anemometer was at a height of 1.75 meters above the mean water l e v e l . I t i s seen that the e f f ec t of the waves on the wind was much greater along path B than along path A; therefore the waves must have been t r a v e l l i n g i n a d i r e c t i o n roughly normal to A. The sense, however, i s indeterminate. The d i g i t a l ana lys i s produces for each data b lock , as w e l l as spec tra , the mean values of the modulus and d i r e c t i o n of the wind speed vector recorded by the sonic anemometer (see "Data Ana lys i s and Inter -p r e t a t i o n " , p. 86). These were p lo t t ed and are shown i n F igure 9 5 , i n the hope that they could be used to determine the d i r e c t i o n of the waves on the assumption that there would be a "wave-driven wind" which would cause a c l e a r l y recognizable departure from "ambient" condi t ions . This was not found; random f luc tua t ions among block means were large enough to completely hide any such e f f e c t ( which i s only several cm/sec; see p. 228. A3 .3 .1 Phase Information In order to determine the phase of the pressure and the v e r t i c a l and h o r i z o n t a l v e l o c i t i e s with respect to , the wave s i g n a l , cross -spectra of a l l the s ignals (pressure, sonic A , sonic B, and sonic w) with •»£ were computed. Cross - spec tra were computed on only the short (the 4-second group las ted about 30 seconds) sec t ion of the t o t a l run i n which the wave group occurred (see Figures 93, 95). Coherences between •»£ and other s ignals are shown i n F igure 96 and phases i n F igure 97. They show a large peak i n coherence near 0.25 Hz and that p leads ^ by 1 6 5 ° , sonic A and B lead "VI by 1 8 ° and sonic w leads V by 9 8 ° . Note that the 224 coherence at 0.25 Hz between the sonic A s i g n a l and i£ i s somewhat less than that between sonic B and . A l l of these phases are i n rough agreement with"the pred ic t ions of p o t e n t i a l theory (Equations A3.12 to A3.15) i f i t i s assumed that the waves were t r a v e l l i n g roughly against the wind. A3.4 Phase Correct ions I f the waves were i n f a c t t r a v e l l i n g into the wind, and for lack of evidence to the contrary th is assumption w i l l be taken as the basis for future c a l c u l a t i o n s , then the problem of phase correct ions to the sonic s ignals because of the 1.2 meter h o r i z o n t a l separat ion of the anemometer from the wave probe and pressure sensor must be considered. I t w i l l be assumed that the wave d i r e c t i o n was perpendicular to the sonic A d i r e c t i o n (see Figure 94). Then from the distances involved the center of measurement of the sonic anemometer was about 30 cm towards the source of the waves from the wave probe. To c a l c u l a t e the c o r r e c t i o n to the phase of the cross - spec tra the wavelength of the waves must be known. The per iod of the waves was four seconds. The water depth during the run was 3 m. The wavelength i n water of depth h can be ca l cu la ted from c 2 = f 2 A 2 = £ tanhtkh = Xg tanh 2Tfh ; A3.16 k T x the r e s u l t of so lv ing th is transcendental equation for f = 0.25 Hz gives a value for \ of 19 meters; th is when put into Equation A3.16 gives a value for c of 480 cm/sec. Knowing the wavelength, the phase angle c o r r e c t i o n to the sonic anemometer s igna l s can now be c a l c u l a t e d . I t i s 225 Sd = _30_ x 36O = 5.7 ^ 6 ° , 1900 and the sonic s igna l s w i l l lead the wave and pressure s igna l s by th i s amount. The corrected phase angles between the wave s i g n a l and the other v a r i a b l e s are d isp layed i n Table A 3 . 1 . TABLE A 3 . 1 Variab les Observed* Expected* Phases Phases 7 >P • 1 6 5 ° - 1 8 0 ° £ , B - 1 2 ° 0 ° - 9 2 ° - 9 0 ° ^Negative phase angles i n d i c a t e that ^ lags the v a r i a b l e concerned. Also shown are the phases from Equations A3.12 to Ae.15 expected from p o t e n t i a l flow theory. The agreement of phases with p o t e n t i a l theory, though not exact, i s seen to be quite good. A3.5 Comparison of Observed Amplitudes with P o t e n t i a l Theory The part p g of the f l u c t u a t i o n s i n a i r pressure p which i s associated with the waves can be estimated from the observed cross - spec tra using the r e l a t i o n (see "Data Ana lys i s and In terpre ta t ion" , Equations 5.2 and 5.3) ( p 2 ) k = z Z { c o 2 n ( f . ) + Q u 2 p t (f.)} h V ( A f \ / ( ( j t y f ^ d y n e A3.17 where the sum i s taken over the three frequencies f^ = 0.195, 0.292, and 226 0.390 Hz. These are the three frequencies which are nearest 0.25 Hz, the observed frequency of the 4-second group; ( A f ) . i s the bandwidth associated with the i frequency. S i m i l a r c a l c u l a t i o n s can be done to obta in from t h e i r spectra the v e r t i c a l and h o r i z o n t a l wind v e l o c i t i e s 2 associated with the a i r flow over the waves. A l l of these quant i t i e s are compared i n Table A3.2 with pred ic t ions from p o t e n t i a l theory computed from Equation A3.12 to A3.15 ( subscr ipt "t") and v i s u a l estimates of amplitudes ( subscr ipt "v") obtained from chart record ings . The funct ions seen by the various sensors i f p o t e n t i a l flow ex is t s are given by Equations A3.12 to A3.15. The height of the anemometer was z = 175 cm; the mean wind speed U 0 2s 200 cm/sec. The waves have wave-number k = 2Tf ~ 3.3 x 10 cm , and speed c = -480 cm/sec; i f the 1900 d i r e c t i o n of U D i s taken (from the sonic ana lys i s ) to be 1 2 5 ° (True) , and of c to be 2 8 5 ° (True) perpendicular to A, then th i s gives ( U o c o s ( 1 6 0 ° ) - c ) Si 670 cm/sec. This when subst i tu ted i n Equations A3.12 to A3.15 gives the f luc tua t ions u 2& 1.2 a cos kct = 1.2 ^ cm/sec w Qt 1.2 a s i n kc t = -1.2 * £ * cm/sec A3.18 o (p - p) -3 £ g dynes/cm for the r e q u i s i t e pred ic t ions from p o t e n t i a l theory (the a s t e r i s k denotes a complex conjugate) . The v i s u a l , s p e c t r a l , and t h e o r e t i c a l values presented i n Table A3.2 are seen to agree to wel l w i th in one order of magnitude, i n d i c a t i n g that / 227 TABLE A3.2 Comparison of T h e o r e t i c a l and Observed Pressure and V e l o c i t y Amplitudes over Swell Moving Against the Wind r> (12)h u (u 2 )^ u w (w 2)^ w p ( V 2 ) k lv S V S t V s t r v r s -2 cm cm/sec ~ cm/sec dyne/cm P t 10 8 6 12 10 6 9 10 30 34 24 Subscr ipts : "v" : v i s u a l estimates from chart recordings "s" : estimates from spectra using r e l a t i o n s analagous to A3.17. "t" : t h e o r e t i c a l pred ic t ions from Equations A3.12 to A3.15, where ~ 2 l-( ^ g ) 2 i s subs t i tu ted for a; see p. 226. 228 to f i r s t order p o t e n t i a l flow theory i s successful i n p r e d i c t i n g the observed r e s u l t s . The d i f ferences between spectra and predic ted ampli -tudes are small enough to be w i t h i n the expected errors i n the c a l c u l a -t i o n s , which are admittedly "quick and d i r t y " . A3.6 The "Wave Driven Wind" v£. H a r r i s (1966) has measured mean- v e l o c i t i e s above waves which appear to be associated with them, i n a wind-water tunnel . He compares t h e i r magnitudes with the Stokes' d r i f t v e l o c i t y i n the a i r , given by - „ 2 „ u 2 -2kz U stokes " 3 C k 6 A3.19. Since th is "mean d r i f t " cannot be observed by an anemometer at a f ixed height and h o r i z o n t a l p o s i t i o n ( i t sees only s i n u s o i d a l motion) , i t i s not c l ear what a s s o c i a t i o n i t has with the "wave dr iven wind" observed by H a r r i s ; s ince he does, however, and s ince the v e l o c i t i e s he observes are about 100 times those ca l cu la ted from A3.19, the c a l c u l a t i o n w i l l be repeated here for the present condi t ions . Using the values of c, k, and z from p.224 and a = 10 cm for the wave amplitude gives Wstokes 2; 0.2 cm/sec, and 100 U . , = 20 cm/sec. S COtC3S Thus i t i s expected that the mean winds observed by H a r r i s would,be too small to be observed i n th is experiment, s ince ambient f luc tua t ions i n wind speed (and d i r e c t i o n ) were large (see F igure 95). A3.7 Wave Damping The phase of the pressure r e l a t i v e to the waves (p leads W by 1 6 5 ° , 229 a 1 5 ° discrepancy from the p o t e n t i a l flow pred ic t ion) ind icates that wave damping was o c c u r r i n g . The analys i s system (see "Data Ana lys i s and In terpre ta t ion" , p. 87) ca lcu la tes E , the rate of change of wave energy per u n i t area per u n i t time, for each block of data analysed; E for the blocks i n which the four-second wave group appeared was negative, - 2 - 1 and averaged about -20 erg cm sec . Since the t o t a l wave energy i s given by E = h ? w g a 2 (Lamb, § 230) A3.20; where £ w i s the densi ty of water, then for an amplitude of 10 cm E 2= 5 x 10 4 erg c m - 2 , and the damping c o e f f i c i e n t E / E for the wave group i s 4 x 1 0 - 4 sec --' -, g iv ing the waves a time of about 40 minutes to reach 1/e of the i r i n i t i a l amplitude, or a Q of 3900. Mi les (1957) puts h i s r e s u l t s i n terms of 1 = C = E / C J E , A3.21 Q the " f r a c t i o n a l increase i n wave energy per radian"; here £ works out to about -2 .5 x 10" 4 . This negative value i s s l i g h t l y less than p o s i t i v e values obtained i n s i tua t ions of ac t ive wave generation; see "Results", Table 6.6 ( i t i s r e a l i z e d , of course, that M i l e s ' (1957) mechanism i s i n a p p l i c a b l e here; the comparison i s made with measured values , not t h e o r e t i c a l l y predic ted ones). P h i l l i p s (1966) makes an estimate of the "attenuation coe f f i c i en t" of waves i n an adverse wind on the assumption that the at tenuat ion i s caused by the i n t e r a c t i o n of the turbulent Reynolds stresses with the wave-induced undulations i n the flow. The attenuat ion caused by th is 230 mechanism i s exponential ; he wri tes i t as \ E ( t ) = E(o)e" 2 Tw t A3.21, g i v i n g for % , from Equation; A3.21 , C = -2_T_w_ A3.22. P h i l l i p s p lots - T W / C J against c / u * i n h is F igure 4.1.19 for the range 5 4s c / u A Z 50, u for the per iod of time while the wave group was pass-ing the sensors was measured with the sonic anemometer, g iv ing u* = (TEW") ^ = ( T s o n i c / e a ) ^ — 1.3 cm/sec; t h i s , with a c for the waves of 480 cm/sec gives a value for c/u,v of 375, f a r outs ide of P h i l l i p s ' range. The !> ca lcu la ted from P h i l l i p s ' f i gure for c /u* St 50 i s about - 6.5 x 1 0 - 4 ; i t increases throughout the range with increas ing c /u* . Since at c /u* = 50 i t i s a lready more than double the observed £ , the ex trapo la t ion of P h i l l i p s ' r e s u l t s to higher values of c /u* i s unreasonable. The damping regime discussed by P h i l l i p s involves i n t e r a c t i o n of turbulent Reynolds stresses with the waves; the small value of u^ measured i n th is experiment indicates that these stresses were very small and suggests that the mechanism which was damping the waves was of a d i f f e r e n t sort ; the fac t that the t h e o r e t i c a l pred ic t ions are higher than the observed values of ^ (whereas i n the case of wave generation the pred ic t ions of th is same theory underestimate observed rates of wave growth) s trongly suggests that P h i l l i p s ' t h e o r e t i c a l curves 231 for -Tw/cj w i l l reach a maximum and decrease at values of c / u * higher than the range p l o t t e d . A3.8 Momentum Flux The r e l a t i v e phases of the pressure and sonic B s ignals show the greatest deviat ions from p o t e n t i a l theory; p leads r[ by 1 6 5 ° , 1 5 ° less • than the predic ted 180° , while sonic B leads ^ by 12°, 1 2 ° more than the predic ted 0 ° . This f a c t observed alone seems contradic tory; the phase of the pressure s i g n a l ind icates that the non-potent ia l flow process sh i f t ed the streamlines of the mean flow so they lagged the waves; the sonic B s i g n a l ind icates the opposite . At the same time the sonic w s i g n a l has no s i g n i f i c a n t phase s h i f t from 9 0 ° . This means that sonic w leads sonic B by 8 0 ° ; th is d i f ference from quadrature ind icates the presence of momentum trans fer . Further , the 1 0 ° s h i f t i s i n the d i r e c t i o n of a p o s i t i v e uw covariance or a negative wind s tress T*= - £ Tiw, i n d i c a t i n g that momentum transfer from waves to wind was o c c u r r i n g . I f the s tress i s ca lcu la ted from the co-spectrum between sonic B and w i n the frequency range of the waves, T s o n i c = - f a w = - ,0 .002 dyne c m - 2 . T h i s , then, i s to be compared with that ca lcu la ted from the pressure-waves c o r r e l a t i o n (for the frequency range of the four-second group) *>- = ,E = - 0.04 cos 2 0 ° ^ - 0.04 dyne cm" 2 . C w " C The stresses ca l cu la ted from the two methods have the same s ign but d i f f e r i n magnitude by a factor of 20. 232 The fac t that the stresses are of the same sign indicates s e l f -consistency i n the measurements; i f energy was being trans ferred from waves to wind the momentum transfer would;?be expected to be i n the same d i r e c t i o n . The d i f ference i n s tress measured at the two l eve l s i s outs ide the l i m i t s of expected experimental e r r o r . Even quite unreasonable s h i f t s i n the angle of the incoming waves with respect to the wind would not br ing the two in to agreement. Therefore i t appears that a s tress gradient may have ex is ted over the waves. I f this were so, then i t s approximate s i ze would be A 7T / N . -2 x 1 0 - 4 dyne c m - 2 / c m . A z This gradient would create a h o r i z o n t a l a c c e l e r a t i o n d u = - _1 3T i i 0.17 cm sec" 2 . a t ^ a z I f A t i s taken as 30. seconds (the time for the group to pass by) , then the maximum change i n h o r i z o n t a l v e l o c i t y caused by the gradient would be 5 cm/sec, which i s a f a i r l y small va lue . Therefore although the s tress gradient appears to have ex i s t ed , i t s e f f ec t on the flow was pro-bably not l arge . Because s tress gradients have not often been observed, i t would be i n t e r e s t i n g i n any future work to make attempts to define them i n d e t a i l . A3.9 Conclusions A sec t ion of record of about 30 seconds durat ion , containing s imultan-eous information on s t a t i c pressure and the three components of the f l u c t u a t i n g wind speed over a group of about seven waves of 4 second 233 period which were running against the mean wind, has been analysed i n d e t a i l . The r e s u l t s of the analys i s show that to f i r s t order the flow over the waves i s predic ted by simple p o t e n t i a l theory. They show further that deviat ions from p o t e n t i a l flow, although smal l , are of c r u c i a l importance to the time h i s t o r y of the waves. Three phenomena were inves t iga ted . The f i r s t , the "wave-driven wind" observed by H a r r i s (1966) proved to be hidden i n ambient mean wind speed and d i r e c t i o n f l u c t u a t i o n s , and was not inves t igated f u r t h e r . The second was energy t rans fer . The p a r t i c u l a r wave group studied was i n a regime of damping caused by adverse winds. The damping co-e f f i c i e n t ca l cu la ted from the pressure, waves c o r r e l a t i o n gives a Q for the wave group of 3900. This represents a rate of damping s l i g h t l y smaller than rates of growth observed i n s i tua t ions of weak but ac t ive wave generat ion. The rate of damping i s also much smaller than that pre-d ic ted from t h e o r e t i c a l considerat ions by P h i l l i p s (1966); i t i s thought that the flow regime i n the experiment was quite d i f f e r e n t from that assumed by P h i l l i p s . The t h i r d phenomenon inves t igated was momentum trans fer . I t was found to be negative (from waves to wind) both by c o r r e l a t i o n s of pressure with wave e levat ions at the sea surface and by sonic B - sonic w c o r r e l a t i o n s measured at 1.75 m above the surface . The s tress at 1.75 m proved to be less than that at the surface , i n d i c a t i n g a weak v e r t i c a l s tress gradient of about 2 x 10" 4 dyne c m - 2 / c m . While the phase of the pressure s igna l r e l a t i v e to that of the waves 234 was s h i f t e d from 1 8 0 ° to + 1 6 5 ° , the phase of the sonic B s igna l with respect to the waves was sh i f t ed from 0 ° to + 1 2 ° , both by the wave damp-ing mechanism. The sonic w s i g n a l remained at about 9 0 ° with respect to the waves. These facts i n d i c a t e (and again i t must be stressed that the conclusions are h i g h l y speculat ive) that whereas the pressure phase ind ica tes that the streamlines c lose to the water surface are sh i f t ed so they lag the waves, at only 0.1 wavelength above the surface the stream-l ines appear to be arranged so that w remains i n quadrature with the waves while u leads them. The question of the mechanism of energy and momentum transfer from waves t r a v e l l i n g against the wind remains an open one. M i l e s ' (1957). theory of shear flow i n s t a b i l i t y i s obvious ly i n a p p l i c a b l e , s ince i n this case there i s no " c r i t i c a l height" at which wave speed equals wind speed; th is does not of course r u l e out the p o s s i b i l i t y that other shear flow i n s t a b i l i t y mechanism may be operat ive . I f the damping has anything to do with mean shear flow i n s t a b i l i t y , then i t w i l l be necessary to know what the s tructure of the mean flow i s l i k e over such waves, perhaps very close to the water surface . I f the mechanism involves turbulence, then the s t ruc ture of the turbulence must also be known and separated from the "undulance" discussed here. 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F l u i d Mech. .13, 433-48. (1965) A note on the i n t e r a c t i o n between surface waves and wind p r o f i l e s . J . F l u i d Mech. 22, 823-7. (1967) On the generation of surface waves by shear f lows. Part 5. J . F l u i d Mech. 30, 163-75. M i t s u t a , Y . , M. Miyake and Y. Kobori (1967) Three dimensional sonic anemometer-thermometer for atmospheric turbulence measurement. Kyoto U n i v e r s i t y In terna l Report; 10 pp. Miyake, M. M. _et al (1969) Intercomparison of turbulent f luxes over water determined by aerodynamic and eddy c o r r e l a t i o n techniques. Quart. Jour . Roy. Met. Soc. ( i n the p r e s s ) . Motz fe ld , H. (1937) Die turbulente stromung an we l l i gen wanden. Z . angew Math. Mech. _7 , 193-212. Munk, W. H. (1955) High frequency spectrum of ocean waves. J . Mar. Res. 14, 302-14. Neumann, G. (1953) On ocean wave spectra and a new method of forecas t -ing wind-generated sea. Beach Eros ion Board, U. S. Army Corps of Engineers , Tech. Mem. No. 43, 42 pp. Panofsky, H. A . , N. Busch, B. Prasad, S. Hanna, E . Peterson and E . Mares (1967) Propert ies of wind and temperature at Round H i l l , South Dartmouth, Mass. Res. and Dev. Tech. Rep. ECOM-0035-F, U. S. Army E l e c t r o n i c s Command, F o r t Huachuca, A r i z o n a . Patton, R. S. and H. A . Marmer (1932) The waves of the sea. '^Physics of the e a r t h , " Volume 5_, Oceanography Nat. Res. C o u n c i l , pp. 207-28. P h i l l i p s , 0. M. (1957) On the generation of waves by turbulent wind. J . F l u i d Mech . .2 , 417-45. _ _ _ _ _ (1966) "The Dynamics of the Upper Ocean." Cambridge U n i v e r s i t y Press . 239 (1967) The maintenance of Reynolds s tress i n turbulent shear flow. J . F l u i d Mech. 27, 131-44. (1961) On the dynamics of unsteady g r a v i t y waves of f i n i t e amplitude. Part 2. J . F l u i d Mech. _11, 143-155. P l a t e , E . J . and G. M. Hidy (1967) Laboratory study of a i r f lowing over a smooth surface onto small water waves. J . Geophys. Res. ]_2 (18)5 4627-41. , P. C. Chang and G. M. Hidy (1969) Experiments on the generation of small water waves by wind. J . F l u i d Mech. _35 (4), 625-56. Pond, S . , R. W. Stewart and R. W. B u r l i n g (1963) Turbulence spectra i n the wind over waves. J . Atmos. S c i . ^0 (4), 319-24. P r i e s t l e y , J . T. (1965) C o r r e l a t i o n studies of pressure f luc tua t ions on the ground beheath a turbulent boundary l a y e r . U. S. Nat. Bur. Stand. Rep. 8942; 92 pp. Shemdin, 0. H. (1968) Wind Generated Waves: recent and future developments. F i f t h Space Congress, Oceanography Sess ion, Cocoa Beach, F l a . and E . Y. Hsu (1966) The dynamics of wind i n the v i c i n i t y of progress ive water waves. Stanford U n i v . , Dept. of C i v i l Eng. Tech. Rep. No. 66. (1967) The dynamics of wind i n the v i c i n i t y of progress ive water waves. J . F l u i d Mech. _30, 403-16. Smith, S'. D. (1967) Thrust anemometer measurements of w ind-ve loc i ty spectra and of Reynolds s tress over a t i d a l i n l e t . J . Mar. Res. 25 (3), 239-62. Snyder, R. L . and C. S. Cox (1966) A f i e l d study of the wind generation of ocean waves. J . Mar. Res. _24 (2) , 141-78. Stanton, S i r T . E . , D. M a r s h a l l , and R. Houghton (1932) The growth of waves on water due to the ac t ion of the wind. Proc . Roy. Soc. A 137, 283-93. Stewart, R. W. (1961) The wave drag of wind over water. J . F l u i d Mech. _10, 189-94. (1967) Mechanics of the a i r - s e a i n t e r f a c e . Phys. F l u i d s , Supplement: Boundary Layers and Turbulence, S47-S55. Sverdrup, H. U . , and W. H. Munk (1947) Wind, Sea, and Swell : Theory of Relat ions for Forecas t ing . U. S. Hydrogr. O f f i c e , Wash., Pub. No. 601. 44 pp. 240 T a y l o r , R. J . (1960) A new approach to the measurement of turbulent f luxes i n the lower atmosphere. J . F l u i d Mech. _10, 449-58. T h i j s s e , J . T . (1951) Growth of wind-generated waves and energy trans-f e r . "Grav i ty Waves." Nat. Bur. Stand. Washington c i r c u l a r No. 521, 281-7. Thomson, S i r W. (1871) Hydrokinet ic so lut ions and observat ions . P h i l . Mag. (4) 42, 368-72. U r s e l l , F . (1956) Wave generation by wind. "Surveys i n Mechanics" (ed. G. K. B a t c h e l o r ) . Cambridge U n i v e r s i t y Press . Pp. 216-249. Wei l er , H. S. and R. W. B u r l i n g (1967) D i r e c t measurements of s tress and spectra of turbulence i n the boundary layer over the sea. J . Atmos. S c i . 24 (6), 653-64. Wiegel , R. L . and R. H. Cross (1966) Generation of wind waves. J . Waterways and Harbors D i v . , ASCE 92, 1-26. Wi l lmarth , W. W. and C. E . Wooldridge (1962) Measurements of the f l u c t u a t i n g pressure at the wa l l beneath a th ick turbulent boundary l a y e r . J . F l u i d Mech. 14, 187-210. Wu, J i n (1968) Laboratory studies of wind-wave i n t e r a c t i o n s . J . F l u i d Mech. 34 (1) , 91-111. Wuest, W. (1949) Be i t rag zur entstehung von wasserwellen durch wind. Z . angew. Math. Mech. 29, 239-52. FR"OK""CANADlAN "HYDROG RAPHl'C SERVICE CHART No. 3W6 FIGURE 1 MAP AT- SITE OF EXPERIMENT FIGURE 2. AVAILABLE. FETCH. AT. SITE FIGURE 3. PHOTOGRAPH OF RECORDING PLATFORM AND INSTRUMENT MASTS (LOOKING NORTHEAST) FIGURE 77. PHOTOGRAPH OF EQUIPMENT DEPLOYMENT PRIOR TO A RUN AT BOUNDARY BAY ( F R E Q U E N C Y ) " ' ( K H Z ) ' ACCESS -to SPACE BEH IN D' DIAPHRAGM'/' FIXED PLATE OF / STAIN LESS CAPACITOR - / STEEL CASE FIGURE 6. SCHEMATIC CROSS-SECTION OF MICROPHONE CEAK JZ. xz. ' ,1 11 i • ' p a MICROPHONE _PIAPHRA6M ; F O R E -VOLUr'lE "POXY ET H Y L E N E TOPING do cm) BOBBER. 'SSATERERO.ftE.ING D I A P H R A G M " BACKUP NES-S.UC.ANT_ FIGURE 7. SCHEMATIC DIAGRAM OF PRESSURE MEASUREMENT SYSTEM BUOY I N S T R U M E N T M A S T r MICROPHONE OSCI L L A T O R (IOO m H z ) B U F F E R RG 174/U (7 METERS) R F A M P U R E R SHIELDED 300X1 TWIN - LEAD (feO METERS) FM" TUNER M A T C H I N G A M P L I F I E R FIGURE 8. PRESSURE RECORDING ELECTRONICS: BLOCK DIAGRAM - I 5 " V D C O S C I L L A T O R (100 m H a C L A P P ) 13TURNS Cu FOIL £200 - O U T T /777 M I C R O P H O N E -' 1 1. RES I STANCES IN _ K £1. 2. CAPACITANCES IN PF. "3 e O U T-e l N FE.EDTH ROUGH <. IO P.F. 4. ALL GROUNDS TO Al.. CASE. • I 5 Y D C B U F F E R AMPLIFIER 0 T U R N S 2" D 8 C 3 RG 174/U :(50Xi) TO RF AMPLI-FIER OM-M.ASX FIGURE 9.' WIRING DIAGRAMS OF BUOY OSCILLATOR AND AMPLIFIER .6 FIGURE 10. FM TUNER RATIO DETECTOR RESPONSE CURVE T A U T R U B B E R D l A P H R A & M PRIV1MG PLATE' DRIVE TO SINE WAVE OSCILLATOR. SECURING STRAPS B A C K P L A T E  SEALAMT T E N G A L L O N OIL D R U M CABLE B U Q Y \ / V V y , B A R O C E L . WOODEN TO C H A R T RECORDER, and OSCILLOSCOPE ~7 7 7 — 7 -FIGURE 11. PRESSURE SENSOR CALIBRATION SETUP or to cv2 _ O - t r -c\2 I , ul (A — in - J p. LU P s _ ;< A i t LUi 4-4- M- x-4 44-+ +x I i + h < X CL o o o o od oo / o / 0 0 0 o o 0 0 / o 0 / 0 HI ZL >r — — _ _ II L o AXrA\"XVSN35 o 3 o + lo ( S 3 3 2 1 9.3<1.) Q V 3 1 3 W H B FIGURE 1 2 . LABORATORY CALIBRATION OF THE BUOY PRESSURE SENSOR P R O F I L E % C R O S S - S E C T I O N A L V I E W HIGH-DRAG WAVE--. PROBE. . I N L E T TO . M I C R O P H O N E P T A T P M R A G M PERSPEX. "PLATE T VSAtER-X l N E T . FIGURE 13. DIAGRAM OF THE BUOY FIGURE 15. SCHEMATIC DIAGRAM OF WIND TUNNEL SETUP FIGURE 16. PROFILE OF BUOY (TO SCALE) 15" M n Vi o s > w M o w s: *i H O M O S3 < > O RI * | o w a d co o HS > H > O CO > CO O ?d f fcd M < (CO O M CO H P> 2! n '0 10 <\1 I r o 2 5 >-P h Z w 0 CD r < > o < at 3 on v> lil £ 0. -5 -40 -15 AERODYNAMIC CALIBRATION OF THE BUOY: PRESSURE VS. . DISTANCE FIGURE 19. AERODYNAMIC CALIBRATION OF THE BUOY: FRACTION OF STAGNATION HEAD AT PRESSURE PORT VS. WIND SPEED FIGURE 20. SCHEMATIC REPRESENTATION OF THE EFFECT OF ATTACK ANGLE ON THE PRESSURE MEASURED BY THE BUOY FIGURE 21. EFFECTS ON p + f g ^ PHASOR OF 50% ERROR IN PRESSURE CALIBRATION FIGURE 22; PHASE CORRECTIONS FOR TIME SHIFT OF ^ SIGNAL COMPARED WITH FULL CORRECTION CURVE . m '. M E A S U R E ^ -a. : PHASE-CORRECTION A P P R O X I M A T E D C : PHASE CORRECTION EXACT q0° NO^suS^cRiPTvM'EXStJREP IF NO WAVE" GENERATION WAVE DAMPING (HA GH ..FLREQUENC\ES) (.LOW FREQUENCIES) FIGURE 23. EFFECTS ON p + PHASORS OF LOW AND HIGH-FREQUENCY . APPROXIMATIONS TO FULL PHASE CORRECTION CURVE FIGURES 24-27. POWER SPECTRA OF p , p g + f^S^ > FOR RUNS 1, 2, 2a, and 3 FIGURES 28;-31. POWER SPECTRA OF p, p + >y and p s , p s + f a ^ 7 » ^ FOR RUNS 4a, 4b, 5 and 6 RUN 1 U-=220 •» j 3 cr 'i .* ^ C O H E R E N C E C p=370 u 6 h o u \ 1 V \ / \ ••\ J%.f>gr;)\ P H A S E \ f k * 7} ADVANCED 2 S A M P L E S FOR PVIJ ADDITION 1.0 • 8 0 ' O UJ •100 Q -120; - 1 4 0 o -180 °-.140 1.0 1.5 2.0 f (Hz) 2.5 3.0 3 2 1-% RUN 2a ° o Uj=320 c m sec" C =270 ^ P s . 7 s P H A S E k. 8 0 g _ •120 £ -140 < •160 •160 1.0 1.5 2.0 f (Hz) 2.5 3.0 3 3 S.7s C O H E R E N C E RUN 3 U^=340 1.5 2.0 f (Hz) 2.5 3.0 3 5 o , u .4 '» C O H E R E N C E RUN 2b U_=310 , 5 c m s e c ' CD= 275 (Ps'pg>/). 7 P H A S E / »| 11 -100 § -120 Q -140 < 0.5 1.0 1.5 2.0 f (Hz) 2 .5 3.0 3 4 FIGURES 32-35. COHERENCE SPECTRA BETWEEN p s , ^ s ; PHASE SPECTRA BETWEEN p s , ^ s and ( p s + f f t l ^ ) , s FOR RUNS 1, 2a, 2b, and 3 1J0| 1.5 2 .0 f (Hz) 3 6 ,«•. COHERENCE? RUN 5 Us=150 C D =330 P*pQ7p, tj P H A S E 0 . 5 1.5 2 . 0 f (Hz) 3 . 0 38 1 \ C O H E R E N C E RUN 4b U==700 5 cm C p =305 F; rj P H A S E (P'pgrj). -q P H A S E - 8 0 S o. - 1 0 0 = o Q H-140 v ii 0 . 5 1.5 2 . 0 f (Hz) 2 . 5 3 . 0 1.140 37 u UJ cc - 8 0 O UJ Q U - 1 0 0 § l/) (/) UJ u CC 2 - 1 2 0 ° -LlJ OL L_ W o I O Q O - 1 4 0 < _j - 1 6 0 3 I 0. 1 8 0 • 1 6 0 • 1 4 0 RUN 6 ^ C - 2 7 0 C O H E R E N C E U p - ^ / u f x / V<ps'/°9^7sPHASE <i ( A P s . % P H A S E - 8 0 2 -120 o-Li. o 0 . 5 1.5 f (Hz) 39 2 . 5 3 . 0 FIGURES 36-39. COHERENCE SPECTRA BETWEEN p s , PHASE SPECTRA BETWEEN p s , 7 £ s and ( p s + ),V.s FOR RUNS 4 a , ; : 4 b , 5 , and 6 RUN 1 L70 U==220 c m / s e c J-2 A-.2 f (Hz) 4 0 -E(») RUN 2a U==320 , 5 c m / s e c CM c > | . 2 ~ 3.0 10Cf RUN 3 U R ; 3 4 0 cm/sec 100 ^ 0 RUN 2b U—310 0 c m / s e c C p =275 2.0 f (Hz) 4 3 f (Hz) 4 2 3.0 FIGURES 40-43. ENERGY AND MOMENTUM FLUX SPECTRA AND WAVE POWER SPECTRUM FOR RUNS 1, 2a, 2b, and 3 44 RUN 5 U==150 C p =330 cm/sec | 5 0 « C E / L30 / ( (Hz) 4 6 2 5 * E •UJ 50 f(Hz) 45 ; 6 0 •Ul o RUN 6 E(f) £ 0 / \ \ U_=570 r ° cm /sec . A C p = 2 7 0 - /i \ Q N / / \ \ 9 £ 0 I // \ \ u /* \ / 4 0 S // V - ^ J 9 / p " M 20 / \* 10 / \ / \ ' • 1 1 1 1 E 2 " -.1 -.2 30 47 FIGURES 44-47. >E^RGY.;AND:M0MENTUM1FLUX SPEGTRKYAND MVECPOWER SPEGTRUM FOR RUNS 4'a, 24b, 25, and 6 FIGURES 48-51. SPECTRA O F ^ : COMPARISONS WITH SNYDER AND COX, MILES FOR RUNS 1, 2a, 2b, and 3 FIGURES 52-54. SPECTRA OF f: COMPARISONS WITH SNYDER AND COX, MILES FOR RUNS 4a, 4b, and 6. FIGURE 55. DIMENSIONLESS PLOT OF £ VERSUS U 5 / c -f-•f-+ + •ir •fc .0 'FREQUENCY 1.2 1.6 PRESSURE! (1 S P I K E / C Y C L E ) FIGURE 56. POWER SPECTRUM OF p (t) ~~i r 'FREQUENCY 1.2 1. PRESSURE (1 S P I K E / C Y C L E ) HRVES FIGURE 57. . COHERENCE BETWEEN p (t) a n d ^ ( t ) o d _ CO — o Q _ — o — — Q _ a* o 30.o* ~ — PHASE 20.0 — rmmm PHASE 20.0 — a CD i .1 3 1 1 1 ^ -8 1.2 1.6 FREQUENCY PRESSURE; : ( rSPIKE/CYCLE) WRVES ; ( N O S P I K E S J . FIGURE 58. PHASE ANGLE BETWEEN P_(t) andlg(t) CO CM 2 - o CJ l o <—I CD O —1 -s--f-•f-X i i I I .0 'FREQUENCY 1.2 l .B NRVES ( N O SPIKES J FIGURE 59. POWER SPECTRUM OF J £ ( t ) . J K J \— az L U Q CM az Q - l UUo Q_ • c o r (JD O I O -I-.0 -f-4-.4 .B FREQUENCY 1.2 1.6 PRESSURE {1 S P I K E / C Y C L E ) FIGURE 60. POWER, SPECTRUM OF P (t) ~1 1 1 1 .4 .8 1.2 1. FREQUENCY P R E S S U R E WRVES i U S P I K E / C Y C L E ) ; FIGURE .61. COHERENCE BETWEEN p s ( t ) a n d ^ s ( t ) CM. O =*•. CM O CO . LU zcco o CO. 30.0 r~ FREQUENCY 1.2 P R E S S U R E NAVES ;tlSR.lK.E/CYELEX 1.6 FIGURE 62. PHASE ANGLE BETWEEN p s ( t ) a n d ^ s ( t ) f' K J 1— DC L U Q i •—, CM z : Q - 5 --*-LOGiO SPECTRUM -2.0 -1.0 .0 -s-- i -• ~\ . o cn i 1 1 1 1 D ,4 .8 1.2 1.6 FREQUENCY NAVES (1 S P I K E / C Y C L E ] POWER FIGURE 63. . ; . SPECTRUM OF £ s ( t ) z R U N 1 COM P A R I S O M OF S P E C T R A OF j M ^ J t ) F I G U R E 64-O _ l -3 - 2 •1 O LOG F R E Q U E N C Y (HH) > il I — .—4- : : : . , -1 -z - i o ; i • . LOG. F R E Q U E N C Y ( R2) , , v ; v -v , - / r LOG F R E Q U E N C Y . ( H Z ) z R U N 6 COMPARISON OF \ LOQ, F R E Q U E N C Y (HZ) •\zo FIGURE 68 RUN i 6 n ( 0 -140 80 + .1 , 1 1 1 1 • . • LOG FREQUENCY (HZ) C A P A R I S O N OF S P I K E - C O N T A M I N A T E D A N D ^ L E A R " $>X P H A S E S P E C T R A 1 6 0 F I G U R E . R U N 2b J L •140 180 J L 160 LOG F R E Q U E N C Y CHB) "COfHFARISON OF S P I K E - C O N T A M I N A T E D AN-P-. "CLEAR." P > i P H A S E S P E C T R A 180 -1 I I I I I I I o L O G F R E Q U E N C Y ( H 2 ) 160 O +1 COG FREQUENCY (H£) COMPARISON OF SPIKE-CONTAMINATED AND "CLEAR" ENERGY FLUX SPECTRA j r - i f c . e s F I G U R E 7 3 / \ RUN 2b o LOG FREQUENCY (Hz) C O M P A R I S O N O F S P I K E - C O N T A M I N A T E D A N D " C L E A R " E N E R G Y F L U * S P E C T R A " LOG F R E Q U E N C Y - CH-a) z >-: p 5 L > h 5.4.1 S T A N D A R D - D E V I A T I O N S NUMBER OF DETE.RMlNATIO.iMS ^CALIBRATION MADE" B Y _ r . M O V J N G S E N S O R " U P 'AND1SOWN -o o- -O o—O—O-J_L I 1 I I I I I I I I I I I - T - .ncr LOG F R E Q U E N C Y ( H i ) 9 _i[ Ul' io I 0 if) Ld UJ a LU z IOSJI - U .n+i i£0_ >-O o CD FIGURE 78. BUOY SENSOR CALIBRATION FOR BOUNDARY BAY EXPERIMENT FIGURE 79. FREQUENCY RESPONSE OF CALIBRATION DRUM L.OG fc.s FIGURE 80. TYPICAL POWER SPECTRUM OF DOWNWIND VELOCITY FLUCTUATIONS o -TTTj | """J 1"'I I. I"l I [ FIGURE 81 COMPARISON O F SPECTRA FROM THREE PRESSURE SENSORS RUN i a SEPT. 5", l^&S BOUNDARY B A Y I M I L 2 o REFERENCE SENSOR o BUOY SENSOR x AIR SENSOR o O o 04-A -e-O x a -0 9 x © G J J _ _LL 3 J I I I I I I LOG F R E Q U E N C Y ( H z ) - 1 1 1 1 1 II 1 j 1 • 1 1 1 1 1 1 1 j 1 I I 1 1 1 L , F I G U R E 81 -C O M P A R I S O N OF S P E C T R A -- F R O M T H R E E P R E S S U R E SENSORS -* R U N 5"CL S E P T . £7 , BOUNDARY B A Y o -o R E F E R E N C E SENSOR -X . • B U O Y SENSOR. - x AIR. S E N S O R — o X * • — ° o — — X • _ - s X — - A — - o -8 o X • • X -o ° X « - -X 6 • - X * -0 0 - X _• 1 1 1 1 M i l l - 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 l c - 2 -1- o „ X LOG F R E Q U E N C Y ( H Z ) o a hi 'A Lu -D_ V 2 o, Z Lo O in <D <M 00 Lo in o o o in Is-|0 io to 16 fj> ,0>, in L O to • o < o D 9 a B < 19 <P ° SO <T H< 0° in <Pa is <P D n $ • D n < • • £ o o In \ <4-*= ti I — 1 v0 i D 0 .1 I FIGURE 83. NON-DIMENSIONAL PRESSURE SPECTRUM FROM BOUNDARY BAY 1.0 - 1 IO: + 1 -IIZOG.. F.R E.QU E N CY. ( Hjt) FIGURE 84. COHERENCE BETWEEN "BUOY" AND "AIRM PRESSURE SENSORS LOG FREQUENCY (Hz) FIGURE 85. PHASE ANGLE BETWEEN "BUOY" AND "AIR" PRESSURE SENSORS RU"N SYMBOL f a + 5 a . 5 b a 5 c • 5 d o 5 e A _LOG FREQUENCY (Hz) FIGURE 86. COHERENCE BETWEEN "BUOY" AND "REFERENCE" PRESSURE SENSORS FIGURE 87. PHASE ANGLE BETWEEN "BUOY" AND "REFERENCE" PRESSURE SENSORS .8 .8 RUN 5YMBOL 5<x a 5b a 5c • 5d o S.e A o Z X o o XO'G F R E Q U E N C Y ("Hi)' FIGURE 88. COHERENCE BETWEEN "BUOY" PRESSURE SENSOR AND HOT-WIRE AIR SPEED 80 T-OG F R E Q U E N C Y " (Hz) + 1 FIGURE 89. PHASE ANGLE BETWEEN "BUOY" PRESSURE SENSOR AND HOT-WIRE AIR SPEED .8 .6 _ .4 _ X .8 o • A RUN SYMBOL 5a a 5b m 5c a S6 o 5e A r l ' "O . L O G "FREQUENCY (H2) FIGURE 90. COHERENCE BETWEEN "AIR" PRESSURE SENSOR AND HOT-WIRE AIR SPEED • A A A • + S O L A at ££) II O • O R Q • n n O g| H o o • a A O • o 03 2 • D e O • A* D A • O O • 2_ A A " ^ A * • UJ _ 160 P o d i e o <" z UJ <j J 4 0 Lj-a 4] ui J-40 <C a; Q. r80 +1 _ A O I" I I M i l l I I XQG FREQUENCY CHz) FIGURE 91. PHASE ANGLE BETWEEN "AIR" PRESSURE SENSOR AND HOT-WIRE AIR SPEED — ROSJ SYMBOL. — _5Q. ® - 5 b — _.5c. • Sd O — 5e A - -1 LOG FREQUENCY (WTL) FIGURE 92. COHERENCE BETWEEN.'REFERENCE"• PRESSURE SENSOR AND HOT-WIRE AIR SPEED 400 420 4-4-0 460 480 500 J_ T I M E I M S E C O N D S FIGURE 93. CHART RECORDING OF WAVE, PRESSURE AND SONIC ANEMOMETER SIGNALS DURING PASSAGE OF 4-SECOND SWELL GROUP i .151. ' SCALE. 1:10 FIGURE ,94. SCALE DRAWING OF INSTRUMENT SETUP FOR RUN 5 FIGURE,95. TIME VARIATION OF U , 6 FROM SONIC ANEMOMETER (MEANS OVER TEN-SECOND INTERVALS) FOR RUN 5 O 0.5" l .O F R E Q U E N C Y ( H e ) FIGURE 96. COHERENCE SPECTRA DURING PASSAGE OF SWELL GROUP 0. 0,5 1-0 F R E Q U E N C Y ( H 2.) FIGURE 97. PHASE SPECTRA DURING PASSAGE OF SWELL GROUP 

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