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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  by  FREDERIC WILLIAM DOBSON  B.Sc,  Dalhousie  University,  1959  M.Sc,  Dalhousie  University,  1961  A THESIS  SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in  the  Department of  Physics  We accept t h i s required  t h e s i s as conforming to  standard  THE UNIVERSITY OF BRITISH COLUMBIA August,  1969  the  In p r e s e n t i n g an  this  thesis  in partial  advanced degree a t the U n i v e r s i t y  the  Library  I further for  shall  make i t f r e e l y  agree that  permission  f u l f i l m e n t of the requirements f o r of B r i t i s h  available  Columbia,  I agree  that  f o r r e f e r e n c e and S t u d y .  f o r extensive  copying of this  thesis  s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r  by  h i s representatives.  of  this  written  thesis  It i s understood  for financial  permission.  PHYSICS  Department o f  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  Date  gain  AUGUST.  1969  Columbia  shall  that  copying or p u b l i c a t i o n  n o t be a l l o w e d w i t h o u t  my  Supervisor:  Professor Robert W. Stewart ii  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 l a c k of adequate o b s e r v a t i o n a l  t i o n on the normal p r e s s u r e s which t r a n s f e r energy to the waves. p r i n c i p a l o b j e c t of  this  experiment has been to p r o v i d e some of  informaThe the  missing data. A system f o r making simultaneous measurements of normal  pressure  and wave h e i g h t was developed and t e s t e d i n the l a b o r a t o r y and i n the field.  The system c o n s i s t e d of a d i s c - s h a p e d buoy 23 cm i n diameter  which was embedded the p r e s s u r e sensor)  which rode up and down on a  v e r t i c a l r o d , which was the wave s e n s o r . r e j e c t i n g s o - c a l l e d "dynamic" p r e s s u r e s  (in  Careful  a t t e n t i o n was p a i d  to  a s s o c i a t e d w i t h 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 s p e c t r a of  the p r e s s u r e and wave s i g n a l s .  Momentum ( t " ) w  i n c r e a s e of  from the experiment are p r e s e n t e d as power and c r o s s -  fluxes  to the waves,  the waves per r a d i a n ,  S p e c t r a of Energy (E) and  and of £  are a l s o  , the f r a c t i o n a l energy  presented.  Wave power s p e c t r a are found to be normal f o r the s i t e ; power s p e c t r a c o n s i s t land,  on which i s  of a " b a s i c " spectrum s i m i l a r to t h a t observed over  superimposed a wave-induced "hump".  The phase angle between of  the peak of  (pressures  the p r e s s u r e  the waves and the p r e s s u r e a t the frequency  the wave spectrum i s  found to be s h i f t e d from - 1 8 0 °  h i g h over wave troughs) by amounts which exceed  p r e d i c t i o n s of M i l e s  (1957) by an average of  the  theoretical  20 ^ 5 ° over a wide range of  iii  conditions. The E and 7* the frequency of  s p e c t r a are found to be s h a r p l y peaked at or above  w  the peak of  the wave spectrum.  fluxes  E show l a r g e s c a t t e r ,  varies  c o n s i d e r a b l y i n time (and s p a c e ) .  7T  W  The i n t e g r a t e d  i n d i c a t i n g t h a t the wave g e n e r a t i o n  energy process  The i n t e g r a t e d momentum f l u x e s  to the waves show no s i g n i f i c a n t d i f f e r e n c e from t o t a l f l u x e s  from -3  air it  to water computed assuming a c o n s t a n t drag c o e f f i c i e n t appears t h a t about 80% of  the t o t a l drag of  caused by the wave g e n e r a t i o n The ^ s p e c t r a exceed f a c t o r s of  5 to 8,  process.  the p r e d i c t i o n s of M i l e s '  except  A dimensionless presented;  f where ^ / ^  is  w  (f)  The p r e s e n t  not  results  curve suggested by Snyder and Cox (1966)  at h i g h f r e q u e n c i e s ,  speed i s  (1957) theory by  i n d i c a t i n g t h a t h i s " i n v i s c i d laminar" m o d e l . i s  c l o s e to an e m p i r i c a l  where they are c o n s i d e r a b l y lower.  p l o t of  C versus  the r a t i o of wind speed to wave  the observed d a t a i s  f i t t e d by the simple r e l a t i o n  =  -1),  e /e a  ( 2/c u  w  the r a t i o of the d e n s i t i e s  U2 i s  of a i r and water,  two meters,  of  o n l y c o n s i d e r e d a p p l i c a b l e f o r U 5 / c 4.  where U5 i s  T h i s formula i s  the mean wind speed at 5 meters  A l s o p r e s e n t e d are the r e s u l t s  and c i s  this  velocity 6,  height.  of a d r y - l a n d comparison of the buoy  p r e s s u r e sensor w i t h two o t h e r p r e s s u r e s e n s o r s ; the buoy sensor was adequate,  the phase  the  mean wind speed at a h e i g h t of the waves.  ;  the water on the wind i s  adequate to e x p l a i n observed r a t e s of wave growth. fall  of 1.2 x 10  besides i n d i c a t i n g that  comparison produced some  interesting  iv  preliminary  information  turbulent pressure f i e l d  on the i n the  vertical  and h o r i z o n t a l  atmospheric boundary  s t r u c t u r e of layer.  the  'TABLE OF CONTENTS  PAGE  Abstract T a b l e of  . . . . .  ii  Contents  v  L i s t of T a b l e s  vi  L i s t of F i g u r e s  .  Acknowledgements  .  viii xiv  Section  1:  Introduction  1  Section  2:  Theories  4  Section  3:  Observations  17  S e c t i o n 4:  Experiment  38  Section  Data A n a l y s i s and I n t e r p r e t a t i o n  71  5:  S e c t i o n 6:  Results  105  Section  D i s c u s s i o n of R e s u l t s  133  Conclusions  161  7:  S e c t i o n 8: Appendix 1:  Spike Removal  168  Appendix 2:  The Boundary Bay Experiment  184  Appendix 3:  Wave Damping by Adverse Winds  219  Bibliography  NOTE:  A detailed the  235  t a b l e of contents precedes each of  Appendices.  v  the S e c t i o n s and  vi  LIST OF TABLES TABLE  4.1  PAGE  P r e d i c t e d Phase Lag of P r e s s u r e Behind Wave E l e v a t i o n over Wind-Generated Waves  .  .  40  4.2  Values of K/m f o r S i x Wave Probe C a l i b r a t i o n s  5.1  Summary of  5.2  Expected E f f e c t s  48  I n f o r m a t i o n from T e s t H a n d - D i g i t i z e d Data of B a c k s c a t t e r i n g  .  .  on Wave Power  Spectrum 5.3  97  Expected E f f e c t s  of B a c k s c a t t e r i n g  on  Pressure-Waves  Cross Spectrum 6.1  Summary of  98  I n f o r m a t i o n f o r Runs w i t h Buoy on Waves:  October-November  1967  108  6.2  F u r t h e r I n f o r m a t i o n f o r Runs w i t h Buoy on Waves  6.3  Pressure  6.4  Coherence Changes  C a l i b r a t i o n s Used:  Pressure-Waves  7.1  Dimensionless Parameters  1967 i*  .  .  c / u . , TV P *  110  117 .  124  , and L f o r Runs 138  Comparison of Observed Phase Angle of P r e s s u r e to Waves a t 0.6  109  1  of Energy and Momentum from Wind to Waves  w i t h Buoy on Waves  Miles'  . . . .  Cross S p e c t r a  Mean F l u x e s  7.3  October-November  Caused by Removal of - f  6.5  7.2  78  Relative  Hz, w i t h That C a l c u l a t e d from  I n v i s c i d Laminar Model  142  Comparison of Observed Values of E w i t h Those Obtained by K o l e s n i k o v and Efimov (1962)  7.4  F r a c t i o n of  7.5  Observed and P r e d i c t e d T r a n s i t i o n Fetches  7-6  Comparison of Observed and P r e d i c t e d Values  T  =  146  the Wind S t r e s s Supported by the Waves  E/wE  .  .  149 152  of 156  vii LIST OF TABLES  (continued)  TABLE A2.1  PAGE I n f o r m a t i o n Summary f o r Boundary Bay Runs:  September,  1968 A2.2  193  Summary of R e s u l t s of  A n a l y s i s on H o t - W i r e  Anemometer S p e c t r a  196  A2.3  A d v e c t i o n V e l o c i t y of P r e s s u r e - G e n e r a t i n g  A2.4  Calculated Pressure  Differences  Eddies  . . . .  Between A i r and  Ground Sensors f o r Run 5e A3.1  214  C o r r e c t e d Phase Angles Between the Wave S i g n a l and Pressure,  A3.2  204  Sonic " B " , and S o n i c "w"  225  Comparison of T h e o r e t i c a l and Observed P r e s s u r e  and  V e l o c i t y Amplitude over S w e l l Moving A g a i n s t  the  Wind  227  viii LIST OF FIGURES  FIGURE 1.  Map of  S i t e of  Experiment  2.  A v a i l a b l e F e t c h at  3.  Photograph of Recording P l a t f o r m and Instrument  Site Masts  (Looking N o r t h e a s t ) 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)  6.  Schematic  Cross-Section  7.  Schematic  Diagram of P r e s s u r e Measurement  8.  Pressure  9.  W i r i n g Diagrams of Buoy O s c i l l a t o r and A m p l i f i e r  of Microphone  Recording E l e c t r o n i c s :  10.  FM Tuner R a t i o D e t e c t o r  11.  Pressure  12.  L a b o r a t o r y C a l i b r a t i o n of  13.  Diagram of  14.  Pressure  Sensor  ^ vs Immersion Depth  System  B l o c k Diagram  Response Curve  C a l i b r a t i o n Setup the Buoy P r e s s u r e  Sensor  the Buoy  D i s t r i b u t i o n over a P l a n e t o r y  Buoy i n  E l l i p s o i d S i m i l a r to  the  Shape  15.  Schematic  Diagram of Wind Tunnel Setup  16.  P r o f i l e of Buoy (To S c a l e )  17.  Aerodynamic C a l i b r a t i o n of Buoy: f o r V a r i o u s A t t a c k Angles  18.  Aerodynamic C a l i b r a t i o n of  P r e s s u r e vs D i s t a n c e  the Buoy:  P r e s s u r e vs  the Buoy:  F r a c t i o n of  from Bow  Distance  from  Bow f o r ^ 3 0 ° Yaw Angles 19.  Aerodynamic C a l i b r a t i o n of  Head a t P r e s s u r e P o r t vs Wind Speed  Stagnation  ix LIST OF FIGURES  (continued)  FIGURE 20.  Schematic R e p r e s e n t a t i o n  of  the E f f e c t of A t t a c k A n g l e on  the  P r e s s u r e Measured by the Buoy 21.  Effects  22.  Phase  on p + ^ * £ Phasor of 507 E r r o r i n P r e s s u r e C a l i b r a t i o n o  C o r r e c t i o n s f o r Time S h i f t of  Full 23.  Effects  C o r r e c t i o n Curve on p + fg*^  mations 24.  Phasors of Low and High-Frequency A p p r o x i -  to F u l l Phase C o r r e c t i o n 'Curve  Power S p e c t r a of  P '  P  S  25.  7£ S i g n a l Compared w i t h  Power S p e c t r a of  P >  s  P  S  W  +  :  Run 1  :  Run 2a  :  Run 2b  '  :  Run 3  ' t ZM> V.  + S  26.  Power S p e c t r a of  27.  Power S p e c t r a of  P  28.  Power S p e c t r a of  p.  >%  :  Run 4a  29.  Power S p e c t r a of  p, p + f j < 2 ,<£  :  Run 4b  30.  Power S p e c t r a of  v, p + e ^ t > ?  :  Run 5  31.  Power S p e c t r a of  :  Run 6  V  P  P + t ^ t  P '  P  p  +  s  mv.  :  R u n  7  > g  Phase  Spectra  1  Coherence S p e c t r a between p , -w ; Phase S p e c t r a between p ,7? and s *» s s ^s S  S  :  R u n  2 a  S  Coherence S p e c t r a between p , Yi ; Phase S p e c t r a between p ,io and s ts s cs p  s'  p  s  +  ^ 3 ^  :  R  u  n  2  b  s  Coherence S p e c t r a between p , Yf ; Phase S p e c t r a between p ,V and s' s »i ss ' s *s P '  P  S  36.  w  +  s  P > P + (V8"Z  35.  s#  g  s  34.  +  Coherence S p e c t r a between p , ^ p>  33.  w -  +  > P  S  32.  s  s  +  fa3^s  Run 3  ;  Coherence S p e c t r a between P >  Phase S p e c t r a between p ,  g  p ' S  p  s  +  e j n  :•  R  u  n  4  g  a  and  LIST OF FIGURES  (continued)  FIGURE 37.  S p e c t r a between p , i £ , ;  Coherence  g  P s i Ps + ^> g ^ r  38.  e  39.  P,,  e * n ,  +  :  r  to"Z  +  :  and  R  u  n  ; Phase S p e c t r a between p , •»£ and  5  Coherence S p e c t r a between p^, P s > Ps  g  Run 4b  Spectra between p , ^  Coherence P »  _:  Phase S p e c t r a between p ,y£  R  u  n  Phase S p e c t r a between p^  '7s  a  n  d  6  s  40.  Energy and Momentum F l u x S p e c t r a and Wave Power Spectrum:  Run 1 :  41.  Energy and Momentum F l u x S p e c t r a and Wave Power Spectrum:  Run 2a  42.  Energy and Momentum F l u x S p e c t r a and Wave Power Spectrum:  Run 2b  43.  Energy and Momentum F l u x S p e c t r a and Wave Power Spectrum:  Run 3  44.  Energyaand Momentum F l u x S p e c t r a and Wave Power Spectrum:  Run 4a  45.  Energy and Momentum F l u x S p e c t r a and Wave Power Spectrum:  Run 4b  46.  Energy and Momentum F l u x S p e c t r a and Wave Power Spectrum:  Run 5  47.  Energy and Momentum F l u x S p e c t r a and Wave Power Spectrum:  Run 6  48.  S p e c t r a of  {  :  Comparisons w i t h Snyder and Cox, M i l e s :  Run 1  49.  S p e c t r a of  if  :  Comparisons w i t h Snyder and Cox, M i l e s :  Run 2a  50.  S p e c t r a of  if  :  Comparisons w i t h Snyder and Coxrj; M i l e s :  Run 2b  51.  S p e c t r a of  f  :  Comparisons w i t h Snyder and Cox, M i l e s :  Run 3  52.  S p e c t r a of  f  :  Comparisons w i t h Snyder and Cox, M i l e s :  Run 4a  53.  S p e c t r a of  f  :  Comparisons w i t h Snyder and Cox, M i l e s :  Run 4b  54.  S p e c t r a of  If  :  Comparisons w i t h Snyder and Cox, M i l e s :  Run 6  55/  Dimensionless p l o t of  56 - 63  Power and  cross-spectra  and'Vave" s i g n a l s p ( t ) cycle is  £  vs U,_/c of h y p o t h e t i c a l  and * £ ( t ) ;  sinusoidal  "pressure"  removal of a "spike" once per  i n d i c a t e d by the s u b s c r i p t  "s".  xi LIST OF FIGURES  (continued)  FIGURE 56.  Power Spectrum of  P (t)  57.  Coherence between P ( t )  58.  Phase Angle between P ( t )  59.  Power Spectrum of  60.  Power Spectrum of  61.  Coherence between P ( t )  62.  Phase Angle between p ( t )  63.  Power Spectrum of  ^  64.  Comparison of  the  "^(t)  and  65.  Comparison of  the  ^(t)  and  ^g^)  Rower S p e c t r a f o r Run 2b  66.  Comparison of  the  ^ (t)  and  ^ (*-)  Power S p e c t r a f o r Run 3  67.  Comparison of  the ?^(t)  and  ^  Power S p e c t r a f o r Run 6  68.  Comparison of p , Y£ Phase S p e c t r a from " C l e a r " and " S p i k e -  g  g  ^  g  g  Comparison of p , ^  g  and  ^|t)  7  2 ( ) t  s  s  s  ^  Power S p e c t r a f o r Run 1  Run 1  Run 2b  Phase S p e c t r a from " C l e a r " and " S p i k e Run 3  Comparison of p,?£ Phase S p e c t r a from " C l e a r " and " S p i k e Run 6  Comparison of Energy F l u x S p e c t r a from " C l e a r " and " S p i k e -  Comparison of Energy  Run 1  F l u x S p e c t r a from " C l e a r " and " S p i k e -  Contaminated" d a t a : 74.  ?£ (t)  Comparison of p , J£ Phase S p e c t r a from " C l e a r " and " S p i k e -  Contaminated" d a t a : 73.  and  (t)  Contaminated" d a t a : 72.  (t)  g  Contaminated" data: 71.  and ^  P (t)  Contaminated" data: 70.  ^(t)  (t)  Contaminated" d a t a : 69.  and  g  Run 2b  Comparison of Energy F l u x S p e c t r a from " C l e a r " and " S p i k e Contaminated" d a t a :  Run 3  xii LIST OF FIGURES  (continued)  FIGURE 75.  Comparison of Energy F l u x S p e c t r a from " C l e a r " and " S p i k e Contaminated" d a t a :  Run 6  76.  Map of  S i t e Area f o r Boundary Bay Experiment  77.  Photograph of Equipment Deployment P r i o r  78.  Buoy Sensor  79.  Frequency Response of  80.  T y p i c a l Power Spectrum of Downwind V e l o c i t y F l u c t u a t i o n s  81.  Comparison of Run l a  the Power S p e c t r a from Three P r e s s u r e  Sensors:  82.  Comparison of  the Power S p e c t r a from Three P r e s s u r e  Sensors:  to a Run  C a l i b r a t i o n f o r Boundary Bay Experiment C a l i b r a t i o n Drum  Run 5 a 83.  Non-dimensional P r e s s u r e Spectrum from Boundary Bay  84.  Coherence between "Buoy" and " A i r " P r e s s u r e  85.  Phase Angle between "Buoy" and " A i r " P r e s s u r e  86.  Coherence between "Buoy" and "Reference" P r e s s u r e  87.  Phase Angle between "Buoy" and "Reference" P r e s s u r e  88.  Coherence between "Buoy" P r e s s u r e Sensor and H o t - w i r e A i r Speed  89.  Phase A n g l e between "Buoy" P r e s s u r e Sensor and H o t - w i r e A i r Speed  90.  Coherence between " A i r " P r e s s u r e Sensor and H o t - w i r e A i r Speed  91.  Phase Angle between " A i r " P r e s s u r e Sensor and H o t - w i r e A i r Speed  92.  Coherence between "Reference" P r e s s u r e Sensor and H o t - w i r e A i r  Sensors Sensors Sensors Sensors  Speed 93.  Chart Recording of Wave, P r e s s u r e ,  and Sonic Anemometer  d u r i n g Passage of 4-Second S w e l l Group 94.  S c a l e Drawing of Instrument Setup f o r Run 5  Signals  xiii LIST OF FIGURES  (continued)  FIGURE 95.  Time V a r i a t i o n of U, Q Second I n t e r v a l s )  from Sonic Anemometer (Means over T e n -  f o r Run 5  96.  Coherence S p e c t r a d u r i n g Passage of Swell Group  97.  Phase S p e c t r a d u r i n g Passage of S w e l l Group  xiv  ACKNOWLEDGEMENTS  T h i s work has been p a r t of of  the A i r - S e a I n t e r a c t i o n r e s e a r c h program  the I n s t i t u t e of Oceanography a t  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 s t a y a t IOUBC I have been supported by the  I n s t i t u t e of Oceanography, a UBC Graduate F e l l o w s h i p , Research C o u n c i l Postgraduate I would l i k e  Scholarship.  to thank D r . R. W. Stewart  many s t i m u l a t i n g d i s c u s s i o n s ,  and a N a t i o n a l  and D r . R. W. B u r l i n g  and D r . B u r l i n g f o r h i s  painstaking  for examina-  t i o n of my t h e s i s . All  of  the graduate  students and s t a f f  me i n my work one way or a n o t h e r , p a r t i c u l a r I am deeply indebted amount of h i s  spent some of my most p l e a s a n t  the I n s t i t u t e have  and I thank them c o l l e c t i v e l y .  helped In  to John G a r r e t t , who s p e n t an i n o r d i n a t e  time on my b e h a l f ,  v a l u a b l e p i e c e of d a t a f o r me.  of  and to Ron W i l s o n , who h a n d - d i g i t i z e d a I a l s o thank Jim E l l i o t t , w i t h whom I  and p r o d u c t i v e s c i e n t i f i c  hours  at  Boundary Bay. L a s t l y I thank my w i f e E v e l y n , many  who l o v i n g l y  typed t h i s  times.  F.  W. Dobson  thesis  so  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: 1.1  Purposes of  INTRODUCTION  the Research  The p r i n c i p a l  concern of  this  t h e s i s i s w i t h the d e t e r m i n a t i o n of  the c o r r e l a t i o n between f l u c t u a t i o n s  i n s u r f a c e p r e s s u r e and s u r f a c e  e l e v a t i o n on wind-generated g r a v i t y waves. of and  two important parameters:  T h i s permits the  the v e r t i c a l f l u x e s  estimation  of m e c h a n i c a l  energy  of momentum from the wind to the waves. As such the r e s e a r c h 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 t h e r e f o r e has been c a r r i e d out as p a r t of A i r - S e a I n t e r a c t i o n program at the I n s t i t u t e U n i v e r s i t y of B r i t i s h A knowledge of insight into  1960)  the energy f l u x from the wind to the waves p r o v i d e s  the wave g e n e r a t i o n process the " c l a s s i c a l "  itself.  growth of wind waves a t sea.  Even today t h i s  process  theory (the M i l e s - P h i l l i p s theory:  i s now thought to be inadequate  to e x p l a i n the  observed  Both M i l e s and P h i l l i p s (1957) have  t h a t the least-known v a r i a b l e i n t h e i r equations at the sea s u r f a c e .  the  Columbia.  i s not w e l l - u n d e r s t o o d ; Miles,  of Oceanography of  the  is p ( x , t ) ,  The p r e s s u r e measured i n t h i s  the time v a r i a t i o n of p r e s s u r e a t one  the  stated  pressure  experiment i s  p(o,t),  location.  The M i l e s theory d e a l s w i t h energy t r a n s f e r from the mean flow the a i r to the waves v i a a p o s i t i v e flow s t r e a m l i n e s energy t r a n s f e r .  feedback p r o c e s s ,  whereby the mean  are m o d i f i e d by e x i s t i n g waves so as to i n c r e a s e The end r e s u l t of  the theory i s  normal p r e s s u r e a t the water s u r f a c e . 1  the  a prediction for  energy t r a n s f e r i n terms of the magnitude and phase of An o b j e c t of  in  the  the wave-induced  t h i s work i s  to  2  measure the energy t r a n s f e r by measuring t h i s i t w i t h the t h e o r e t i c a l Stewart  pressure,  and to  compare  predictions.  (1961) has p r e d i c t e d t h a t a l a r g e f r a c t i o n of  the  total  wind s t r e s s on the water s u r f a c e goes d i r e c t l y i n t o wave momentum. knowledge  of  the energy f l u x  to the waves i m p l i e s ,  almost i r r o t a t i o n a l , a knowledge instance,  Stewart  1961).  of  assuming they  the momentum f l u x  T h e r e f o r e a secondary o b j e c t  momentum f l u x measured i n t h i s way w i t h t h a t e i t h e r logarithmic v e r t i c a l profiles  are  to them (see, is  for  to compare  the  i n f e r r e d from assumed  of mean wind speed o r measured by measuring  the Reynolds s t r e s s 7j* = _ putw and thus o b t a i n an e s t i m a t e of Stewart's  A  the s i z e of  fraction.  In s p i t e of a g r e a t d e a l of r e c e n t work ( M i l e s 1957, 1960,  1962,  1965,  1959;  Lighthill  1967;  1962;  P h i l l i p s 1957,  1966,  1967;  1959 a and b ,  Bryant 1966;  Stewart 1967), no theory e x i s t s  Benjamin  today which p r e -  d i c t s w i t h i n an o r d e r of magnitude the energy t r a n s f e r from the wind to the waves i t theories  is  generates.  the l a c k of good e x p e r i m e n t a l  the a i r over waves. v i d e some of  A p r i n c i p a l reason f o r the f a i l u r e of  the  i n f o r m a t i o n on the flow of  The p r i n c i p a l o b j e c t of  this  experiment i s  to  pro-  the m i s s i n g i n f o r m a t i o n , and i n so doing to throw some l i g h t  on the b a s 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 p u t t i n g a p r e s s u r e sensor on the water  s u r f a c e and of g e t t i n g negligible.  it  to measure normal p r e s s u r e s  The f i r s t d e c i s i o n  to be made i s whether  there  the buoy c a r r y i n g  the p r e s s u r e sensor should be allowed to d r i f t as f r e e l y as (Lagrangian measurement)  are f a r from  possible  or to be c o n s t r a i n e d to move o n l y 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 a n a l y s i s d i f f i c u l t i e s  insurmountable i n the former measurement,  the l a t t e r  is  felt  seem  to be  mandatory. U n f o r t u n a t e l y 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 ; it  f a i r l y b r i s t l e s w i t h problems.  The p r i n c i p a l one i s k e e p i n g water  away from the p r e s s u r e s e n s i n g p o r t ; a f i l m of water 1 mm t h i c k causes a s p i k e i n the p r e s s u r e r e c o r d l a r g e r by a f a c t o r of f i v e  than the  a i r p r e s s u r e amplitudes a c t u a l l y observed i n the d a t a .  In a Lagrangian  measurement i t would be p o s s i b l e time i n a f a i r l y not.  largest  to keep water from the sensor 90% of  s t r o n g wind; w i t h the q u a s i - E u l e r i a n measurement i t  T h e r e f o r e s p e c i a l d a t a p r o c e s s i n g must be used to e x t r a c t  the is  useful  i n f o r m a t i o n from a p r e s s u r e s i g n a l on which are superimposed numerous water-induced  spikes.  Another major cause f o r c o n c e r n , s e p a r a t e from the problem,  is  the e f f e c t of  the p r e s s u r e s  i t measures.  the shape of  the v e h i c l e  1  flows  c a r r y i n g the sensor on  Care must be taken to ensure r e j e c t i o n of  s o - c a l l e d "dynamic" p r e s s u r e i - P "U' , where f. Vet wind speed,  Lagrangian-Eulerian  p  is  the  a i r d e n s i t y and U  caused by the s t r e a m l i n e c o n f i g u r a t i o n s e t up as the a i r  over the buoy. The l a s t  ographers  problem to be mentioned i s  the one which haunts a l l ocean-  (except perhaps the Ivory-Tower b r e e d ) :  e l e c t r o n o m i c s work i n the presence of s a l t  water.  t h a t of making s e n s i t i v e  SECTION 2:  THEORIES  2.1  The K e l v i n - H e l m h o l t z I n s t a b i l i t y  4  2.2  Jeffreys'  S h e l t e r i n g Hypothesis  5  2.3  The S t a b i l i t y A n a l y s e s of Wuest and Lock  7  2.4  E c k a r t ' s Model  8  2.5  Phillips  9  2.6  Miles'  2.7  (1957)  (1957) I n v i s c i d Laminar Model  11  2.6.1  Assumptions  11  2.6.2  Theory  12  Miles'  .  .  (1962) V i s c o u s Laminar Model  14  SECTION 2:  THEORIES  From the b e g i n n i n g , man's encounter w i t h the sea has been a stormy one. of  Extended sea voyages,  the winds, were s e r i o u s l y  generated. The  2.1  "How?" has no s a t i s f a c t o r y  the c o m p l e x i t y of the p r o c e s s .  theoretical  w i t h the a i d  impeded by the waves which those winds  That the wind generates sea waves has  question,  indicates  be  f o r a long time o n l y p o s s i b l e  answer;  long been  recognised.  i n some sense  this  In the f o l l o w i n g paragraphs  p r o g r e s s made up to the time when t h i s work was begun w i l l  reviewed.  The K e l v i n - H e l m h o l t z I n s t a b i l i t y The  first  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 d e n s i t y  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 (1868).  to each other was H . Helmholtz  The problem was t a c k l e d i n more d e t a i l  (Lord K e l v i n ) , i s known as  who i n c l u d e d the e f f e c t s  i n 1874 by S i r W. Thomson  of s u r f a c e  the " K e l v i n - H e l m h o l t z I n s t a b i l i t y " ,  tension.  and was  His  solution  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, §  s i d e r e d as i n v i s c i d . to be of n e g l i g i b l e  268).  Both f l u i d s  The boundary l a y e r between the two f l u i d s thickness  The mean flow of both f l u i d s speed U w i t h d i s t a n c e  compared w i t h wave h e i g h t s is  to be  water speed  is  tension T i s  Solutions  ^  is  assumed  considered.  from t h e i r common b o u n d a r y ) , but d i f f e r e n t  For the case c o n s i d e r e d here the a i r speed i s U  surface  is  taken, to be u n i f o r m (no v a r i a t i o n i n  two media.  zero.  are con-  a i r density  i n c l u d e d i n the  and  ^  w  i s water  a  and the  density;  analysis.  are looked f o r i n the case f o r which the i n t e r f a c e 4  in  is  the  5 deformed w i t h a t r a v e l l i n g wave of  the form exp i k ( x - c t )  number k i s known and the wave phase v e l o c i t y c i s The r e s u l t i n g c i s  C =  is  ±  r J 3 (Cw-g>)  . Tfc  fwg>U> j  _  l a r g e enough to make the c e x p r e s s i o n complex;  for instance,  the p o s i t i v e  root  Because water waves have a minimum v e l o c i t y  Lamb 1932,  § 267)  there i s  which generate no waves of any wave l e n g t h ; s t a b i l i t y c r i t e r i o n yields can be e f f e c t i v e ,  1  2  grow e x p o n e n t i a l l y i f f o r a g i v e n k the a i r speed  s i g n produces growth. (see,  to be s o l v e d f o r .  complex i n g e n e r a l :  The waves w i l l Ua  where the wave  a range of wind speeds C  s u b s t i t u t i o n of  w  into  the minimum wind speed a t which t h i s mechanism  about 650 cm/sec.  S i n c e 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 a t much lower speeds,  and s i n c e  in fact  assumption of a v e r y t h i n i n t e r f a c i a l boundary l a y e r i s u n r e a l i s t i c , mechanism i s not thought to be v e r y e f f e c t i v e wind speeds and wavelengths  over most of  commonly observed a t sea  The most t e l l i n g argument a g a i n s t  the e f f i c a c y  Helmholtz mechanism has been g i v e n by Stewart If  the  e x p o n e n t i a l wave growth i s  the  this  the ranges of  (see M i l e s of  the  1959b).  Kelvin-  ( p e r s o n a l communication).  o c c u r r i n g then the b r a c k e t e d term on the  r i g h t hand s i d e of e q u a t i o n 2.1  is  pure i m a g i n a r y .  p r o p a g a t i o n v e l o c i t y of the waves b e i n g generated i s  T h i s means that g i v e n by  which i s v e r y s m a l l f o r the case of a i r f l o w i n g over water.  the Ua.  The wave  must grow s t r a i g h t up and h a r d l y propagate at a l l , a h i g h l y u n r e a l i s t i c s t a t e of  2.2  affairs.  Jeffreys'  S h e l t e r i n g Hypothesis  S i r Harold Jeffreys  (1925) re-examined the problem and produced  6 another mechanism which c o u l d generate waves. a i r flows s i d e of  over the wavy water s u r f a c e ,  the wave c r e s t s  leeward s l o p e of  s e p a r a t i o n occurs on the  the  leeward  w i t h r e - a t t a c h m e n t somewhere f u r t h e r down on the  the wave.  downwind s l o p e s of  He assumed t h a t as  T h i s produces low ambient p r e s s u r e s on the  the waves,  and hence a p r e s s u r e f i e l d coupled to  the  wave p r o f i l e i n such a way t h a t the c o r r e l a t i o n  of p r e s s u r e and wave  vertical velocity is  energy t r a n s f e r  the a i r flow to the  positive,  producing a p o s i t i v e  waves.  He assumed t h a t t r a n s f e r s In h i s  caused by shear s t r e s s e s are n e g l i g i b l e .  c a l c u l a t i o n s he r e p l a c e d the sea s u r f a c e w i t h a s i n g l e  component, on t h i s  a l o n g - c r e s t e d s i n e wave.  where s i s  less  coefficient".  which i s  He then c a l c u l a t e d the work done  g i v e n by  than one arid was c a l l e d by J e f f r e y s He c a l c u l a t e d s by f i r s t  l o s s due to m o l e c u l a r v i s c o s i t y J  m a i n t a i n waves a g a i n s t  this  l e a s t winds which seemed Since i n J e f f r e y s ' c h o i c e of  and compared t h i s w i t h "observed"  j u s t capable of g e n e r a t i n g  m  argument.  n  He chose U ± m  (1956) reviews  n  his  waves.  c a l c u l a t e d s v a r i e s as this represents  (U ^ )^j m  n  his  a weak p o i n t  to be 110 cm sec ~ \ g i v i n g an s of  0.3.  the d a t a a v a i l a b l e a t t h a t time on p r e s s u r e  v a r i a t i o n s over s o l i d model waves i n wind t u n n e l s : (1932), M o t z f e l d  energy  and then the l e a s t wind U . t h a t can mm  loss,  theory,  a "sheltering  c a l c u l a t i n g the r a t e of  the " c o r r e c t " U ^ was c r i t i c a l ;  Ursell  et a l  Fourier  s i n e wave by the component of normal p r e s s u r e i n quadrature w i t h  7£ , the wave h e i g h t ,  in his  from  (1937),  the works of  and T h i j s s e (1951), and f i n d s  Stanton  little  evi-  7 dence to support a v a l u e f o r s as l a r g e as t h a t suggested by J e f f r e y s : "The evidence  of  the three s e t s of measurements  the c o n c l u s i o n t h a t the p r e s s u r e d i f f e r e n c e s  . . . on the whole  favours  over a s o l i d p r o f i l e com-  posed of a number of waves are an o r d e r of magnitude s m a l l e r than the differences  p o s t u l a t e d by J e f f r e y s . "  He goes on to p o i n t out t h a t no  experiments had been done (1956) over l i q u i d p r o f i l e s , f o r e the q u e s t i o n of  the a c t u a l s i z e of s i s  some such o b s e r v a t i o n s ; It  is  they w i l l be d i s c u s s e d  i s not the o n l y type of  waves and produce the n e c e s s a r y  Lighthill  open.  there-  There are now  i n "Observations".  perhaps worth n o t i n g here t h a t the type of s e p a r a t i o n suggested  by J e f f r e y s  pressures.  still  and t h a t  There i s (1962),  also  s e p a r a t i o n which can e x i s t over  the  phase quadrature between waves and  the p o s s i b i l i t y ,  first  c l e a r l y p o i n t e d out by  t h a t flow r e v e r s a l can occur over the waves i n  c o - o r d i n a t e system moving a t the wave phase speed,  and t h a t t h i s  the type of  " s e p a r a t i o n " can produce the r e q u i r e d phase q u a d r a t u r e .  2.3  The S t a b i l i t y A n a l y s e s of Wuest and Lock Wuest (1949) and Lock (1954) both t r e a t e d the problem of  f l o w over a s e m i - i n f i n i t e  plate.  T h e i r work i s  A l t h o u g h t h e i r a n a l y t i c a l procedures d i f f e r , conditions necessary  reviewed by U r s e l l  they both looked f o r  f o r the onset of i n s t a b i l i t i e s  They used the same p h y s i c a l model:  a i r flows  the i n f l u e n c e of g r a v i t y and w i t h s u r f a c e t e n s i o n ; pressure gradient.  and grow downstream.  i n the  (1956). the  flow.  over deep water under the a i r flow has no  V i s c o s i t y causes boundary l a y e r s  Both l a y e r s are assumed to have i n s t a b i l i t i e s  the laminar  i n a i r and water.  which s t a r t a t some l o c a t i o n  O u t s i d e the boundary l a y e r i n the a i r the flow  i n i t i a l l y u n i f o r m and the water a t r e s t .  The i n t e r f a c e  is  is  then p e r t u r b e d .  8 with a small  (much s m a l l e r  d i s t u r b a n c e of wavelength s t a r t of  the  layer.  the s t a b i l i t y X  of  than the boundary l a y e r X  much l e s s than the d i s t a n c e  The problem i s  x  sinusoidal from the  then reduced to the d e t e r m i n a t i o n of  t h i s flow f o r v a r i o u s wind speeds and v a l u e s of  X  and  • Their results,  situation  as U r s e l l p o i n t s  (or f o r t h a t m a t t e r ,  What was p r e d i c t e d i n f a c t was from laminar f l o w , turbulent.  while  The models,  out,  are hard to r e l a t e  to any simple l a b o r a t o r y the c o n d i t i o n f o r  being determinations  of  a i r or the  to any  field  experiment).  t r a n s i t i o n to  the flow over wind-generated  s a i d n o t h i n g about the subsequent behaviour of  2.4  thickness)  turbulence  waves i s  the onset of  invariably instability,  the i n s t a b i l i t i e s  in  the  water.  E c k a r t ' s Model Eckart  (1953) proposed a model which was  the f o r e r u n n e r of P h i l l i p s '  (1957) T h e o r y . 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 "storm" e x i s t s . of  The storm c o n s i s t s of a s t a t i o n a r y  circular  random d i s t r i b u t i o n  s i m i l a r "gusts" ( r e g i o n s of r e l a t i v e l y h i g h normal p r e s s u r e ) .  model n e g l e c t s shear s t r e s s e s and any d i s t o r t i o n of produced  the flow by wave-  feedback.  The p r e s s u r e v a r i a t i o n s equations coupled  The  i n the gusts are s m a l l enough so  of motion can be made l i n e a r .  (i.e.  they move over  through a feedback mechanism) the water  long enough t h a t stationary.  The gusts are not  at  conditions  the wind speed,  that  the  directly  to any waves a l r e a d y and have been p r e s e n t  w i t h i n the "storm" and o u t s i d e i t  are  present; for  9 The problem was posed i n terms of F o u r i e r integrals,  and the r e s u l t s  are s t a t i s t i c a l .  Stieltjes  He o b t a i n e d an e x p r e s s i o n  f o r mean-square wave h e i g h t i n terms of mean-square p r e s s u r e a t "storm" c e n t r e , storm.  "storm" d i a m e t e r , "gust" d i a m e t e r , and d i s t a n c e from the  He then used known wave h e i g h t s  mean-square p r e s s u r e s h i s  from a r e a l  theory p r e d i c t e d .  storm to see  2  ,  the f u l l  what  The p r e s s u r e s r e q u i r e d are  an o r d e r of magnitude too l a r g e (they would have to be g r e a t e r 1 p U.  the  dynamic head f o r a i r moving at a speed U  a  than  ).  This  Va  leads him to b e l i e v e  t h a t randomly d i s t r i b u t e d normal p r e s s u r e s  a i r are p r o b a b l y i n s u f f i c i e n t his  conclusion is  i n the e s t i m a t i o n o f p r e s s u r e s ;  measurements were a v a i l a b l e o f the open ocean.  to generate waves.  i n the  The weakest p o i n t i n a t t h a t t i m e , no  the mean-square p r e s s u r e over waves i n  I n the l i g h t of r e c e n t measurements,  his  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 g e n e r a t i o n theory which  a l o n g w i t h t h a t of M i l e s ten y e a r s .  (1957) has r e c e i v e d much a t t e n t i o n over the  S i n c e the p r e d i c t i o n s of  past  the theory are not d i r e c t l y a p p l i -  c a b l e to the p r e s e n t e x p e r i m e n t a l r e s u l t s ,  only a brief  outline is  given  the subsequent motion on an i n i t i a l l y f l a t  water  below. Phillips surface after  considers  the onset of a t u r b u l e n t wind.  assumed to be i n v i s c i d and i r r o t a t i o n a l . as the t u r b u l e n t p r e s s u r e f l u c t u a t i o n s water s u r f a c e ,  forced o s c i l l a t i o n s  The water motions  are  Stated s i m p l y , he f i n d s  i n the a i r are c a r r i e d over  that the  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 p r e s e n t i n the a i r p r e s s u r e  10 spectrum.  Of t h i s  initial  broad wave spectrum, two components f o r each  F o u r i e r component i n the p r e s s u r e f i e l d correspond to f r e e g r a v i t y wave oscillations:  those two which move a t angles  to the wind such that  p r o p a g a t i o n speed i n the wind d i r e c t i o n equals of  t h e i r generating pressure f l u c t u a t i o n s .  long as the p r e s s u r e f l u c t u a t i o n s waves;  that i s ,  the " a d v e c t i o n v e l o c i t y "  These components grow as  r e t a i n t h e i r phase r e l a t i v e to  they "resonate" w i t h the p r e s s u r e  The development of the waves f a l l s  their  the  field.  i n t o two s t a g e s ,  depending on  whether the "time s c a l e " f o r which the atmospheric p r e s s u r e  fluctuations  e x i s t and m a i n t a i n t h e i r phase r e l a t i v e to the waves i s g r e a t e r or l e s s than the e l a p s e d  time from the onset of  the wind.  I n 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 =[4jj[jM m  V ?w/  (where g i s  the s u r f a c e t e n s i o n of the water, angles  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  and  ^  is  to the wind g i v e n by.O^s COS*(c«,/U j) a  v e l o c i t y of  the g e n e r a t i n g p r e s s u r e  its  density).  > where U a i s a  They move a t the  advection  fluctuations.  I n the " p r i n c i p a l " development stage the phase of a g i v e n F o u r i e r component of the p r e s s u r e wanders r e l a t i v e to the wave phase, result  the wave amplitude grows a s V t  derives where a and  and as a  , as i n a random walk problem.  f o r the mean-square wave amplitude the e q u a t i o n a * f^t/z^ 1  i s wave a m p l i t u d e , p i s  U,j i s  atmospheric p r e s s u r e ,  the a d v e c t i o n v e l o c i t y of  To v e r i f y h i s  p  He  <} U j ,  i s water  a  density,  the p r e s s u r e f i e l d .  theory he e s t i m a t e s the approximate s i z e of the rms  pressure f l u c t u a t i o n s  to be  O.lp^U*  2.3  where  ^>  is  a i r d e n s i t y and (J* i s mean wind speed a t "anemometer.  h e i g h t " - - u s u a l l y 10 m e t e r s . magnitude agreement of h i s v a l i d i t y of p.  the e s t i m a t e  He f i n d s  that t h i s  estimate  gives order-of-  theory w i t h observed wave growth.  2.3  is  discussed  i n " D i s c u s s i o n of  The Results",  139. Miles  (1960) has i n c o r p o r a t e d the above model i n t o a composite  involving i t  and h i s own:  Miles  (1957).  T h i s w i l l be d i s c u s s e d  one  i n the  f o l l o w i n g paragraphs.  2.6  Miles'  (1957) I n v i s c i d Laminar Model  The p r e s e n t work p r o v i d e s a t e s t of posed i n 1957 by M i l e s ;  therefore,  this  detail  than the o t h e r s mentioned so f a r .  2.6.1  Assumptions The a i r i s  water;  the wave g e n e r a t i o n theory p r o theory w i l l be d i s c u s s e d  assumed to be i n v i s c i d and i n c o m p r e s s i b l e , as i s  the a i r i s  g i v e n a p r e s c r i b e d mean shear flow  the  ( i n the absence of  waves) which v a r i e s w i t h h e i g h t above the s u r f a c e o n l y . that there i s  i n more  It  is  assumed  a feedback from the waves such t h a t wave-induced a i r  v e l o c i t y and p r e s s u r e p e r t u r b a t i o n s are two-dimensional and s m a l l enougtr to be unimportant i n the n o n l i n e a r processes Turbulence i n the a i r i s n e g l e c t e d  of  the flow i s  Because of  the l a s t  assump-  called "quasi-laminar".  Mean water c u r r e n t s are assumed to be a b s e n t . is  of m o t i o n .  (except f o r the i m p l i c i t assumption  t h a t i t m a i n t a i n s the p r e s c r i b e d shear f l o w ) . tion,  the equations  The water-wave motion  assumed to be i r r o t a t i o n a l , and wave amplitudes and s l o p e s s m a l l  enough t h a t l i n e a r wave theory (see, may be u s e d .  The magnitude of  for instance,  the speed of  Phillips,  the waves i s  1966, £ 3.2)  assumed to be  t h a t of f r e e g r a v i t y waves p l u s a s e c o n d - o r d e r p e r t u r b a t i o n term which is  caused by the component of aerodynamic p r e s s u r e i n phase w i t h  wave  the  slope.  2.6.2  Theory The  wave e l e v a t i o n of a g i v e n F o u r i e r wave component i s  taken to  be  _7£(x,t)  -  a exp[ctecx-ct)]  a m p l i t u d e , . k = ETT/'X  where a i s  x is horizontal distance phase speed,  and t i s  ( X i s wavelength)  where  ^  is  time.  the wave number, c is  the wave  The p e r t u r b a t i o n aerodynamic p r e s s u r e to be  = ( o l + c|3) p U ^ k ^  2  f t  a i r d e n s i t y , U,  2.5 {^/ P,g^^ > where t" i s  is  a " r e f e r e n c e speed"  (later  defined  -> 5  as  the t o t a l momentum f l u x from a i r to w a t e r ) ,  ok. and |3 are d i m e n s i o n l e s s The  is  (the waves are l o n g - c r e s t e d ) ,  a s s o c i a t e d w i t h the mean flow he takes  p  2.4,  and  coefficients.  hydrodynamic equations  are then s o l v e d f o r the wave v e l o c i t y  c,  giving  = Cw + e /e (<*• + ^p) u, *  2.6,  2  a  where C  w  is  the speed of f r e e g r a v i t y waves ( C  k i n deep water) water.  w  and  £  w  is  the r a t i o of  » V g / h . . f o r wave number  the d e n s i t i e s  of a i r and  He then assumes  Xw .» e./f l* 1  and  w  expands c about c , w  w  +  cp)u  t  i  k e e p i n g o n l y the f i r s t  two terms.  This  gives  13  c  c  n  w  +  ^e«/e i^  ip)(u,/cw)  +  w  S u b s t i t u t i n g 2.7 i n t o 2.4 g i v e s f o r the s u r f a c e  ^Cx,t)& where i t  is  aexp[lf./  assumed  |a + c<3| Miles  then  j  2.7.  elevation 2.8,  phc (U / J tJe«p[tk(K-c tg l  e  w  w  1  c  w  that  «  2.9.  e /e (cw/u,)* w  a  defines  2.10 as the f r a c t i o n a l i n c r e a s e i n wave energy E per r a d i a n .  r-(e./ft,)P(u./e«) In  a l a t e r paper ( M i l e s 1960)  wave p r o p a g a t i o n angle In  generalized for  arbitrary  2. to the wind) 6 to / = ( ? « / P ) | 3 (u,COS0/Cw)  the same paper h i s own model i s  Phillips  this  1  the r e l a t i o n i s  (relative  From 2.8  w  g e n e r a l i z e d to i n c l u d e t h a t of  (1957).  He f i n d s  t h a t to o b t a i n JT he must s o l v e the i n v i s c i d form of  Orr-Sommerfeld e q u a t i o n (see, He f i n d s  that  field  a d i r e c t f u n c t i o n of  is  a t the h e i g h t The  for instance,  L i n , 1955,  (the " c r i t i c a l " h e i g h t  z  c  (Conte and M i l e s ,  1959)  U =  C . w  integrated numerically  f o r the commonly-observed l o g a r i t h m i c p r o f i l e  U ( 2 ) = U, l o g ( z / z o ) = 2 . 5 u j o o , (Z/z ) a  where U = .(t'/^ ) * i s  the wave  the v e r t i c a l wind shear  ) above the wave where  i n v i s c i d Orr-Sommerfeld e q u a t i o n i s  #  Equation 1.3.15).  the growth r a t e of a g i v e n F o u r i e r component of the c u r v a t u r e of  the  the " f r i c t i o n v e l o c i t y " , T  2.11 the Reynolds s t r e s s ,  and  14 Z  0  is  103).  a "roughness  length"  (see,  f o r example,  Lumley and Panofsky,  The e q u a t i o n has a l s o been s o l v e d f o r s e v e r a l  files  and i n c l u d i n g v i s c o u s  sinusoidal  p.  d i f f e r e n t wind p r o -  e f f e c t s by Benjamin (1959), u s i n g an o r t h o g o n a l  c o - o r d i n a t e - system which f o l l o w s  the unperturbed flow  stream-  lines . The r e s u l t of M i l e s ' (5  computations  to C / ( j f o r v a r i o u s v a l u e s w  (  of  is  a curve ( M i l e s 1959a)  the wind p r o f i l e  relating  parameter  2.12. Other curves presented  i n the same paper r e p r e s e n t h i s p r e d i c t i o n s  the energy i n p u t r a t e from the wind to the waves and of between a i r p r e s s u r e and wave e l e v a t i o n .  Some of  the phase  the p r e d i c t e d  of angle  phase  angles have been c a l c u l a t e d f o r p r o f i l e parameters expected  i n the  ments done a t the p r e s e n t  i n Table 4.1.  2.7  Miles' Miles  (1962), is  f e r r e d by "viscous  "viscous ture of  of  the  i n s t u d y i n g the growth of  Reynolds s t r e s s e s " from the wind to c r i t i c a l layer is  so s m a l l t h a t i t  s u b l a y e r " immediately above the water the mean wind p r o f i l e i s  curvature,  is  for i t s  ineffective.  waves on  r a t u r e w i t h the waves.  is  surface,  trans-  the waves when i n the  so-called  where the  curva-  l i n e a r and so the i n v i s c i d laminar a c t i o n on the e x i s t e n c e of  The term "viscous  the energy t r a n s f e r occurs  referred  short-wavelength  l e d to propose a theory whereby energy may be  mechanism, which depends  that  they have been presented  (1962) V i s c o u s Laminar Model  shallow water,  the h e i g h t  site,and  experi-  profile  Reynolds s t r e s s e s "  through the a c t i o n of p r e s s u r e s  implies i n quad-  The t h e o r e t i c a l model d e s c r i b e d i n the paper  to as the "viscous  laminar" model.  is  15 The theory c o n s i d e r s  the growth of  s u r f a c e of a s l i g h t l y v i s c o u s finite  depth which i s  liquid  subjected  normal and t a n g e n t i a l  two-dimensional waves on the  (which we w i l l  call  the water)  to p r e s c r i b e d s u r f a c e s t r e s s e s .  s t r e s s e s are c a l c u l a t e d by s o l v i n g the  of  These  viscous  Orr-Sommerfeld e q u a t i o n f o r the a i r above the wavy water s u r f a c e  i n the  o r t h o g o n a l s i n u s o i d a l c o - o r d i n a t e system used by Benjamin (1959). By s u b j e c t i n g  the equations  of motion f o r the water  d i t i o n s which match the c a l c u l a t e d normal and t a n g e n t i a l air-water interface, speed i s  an e i g e n v a l u e  The s o l u t i o n s posed of  the f i r s t  by the wavy motion of  s t r e s s e s at the s u r f a c e and at the bottom;  f o r 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 p e r t u r b e d the water.  M i l e s p o i n t s out t h a t a l t h o u g h the two  i(  wavelengths,  a p o s s i b i l i t y of resonance between them a t 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 where u  computed  s u r f a c e water waves  c l a s s e s of waves are independent a t most wind speeds and there i s  phase  e q u a t i o n f o r the water waves are com-  s o l u t i o n gives free  damped by the a c t i o n of v i s c o u s the second s o l u t i o n i s  the phase speed i s  the  the waves.  to the e i g e n v a l u e  two p a r t s :  stresses at  e q u a t i o n f o r the complex wave  o b t a i n e d and the imaginary p a r t of  to o b t a i n the growth r a t e of  to boundary con-  is  growth of  the f r i c t i o n v e l o c i t y  the f i r s t  mate s o l u t i o n s  (u* = 5 cm sec  ^ and \ s  as d e f i n e d i n e q u a t i o n 2 . 1 1 ) .  c l a s s of waves i s  discussed  5 cm, The  i n some d e t a i l .  Approxi-  are found f o r the growth r a t e and growth r a t e curves  are  presented. S i n c e the measurements made i n the p r e s e n t almost c o m p l e t e l y o u t s i d e model,  its  set  the range of v a l i d i t y of  p r e d i c t i o n s w i l l not be d i s c u s s e d  of experiments the v i s c o u s  are  laminar  i n any more d e t a i l .  The  16 range of v a l i d i t y o f the model w i l l be g i v e n , however, may be d i s c u s s e d i n terms of The t h i c k n e s s of  S where  1)^  is  Si  s  the v a r i o u s wave g e n e r a t i o n  the laminar s u b l a y e r  theories.  5 Dci/U.»  2.13  the k i n e m a t i c v i s c o s i t y of C  to  .  results  is  the a i r .  c h a r a c t e r i s t i c l e n g t h <5 f o r the t h i c k n e s s of ing  so t h a t the  Miles defines  the c r i t i c a l  a  layer accord-  )/.  5 =(V./Ulk)'  s  C  1  where U  c  and k i s  is  the s l o p e of  the mean wind p r o f i l e at the c r i t i c a l  the wavenumber of the waves.  He then d e f i n e s  height  a non-dimensional  height Z and f i n d s  =  Z /<$c c  t h a t the v i s c o u s  laminar model a p p l i e s over the range  0<2<2.3.  He computes n u m e r i c a l v a l u e s f o r wave growth assuming a mean wind p r o f i l e which i s His  p r e d i c t i o n s cover wavelengths  5 - 30 cm sec of  l i n e a r i n the laminar s u b l a y e r and l o g a r i t h m i c above  c/u* less  ^.  from 1 - 10 cm and v a l u e s of u^ from  The energy t r a n s f e r i s  thus a p p r e c i a b l e o n l y f o r v a l u e s  than about 8.  T h i s concludes tion.  it.  the o u t l i n e s of  The d i s c u s s i o n s  t h e o r e t i c a l progress on wave genera-  g i v e n are f a r from e x h a u s t i v e ,  have been advanced s i n c e  the date of  and o t h e r  the 1962 M i l e s p a p e r .  theories  Any of  these  newer ideas which are r e l e v a n t to the p r e s e n t work are mentioned i n context,  d u r i n g " D i s c u s s i o n of R e s u l t s " .  SECTION 3:  3.1  . Sverdrup and Munk (1947) 3.1.1  3.3  17  Comparisons w i t h T h e o r i e s of M i l e s and P h i l l i p s  3.2  OBSERVATIONS  F i e l d Measurements  (1957)  (1957) of Normal P r e s s u r e s  3.2.1  K o l e s n i k o v and Efimov (1962)  3.2.2  Longuet-Higgins,  3.2.3  Summary  18 over Waves  .  .  .  .  20  C a r t w r i g h t and Smith (1963)  . . .  .  22 25  Recent O b s e r v a t i o n s  26  3.3.1  T e s t s of M i l e s ' V i s c o u s Laminar Model T  26  3.3.2  F i e l d T e s t s of M i l e s ' I n v i s c i d Laminar Model  3.3.3  3.3.2a  Snyder and Cox (1966)  3.3.2b  B a r n e t t and W i l k e r s o n (1967)  . . .  27 27  . . . . . . .  29  L a b o r a t o r y T e s t s of M i l e s ' I n v i s c i d Laminar Model  3.4  20  31  3.3.3a  Wiegel and Cross (1966)  3.3.3b  Shemdin and Hsu (1967)  33  3.3.3c  Bole and Hsu (1969)  34  Summary of O b s e r v a t i o n s  .  31  36  SECTION 3:  OBSERVATIONS  The amount of a v a i l a b l e e x p e r i m e n t a l d a t a on p r e s s u r e s over waves and on r a t e s of wave growth under the a c t i o n of i n g r a p i d l y over the p a s t ten y e a r s . total  S i n c e so l a r g e a f r a c t i o n of  i n f o r m a t i o n has been p u b l i s h e d i n the l a s t  necessary  to d i v i d e the f o l l o w i n g reviews of  to t h i s work i n t o  two p a r t s :  i n c e p t i o n and d e s i g n of e a r l y 1965) until  late  results  3.1  the wind has been i n c r e a s -  of  three y e a r s ,  i t has become  the experiments most  those which were p u b l i s h e d p r i o r  relevant  to  the  the p r e s e n t experiment (which was conceived i n  and those which appeared i n the l i t e r a t u r e from that 1968.  the  time  The l a t t e r form the background a g a i n s t which the  the experiment are  final  discussed.  Sverdrup and Munk (1947) The b e s t summary of r e l i a b l e o b s e r v a t i o n s  a v a i l a b l e i n 1957 when  M i l e s and P h i l l i p s p u b l i s h e d t h e i r n o w - c l a s s i c a l wave g e n e r a t i o n  theories  was t h a t made by Sverdrup and Munk (1947) to produce a method of  practical  wave f o r e c a s t i n g f o r the U. S. Navy.  T h i s summary drew h e a v i l y on those  of Krummel (1911), P a t t o n and Marmer (1932), and C o r n i s h (1934) as w e l l as  others. The r e l e v a n c e o f  the theory of wave f o r e c a s t i n g and of the  assumptions  about how waves grow as d i s c u s s e d by Sverdrup and Munk need not be considered here--these  p o i n t s were d i s c u s s e d f u l l y by U r s e l l  (1956).  The  authors summarize the o b s e r v a t i o n s of wave growth on n o n - d i m e n s i o n a l plots  (a)  velocity gt/U , a  of wave steepness H / L versus wave "age" c / U , (b) of wave a  c/U  a  and wave h e i g h t g H / U  2 a  versus  f e t c h g F / U 2 and d u r a t i o n  where H = peak to trough wave h e i g h t of 17  a  the " s i g n i f i c a n t "  waves,  18 L = wavelength of c = wave phase U  a  the " s i g n i f i c a n t " waves,  velocity,  = anemometer wind speed a t 8 m h e i g h t ,  g = gravitational acceleration, F = fetch:  d i s t a n c e over which the wind blows over  the  water, and It  t = duration:  time f o r which wind has blown.  s h o u l d be noted t h a t H and L do not r e f e r to the l o n g - c r e s t e d  soids  considered i n p o t e n t i a l  Sverdrup and Munk d e f i n e  theory and i n the wave g e n e r a t i o n  fully.  The f a c t  theories.  them as the "average h e i g h t and p e r i o d of  o n e - t h i r d h i g h e s t waves", and as such same as t h a t of  sinu-  t h e i r energy budget i s not  long-crested sinusoids;  they d i s c u s s  the  the the  differences  t h a t H and L are not s i m p l y r e l a t e d to the a and \  simple s i n u s o i d a l wave should be borne i n mind d u r i n g the  of a  discussions  which f o l l o w . T h i s work, b a s i c a l l y e m p i r i c a l i n n a t u r e , and the wave theories  of J e f f r e y s  able theories  of  the e x i s t i n g  article  (1925) and E c k a r t (1953), were the most r e c e n t  of wave g e n e r a t i o n when an a r t i c l e by U r s e l l  the a t t e n t i o n of  (1957).  avail-  (1956) turned  t h e o r e t i c a l f l u i d d y n a m i c i s t s to the complete inadequacy explanations  of  the p r o c e s s .  As the d i r e c t r e s u l t of  two t h e o r i e s were p u b l i s h e d w i t h i n a y e a r , by M i l e s  Phillips  generation  These two t h e o r i e s ,  the  (1957) and  as u n i f i e d by M i l e s i n 1960,  form  the " c l a s s i c a l " view of wave g e n e r a t i o n .  3.1.1  Comparisons w i t h T h e o r i e s of M i l e s  (1957,  1960)  and P h i l l i p s (1957)  P h i l l i p s "(1957), h a v i n g o b t a i n e d a v a l u e f o r the mean square wave amplitude a  1  i n terms of  the wind d u r a t i o n and speed and the mean square  19 fluctuations  i n normal p r e s s u r e of  mated the s i z e o f  p  4  the wind f i e l d  (see  pp. 9-11),  as  " ? - 0.1 (e<Xf  3.1,  and compared the r e s u l t i n g a*" w i t h t h a t expected Sverdrup and Munk's paper; he found the p o i n t s be i n good agreement w i t h h i s Phillips  esti-  concluded t h a t h i s  theoretical  e s t i m a t e of  from the r e s u l t s  relating gH/U  curve.  By 1960,  2  in  to g t / U to  however,  was too l a r g e by as much as  p  one o r d e r of magnitude. Miles necessary  (1959a) made, b e s i d e s an e s t i m a t e of  to r a i s e waves, a comparison w i t h the Sverdrup-Munk d a t a as  well.  He computed from h i s  s  Jeffreys'  (see  first, and,  t h e o r y the v a l u e of  wave g e n e r a t i o n t h e o r y , p p . 5-9)  by computing the mean energy t r a n s f e r  secondly,  latter  the minimum wind speed  the " s h e l t e r i n g  coefficient"  by two d i f f e r e n t  methods:  to the waves u s i n g h i s  theory;  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  computation i n v o l v e d 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 p r o p o r t i o n a l to cos 6 i s u s e d ) .  spectrum of  Both assumed a wave  the f u n c t i o n a l form g i v e n by Neumann (1952).  parameters were taken from e m p i r i c a l r e s u l t s  Wind p r o f i l e  quoted by E l l i s o n  (1956). _o  From these two computations, h i s e s t i m a t e s f o r s were 1.1  x 10  and  _2 0.9  x 10  ; the v a l u e found by Sverdrup and Munk to f i t  the  observations  _o is  1 . 3 x 10  .  Munk (1955) a l s o  s h e a r i n g s t r e s s e s of measurements  c a l c u l a t e d a range of v a l u e s  f o r s from  the wind on the water i n f e r r e d from some e a r l i e r  of Van Dorn,  used by M i l e s as evidence  and i t s of  value o f 0 . 8 - l . 2 x l 0  the v a l i d i t y of h i s  also  theory.  Thus the " c l a s s i c a l " theory of wave g e n e r a t i o n , theory as u n i f i e d by M i l e s (1960), r e l i e s  was  the M i l e s - P h i l l i p s  h e a v i l y f o r the e x p e r i m e n t a l  20 proof of i t s  v a l i d i t y on the wave growth o b s e r v a t i o n s  Sverdrup and Munk i n 1947. P h i l l i p s were p u b l i s h e d ,  summarized by  A t the time when the t h e o r i e s  of M i l e s and  the o n l y i n f o r m a t i o n on p r e s s u r e  fluctuations  over waves was t h a t c o n t a i n e d i n the t h r e e experiments  done on v a r i a t i o n s  i n normal p r e s s u r e on s o l i d wave models by Stanton ^ t al  (1932), M o t z f e l d  (1937), and T h i j s s e (1951), a l l of which were c o n s i d e r e d by M i l e s (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.  o b t a i n e d by these three experiments, from c o n s i s t e n t ,  the f i r s t  two experiments  r a n g i n g from 0.006 (Stanton)  to 0.11  t h a t the p r e s s u r e d i s t r i b u t i o n i s expected by J e f f r e y s ,  3.2  reviewed by U r s e l l  fluctuations  are f a r  ( M o t z f e l d ) ; T h i j s s e s work  such t h a t  for s indicates  1  the wave would grow as  i n d i c a t i n g an s n e a r e r  0.27.  over Waves  two accounts were p u b l i s h e d of measurements  of  pressure  over waves, both from f r e e l y f l o a t i n g buoys equipped w i t h  accelerometers pressure.  i n 1956,  g i v i n g measured v a l u e s  F i e l d Measurements of Normal P r e s s u r e s I n 1962  The r e s u l t s  f o r measuring the wave h e i g h t  simultaneously with  Each w i l l be d i s c u s s e d below i n the context  of  the  the  present  work.  3.2.1  K o l e s n i k o v and Efimov K o l e s n i k o v and Efimov  this of  (1962)  (1962) used a buoy about 30 cm i n d i a m e t e r - -  f i g u r e i s not g i v e n i n the p a p e r , but i s  an e a r l i e r model ( K o l e s n i k o v and Kononkova,  o b t a i n e d from a d e s c r i p t i o n 1961).  I t was c o n i c a l i n  shape underwater; the p a r t which was i n the a i r was a segment of a sphere The p r e s s u r e p o r t was a t the  top of  the segment on the v e r t i c a l a x i s  the buoy, and was about 5 cm above the water when the buoy was Wave h e i g h t s  were o b t a i n e d by twice  of  level.  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  l o c a t e d i n the buoy.  The d e s c r i p t i o n of diagram of  the experiment i s  the i n n e r workings of  the buoy i s not dimensioned.  reader must r e l y f o r c r e d i b i l i t y of "In measurements of  t e r s e i n the extreme.  the r e s u l t s  the c a l i b r a t i o n curves of  The  on such statements  the frequency dependence of p r e s s u r e ,  c i t y and wave p r o f i l e ,  A  surface,  velo-  the microbarometer and  the wave r e c o r d e r were l i n e a r . " . F u r t h e r ,  the statement  error exists  caused by the f a i l u r e of  i n the p r e s s u r e measurements  as,  i s made t h a t an the  buoy to remain p a r a l l e l w i t h the s u r f a c e of  the waves w h i c h , f o r  the  measurements made a t sea,  The p r o b a b l e e f f e c t s  of  this  0  e r r o r are d i s c u s s e d more f u l l y below. A c a l c u l a t i o n of  f o r each run u s i n g the E = ± where T i s w(t)  "was about 127 ."  \  the mean i n p u t o f energy E to the waves i s made formula  pet) xu-CO d t  3.2,  the time i n t e r v a l over which the run was taken and p ( t )  are the p r e s s u r e at the s u r f a c e and the v e r t i c a l v e l o c i t y of  waves.  The v a l u e s  of E and of U, the wind speed at 2.5 meters  are p r e s e n t e d — n o t h i n g e l s e - - f o r The f i r s t  the  height^,  seven r u n s .  p o i n t to be made about the experiment concerns the measure-  ment of p r e s s u r e from a buoy which f o l l o w s waves ( L a g r a n g i a n measurement).  s i g n a l from t h i s  accelerometer  the o r b i t a l motions of  The buoy c o n t a i n e d a s i n g l e  which measured v e r t i c a l a c c e l e r a t i o n s .  t i o n of  and  Whether the f i r s t  the  accelerometer  i n t e g r a l of  the  can be c o n s i d e r e d an a c c u r a t e r e p r e s e n t a -  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 d e s c r i b e s r e l a t i v e to the o r b i t a l motion of  the  22 s u r f a c e water p a r t i c l e s . profile  is  Further,  d i s t o r t e d by the a c t i o n of the buoy, which f o l l o w s  surface o r b i t a l v e l o c i t i e s .  since  its  effects  The p r i n c i p a l pressure-wave  to the waves computed by K o l e s n i k o v and  source of e r r o r i n the energy f l u x e s  computed from  c o r r e l a t i o n s measured by the K o l e s n i k o v - E f i m o v buoy  the a i r over the buoy.  p^ — + 1  w i t h the water and about -p^  embedded i n a f l a t  Its  t h a t the dynamic f l u c t u a t i o n s  s u r f a c e v a r y between about  measurement p o r t i s  located  (if  surface,  the buoy i s  e x e r t e d on the buoy  to the water s u r f a c e ,  the hemisphere);  "'/^fj  here U i s  T h i s means t h a t as  the buoy  the t i l t  fluctuations  of  the buoy,  be  pressure, Without a the  spurious  In any c a s e ,  of the energy f l u x from K o l e s n i k o v and Efimov  are compared w i t h the p r e s e n t r e s u l t s  the  tilts  an unknown s p u r i o u s component w i l l  c o u l d not be e x t r a c t e d from the r e c o r d e d s i g n a l .  the computed v a l u e s  for  the mean wind  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 .  pressures  pressure  r e p r e s e n t e d by a hemisphere  added to the quadrature spectrum between the waves and the  time h i s t o r y of  the buoy  the buoy where the  p o t e n t i a l flow theory g i v e s  v e l o c i t y near the water s u r f a c e .  s p h e r i c a l above-  a t the j u n c t i o n of  a t the top of  dynamic p r e s s u r e a t the top of  detailed  arises  the p r e s s u r e s i g n a l w i t h dynamic p r e s s u r e s  w i t h the flow of  water shape i n d i c a t e s  relative  effect  are averaged out over a wavelength.  from the c o n t a m i n a t i o n of associated  the wave  T h i s d i s t o r t i o n p r o b a b l y has l i t t l e  on the v a l u e s of energy t r a n s f e r Efimov,  the p r e s s u r e v a r i a t i o n along a wave  where the wind speed was-;  (1962)  the  same,  i n " D i s c u s s i o n of R e s u l t s " . '  s  3.2.2  L o n g u e t - H i g g i n s , C a r t w r i g h t and Smith (1963) The buoy used by L o n g u e t - H i g g i n s , C a r t w r i g h t ,  and Smith (1963) was  23 much l a r g e r than t h a t of K o l e s n i k o v and Efimov: aluminum d i s c about 1.7 meters when loaded f l o a t e d w i t h i t s (the  30 cm i s  it  consisted  of an  i n diameter and 0.6 meters deep,  top about 30 cm above the water  an e s t i m a t e from t h e i r F i g u r e l a ; no f i g u r e i s  The p r e s s u r e sensor was r a i s e d 6 cm above the top of  which  surface quoted).  the buoy and observed  the p r e s s u r e 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 mated from the same f i g u r e ) centre of  the d i s c .  through twelve o r i f i c e s ,  The c r o s s - s e c t i o n  water was r o u g h l y concave w i t h a f l a t a t the geometric  c e n t r e of  of  the experiment was to measure  tion,  of  motion of  the  change of v e r t i c a l m o t i o n ,  experiment t h e o r e t i c a l l y p r o v i d e d enough  assuming the buoy d i d not move r e l a t i v e  particles,  located  the buoy was p r o v i d e d w i t h  which responded to r a t e s  Thus t h i s  the buoy above  The r a i s e d d i s c was  d i r e c t i o n a l spectrum of w i n d - d r i v e n waves,  p i t c h and yaw.  the  the buoy.  S i n c e the p r i n c i p a l o b j e c t of  three a c c e l e r o m e t e r s  presumably near  the p a r t of  top.  (esti-  to the water  informa-  surface  to c o r r e c t the observed p r e s s u r e s i g n a l f o r the e f f e c t of the buoy along the wave p r o f i l e  (see  p.  2 2  );  this  the  correction  was a p p a r e n t l y not made. L o n g u e t - H i g g i n s et waves,  then f i t  a l , having obtained d i r e c t i o n a l spectra' for  a number of  t h e o r e t i c a l a n g u l a r d i s t r i b u t i o n s to them,  f i n d i n g t h a t one p r o p o r t i o n a l to cos best; r 0.5,  is  where U i = 2.5 u * and u# i s  ones.  to i n c l u d e v i s c o s i t y  appears to f i t  t h e i r data  as U ^ / c v a r i e s from 0.1  the f r i c t i o n v e l o c i t y  estimated  to  from  They then compute the jin phase component of  the t o t a l p r e s s u r e p r e d i c t e d by M i l e s ' (1959)  (V^S)  found to v a r y from 7 to about 0.4  observed mean v e l o c i t i e s .  the  theory as g e n e r a l i z e d by Benjamin  and o t h e r wind p r o f i l e s b e s i d e s l o g a r i t h m i c  24 They f i n d  that  the observed r a t i o s  of  t h e i r p r e s s u r e s p e c t r a to  wave s p e c t r a are p r e d i c t e d moderately w e l l by the t h e o r e t i c a l This fact  they  take to i n d i c a t e  ratios.  t h a t the p r e s s u r e s measured by the buoy  are almost e n t i r e l y produced by "aerodynamical p r e s s u r e changes due the flow of static  the a i r over  pressure  the u n d u l a t i n g s u r f a c e "  term ^ <j  together with  and the waves as measured by the buoy f o r one of  to about + 1 0 ° below a) = 3 r a d / s e c ;  the r u n s .  l a g appears  a l t h o u g h the measurements  U - c increases,  p a r t of  but are  accurate  that  the phase of  inexplicable  Experiment:  wave c o r r e l a t i o n s  of  the  difference theories.  the wave energy they theoretical  t h a t the phase s h i f t s  should remain c l o s e to - 1 8 0 ° .  sons w i t h the d a t a o b t a i n e d from the p r e s e n t i n " D i s c u s s i o n of  o c c u r r i n g as  pressure  the p r e s s u r e s i g n a l remains - 1 8 0 ° over  the v a l i d i t y of M i l e s '  T a b l e 4,1)  lag  frequency,  i n terms of e x i s t i n g  the wave spectrum c o n t a i n i n g most of  of  at  frequency  S i n c e an i n c r e a s e of  t h a t wave damping i s  phase  the wave spectrum i s  are deemed u n t r u s t w o r t h y a t the  the trend i s  as f u r t h e r evidence (see  These  to i n c r e a s e w i t h i n c r e a s i n g  where the i n c r e a s e b e g i n s .  l a g beyond 1 8 0 ° i n d i c a t e s  The f a c t  the peak of  pressure  The phase angles they get are about - 1 8 0 ° ( p r e s s u r e  behind waves) and t h i s  (3 r a d / s e c )  the  between the  angles become l e s s and l e s s r e l i a b l e at h i g h f r e q u e n c i e s  rad/sec.  to  .  They a l s o i n c l u d e a t a b l e of phase d i f f e r e n c e s  0.6  their  the take  predictions  the p r e s s u r e  Where p o s s i b l e ,  versus compari-  experiment are compared,  Results".  As was p r o b a b l y the case w i t h the R u s s i a n buoy,  it  is  expected  that  the p r e s s u r e s i g n a l r e c o r d e d by the E n g l i s h buoy was contaminated by dynamic p r e s s u r e f l u c t u a t i o n s . information of  S i n c e L o n g u e t - H i g g i n s e_t a l o n l y r e p o r t  the pressure-waves  cospectrum, o n l y the e f f e c t s  of dynamic  25 pressure  c o n t a m i n a t i o n on the i n - p h a s e  component of p r e s s u r e w i l l  be  discussed. The h e i g h t meter  is  buoy,  and i t  of  a t which the E n g l i s h buoy f l o a t e d  considerably is  therefore  less significance.  w i t h t i l t i n g of  less  than was a p p a r e n t l y the  Two sources of  the buoy r e l a t i v e  flow r e l a t i v e  to the buoy which i s  not enough i s  known e i t h e r  of  to i t s  case f o r the  expected that dynamic p r e s s u r e  to the water  surface  Russian  that  coherent w i t h the waves.  is  associated  (pitching)  by the p o r t i o n of  the response of  dia-  contamination  contamination e x i s t ;  a r i s i n g from the dynamic p r e s s u r e generated  or of  relative  and  the  that  air  Unfortunately  the buoy to the wave f i e l d  the a i r flow which e x i s t s over r e a l waves to e s t i m a t e the s i z e of  either  effect.  important,  It  is  where the r e l e v a n t  surface  to the buoy which i s  one o r d e r of magnitude  3.2.3.  of  the  two w i l l be more  sensor v a r i e s  as  less  and u  1  ^  U  dynamic p r e s s u r e f o r the  c o n t a m i n a t i o n i s ^ U u ; i n these e x p r e s s i o n s , U i s  wind speed near the water relative  the f i r s t  s i n c e the dynamic p r e s s u r e a t the  the buoy p i t c h a n g l e , source of  expected that  is  the p o r t i o n of  with second  the mean  the wind speed  coherent w i t h the waves, which i s  at  least  than. U o r d i n a r i l y .  Summary  I n summary, both the K o l e s n i k o v and Efimov buoy and the buoy used by L o n g u e t - H i g g i n s ^ t  al  appear to i n t r o d u c e i n t o  they r e c o r d s p u r i o u s dynamic p r e s s u r e s  signals  which .may be as l a r g e as '/g f*^  for  the R u s s i a n buoy and are somewhat s m a l l e r ,  for  the E n g l i s h buoy.  pared,  the p r e s s u r e  The r e s u l t s from the  but f a r from  two experiments  s i n c e K o l e s n i k o v and Efimov r e p o r t i n f o r m a t i o n of  quadrature spectrum and L o n g u e t - H i g g i n s ej: al  the  negligible, cannot be compressure-waves  are a b l e to g i v e  information  26 o n l y on the cospectrum.  Of the two measurements  the one by Longuet-  H i g g i n s j2t al must be c o n s i d e r e d the most s i g n i f i c a n t because they used a p p a r e n t l y had a lower p r o f i l e on the water. t h e i r f i n d i n g t h a t the phase of with respect  touthe waves suggests that M i l e s '  g i v e s at l e a s t a good f i r s t  3.3  t h e i r pressure s i g n a l  For this  reason  is nearly - 1 8 0 °  (1957) i n v i s c i d theory  a p p r o x i m a t i o n to the a i r flow over the waves.  Recent O b s e r v a t i o n s A d i s c u s s i o n of  the o b s e r v a t i o n s  them i n t o two c l a s s e s :  laminar theory  assumed t h a t the h e i g h t of  M i l e s 1962).  the c r i t i c a l  sublayer,  Reynolds s t r e s s e s  Tests of M i l e s '  to the i n v i s c i d theory  In the l a t t e r  layer is  viscous  theory i t  so s m a l l t h a t i t  presumed to occur through the  (seep.  14  is  action  ff.).  V i s c o u s Laminar Model  S i n c e the flow regime p o s t u l a t e d by the l a t t e r  theory i s not  that  commonly observed f o r the range of wind-generated g r a v i t y waves a t which c o n t a i n most of the t o t a l energy,  and s i n c e  o c c u r r e d d u r i n g the p r e s e n t measurements, i t w i l l be mentioned but not commented on. p u b l i s h e d up to l a t e l a t e r references  is  a t h i n s t r a t u m of a i r a d j o i n i n g the w a t e r .  I n the t h e o r y , wave g e n e r a t i o n i s of v i s c o u s  1962 must d i v i d e  and those p e r t a i n i n g to the s o - c a l l e d  (Benjamin 1959;  w i t h i n the v i s c o u s  published since  those which are r e l e v a n t  proposed by M i l e s i n 1957,  3.3.1  the buoy  sea  i t p r o b a b l y never  those experiments  dealing with  The f o l l o w i n g l i s t  of works  1966 has been unabashedly e x t r a c t e d from M i l e s (1967);  are the r e s u l t of a l i t e r a t u r e s e a r c h .  The experiments  d e a l i n g w i t h the v i s c o u s  l a m i n a r theory have  been c a r r i e d out i n the l a b o r a t o r y i n wind-water  tunnels,  where  all  short-  27 wavelength waves are generated boundary l a y e r s . Holmes (1963),  Included i n this  Cohen and H a n r a t t y (1965),  1966), P l a t e and Hidy (1967), H i r e s (1967), and  Chang and Hidy (1969).  straight,  turbulent  category are the works of Hamada (1963),  H a n r a t t y and Woodmansee (1965),  Hidy and P l a t e - ( 1 9 6 5 , Plate,  on smooth water under c o n t r o l l e d  These were conducted i n "normal" i . e .  narrow, wind t u n n e l s ;  a novel,  although e n t i r e l y  long,  qualitative,  approach has been taken u s i n g c y l i n d r i c a l geometry by Auerbach and R i c h a r d s o n (1967). papers mentioned, viscous  With the e x c e p t i o n  of  the f i r s t  a l l show more or l e s s s a t i s f a c t o r y  laminar t h e o r y .  The l a t e s t work,  which exceed and some of which f a l l  3.3.2  than the expected  F i e l d T e s t s of M i l e s ' The experiments  subdivided into  the  Change and Hidy  growth r a t e s are found some of  s h o r t of  the t h e o r e t i c a l  predictions  experimental e r r o r s .  I n v i s c i d Laminar Model relevant  two groups:  l a b o r a t o r y i n wind-wave  two  agreement w i t h  t h a t of P l a t e ,  produces c o n f l i c t i n g r e s u l t s - - e x p e r i m e n t a l  by amounts g r e a t e r  two and l a s t  to M i l e s '  i n v i s c i d model can be  f i e l d measurements,  tunnels.  and those made i n  The f i e l d measurements  w i l l be  the  discussed  first. 3.3.2a  Snyder and Cox (1966)  S i n c e 1962 o n l y two experiments  have been p u b l i s h e d on  of wave growth a t sea under more o r l e s s The f i r s t  is  t h a t of  winds.  c a r e f u l l y monitored  tuned to waves of a s i n g l e wavelength  towed downwind a t the group v e l o c i t y The s h i p ' s  conditions.  Snyder and Cox (1966), who towed behind t h e i r v e s s e l  an a r r a y of wave d e t e c t o r s I t was  observations  of  (17 m).  these waves d u r i n g o f f s h o r e  speed was measured w i t h a t a f f r a i l l o g ,  and wind  speeds w i t h an anemometer on the s h i p a t a h e i g h t of about 6 m above *Not a v a i l a b l e  to the w r i t e r ;  comments are from M i l e s  (1967).  the  28 water s u r f a c e . growth r a t e s  From the r e s u l t i n g i n f o r m a t i o n the l i n e a r and e x p o n e n t i a l  &(k ,t) B  and  |3(fc ,t)  of  0  the 17 m waves have been c a l c u l a t e d from (Hasselman,  . The for  0  a(k t)  time h i s t o r y of F ( t , k „ )  Priestley  is  fitted  If  in Virginia,  of ^ are found to i n c r e a s e  although showing c o n s i d e r a b l e to be f i t t e d  the  fitting  spectrum  then the v a l u e s  scatter.  q u i t e w e l l by the  (1957)  of <X theory.  l i n e a r l y w i t h wind  The e x p e r i m e n t a l p o i n t s  speed, are found  relation  P =(e./e„)U-y - «.) where k IT i s  is  a  the observed mean wind v e l o c i t y . of  coefficient  mic  a p r o f i l e parameter I~l  roughness  of 0.27,  l e n g t h of lO'^cm.  makes the A used too l a r g e and the (1957)  Jeffreys  ( E q u a t i o n 2.12)  P h i l l i p s 1966, theoretical  too s m a l l by an unknown f a c t o r . |3  weak f u n c t i o n of  jQ ;  this  4  of ^ are com-  (1925)  assuming  and of M i l e s (1957) assuming a l o g a r i t h -  T h i s assumed roughness  h i g h by a f a c t o r of about 20 (see  Miles  of  - '  the 17 m wave.s, and  The measured v a l u e s  the t h e o r i e s  a sheltering profile,  3  the wave number and cj the frequency of  pared w i t h the p r e d i c t i o n s  spectra  assumed to be s i m i l a r to one measured by  (1965) over mown grass  values  3.3.  of (X and ^3 were determined by  found are c o n s i s t e n t w i t h those p r e d i c t e d by P h i l l i p s ' The  . ...  was r e c o v e r e d from the observed  time h i s t o r y w i t h a r e g r e s s i o n of the form 3 . 3 .  pressure flu c tu ation s  of  1960)  + p(feo,t)F(k„,t)  O J  a number of r u n s , and v a l u e s  this of  dF(k ,t)/dt =  F (k„,t)  the s p e c t r a l i n t e n s i t y  overestimate  ofJTI  i n G> which p r o b a b l y does not exceed 257».  of 3 x 10"^ and a l e n g t h appears  to be  or Smith 1967), which  growth r a t e computed from , however,  is  only a  w i l l thus cause an e r r o r  T h e r e f o r e the b a s i c  conclusion  29 reached by Snyder and Cox remains u n a l t e r e d .  T h e i r measurements  growth are about one o r d e r of magnitude g r e a t e r Miles'  (1957)  spatial  than those p r e d i c t e d by  theory.  They a l s o made an e s t i m a t e of wave f i e l d assuming e q u a t i o n 3.4 spectrum;  of  the t o t a l momentum t r a n s f e r  c o n s i d e r a b l y l a r g e r than the v a l u e of 1.0 e.g.  their  to h o l d and u s i n g a simple e m p i r i c a l  they o b t a i n e d a wind drag c o e f f i c i e n t  d i r e c t measurements,  to  - 1.4  of about 7 x 10  -  J  x 10"-* o b t a i n e d from  Smith (1967) or W e i l e r and B u r l i n g (1967).  They  assumed the d i s c r e p a n c y to be accounted f o r by the assumptions used make the e s t i m a t e ,  which "tend to give  t r a n s f e r by normal s t r e s s  ...".  an o v e r e s t i m a t e  Because  of  to  the momentum  they o v e r e s t i m a t e d  C^ by such a  l a r g e amount, t h e i r c o n c l u s i o n t h a t the major p o r t i o n of momentum t r a n s f e r a t sea goes d i r e c t l y i n t o waves must be regarded as h i g h l y  speculat-  ive.  3.3.2b  B a r n e t t and W i l k e r s o n (1967)  The second f i e l d experiment was t h a t of B a r n e t t and W i l k e r s o n (1967). They measured p r o f i l e s o f established  the sea s u r f a c e under the a c t i o n of a w e l l -  o f f s h o r e wind i n the upwind and downwind d i r e c t i o n , u s i n g as  t h e i r wave sensor  a s e n s i t i v e r a d a r a l t i m e t e r mounted i n an a i r p l a n e ,  which flew over the water a t an a l t i t u d e of 340 km.  The a l t i m e t e r averaged h e i g h t s  about 9 m i n d i a m e t e r , than about 0.4 H z .  150 m over a d i s t a n c e  over a c i r c u l a r a r e a of  thus f i l t e r i n g out a l l wave f r e q u e n c i e s  The l a r g e speed of  the p l a n e - - a b o u t 100  of  ocean greater  m/sec--meant  t h a t the e n t i r e wave f i e l d was sampled twice w i t h i n about two h o u r s . Wind speeds near shore were e x t r a p o l a t e d from c o n c u r r e n t v a l u e s stations.  a t land  Those f u r t h e r o f f s h o r e were o b t a i n e d by s u c c e s s i v e f i x e s  with  30 L o r a n - C; wind p r o f i l e i n f o r m a t i o n , as i n the case of Cox measurements, Their results 7):  was not  the Snyder and  taken.  are p r e s e n t e d  first  one f o r the upwind one and one f o r  as  two p l o t s  ( t h e i r F i g u r e s 6 and  the downwind run, which show con-  tours of equal s p e c t r a l d e n s i t y on a graph w i t h wave frequency o r d i n a t e and d i s t a n c e  from shore as a b e i s s a ;  the development w i t h f e t c h of i n themselves p r e s e n t  these graphs t h e r e f o r e  show  the complete measured power spectrum, and  an immense amount of d a t a i n a v e r y e d i f y i n g manner.  They found, as expected, lower f r e q u e n c i e s  as  that  as the f e t c h  the major s p e c t r a l peak moves  increases.  energy i n t h e i r s p e c t r a a t f r e q u e n c i e s  to  They found " c o n s i d e r a b l e "  below the peak,  i n d i c a t i n g a broad-  band energy i n p u t mechanism which they presume to be t h a t of P h i l l i p s . If  attention  is  f i x e d on the behaviour w i t h time of a p a r t i c u l a r s p e c t r a l  component f o r which U / > I , i t  is  c  reaches  found to grow e x p o n e n t i a l l y u n t i l  a maximum and then to l o s e energy u n t i l i t  energy 30 - 707= below t h a t of  the maximum:  reaches  i t "overshoots"  it  an e q u i l i b r i u m i t s maximum  values. The f i n d i n g of most i n t e r e s t a l l y determined v a l u e s p.  13)  of  the e x p o n e n t i a l  are i n c l o s e agreement,  empirical  here i s  the f a c t  that t h e i r  experiment-  growth parameter ^ ( E q u a t i o n  a t l e a s t f o r t h e i r downwind r u n , w i t h  curve suggested by Snyder and Cox ( E q u a t i o n 3 . 4 ,  p.  They thus a r r i v e d a t the same c o n c l u s i o n as d i d Snyder and Cox. i n v i s c i d model proposed by M i l e s p r e d i c t s wave growth r a t e s by an o r d e r of magnitude  the  28.) The  smaller  than those measured.  I n t h e i r upwind r u n , they found p o s i t i v e quencies  2.10,  growth r a t e s a t wave f r e -  f o r which the wave phase speed exceeded the wind speed as  they  measured i t . little of  They suggest p o s s i b l e reasons f o r t h i s ,  j u s t i f i c a t i o n i n making such suggestions  t h e i r wind speed  3.3.3  Laboratory  i n view of  scantiness  Studies s i n c e 1962 have attempted to match c o n d i t i o n s  i n wind-water tunnels w i t h M i l e s '  i n v i s c i d laminar t h e o r y .  The f i r s t ,  (1966), was c a r r i e d out a t the Department of  E n g i n e e r i n g , U n i v e r s i t y of C a l i f o r n i a ,  Civil  B e r k e l e y ; the o t h e r two, Shemdin  and Hsu (1967) and Bole and Hsu (1969) used the e x t e n s i v e facilities  the  information.  Three l a b o r a t o r y s t u d i e s  Wiegel and Cross  but there seems  a v a i l a b l e a t the Department of  Civil  wind-wave  Engineering at  Stanford  University.  3.3.3a  Wiegel and Cross  Wiegel and Cross  (1966)  (1966) measured a i r speed and normal p r e s s u r e above  m e c h a n i c a l l y - g e n e r a t e d waves i n three d i f f e r e n t wind-water t u n n e l s , 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 p r e s s u r e a t the same downwind p o s i t i o n as a r e s i s t a n c e w i r e wave gauge,  u s i n g a t o t a l - h e a d tube f o r a i r  tube f o r the normal p r e s s u r e s . to the s u r f a c e of for  the waves,  the waves under s t u d y .  speed and a p i t o t - s t a t i c  The tubes were p l a c e d as c l o s e  as  possible  but were always above the c r i t i c a l h e i g h t z The s t a t i c h o l e s  of  the p i t o t - s t a t i c  tube  were a t 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 f r e q u e n c i e s of  the m e c h a n i c a l l y - g e n e r a t e d waves ranged from  to 2 H z , g i v i n g wavelengths  of 70 to 40 cm and wave speeds of  cm/sec; wind speeds used were 670 to 1340 cm/sec.  1.5  100 to 80  The water depth i n  c  32 their  tunnels ranged from 15 to 45 cm.  terms of static  The r e s u l t s  are presented  coherence and phase c r o s s - s p e c t r a and power s p e c t r a of  p r e s s u r e and wave e l e v a t i o n .  observed  in  the  A l t h o u g h some i n c o n s i s t e n c i e s  were  ( i n one r u n observed phase lags of p r e s s u r e behind wave h e i g h t  were zero throughout the s p e c t r a l range, not 160 - 1 7 0 ° as p r e d i c t e d by the i n v i s c i d t h e o r y ) ,  i n g e n e r a l agreement w i t h the i n v i s c i d theory was  good. There are two causes of u n c e r t a i n t y i n t h e i r measurements. f i r s t has been d i s c u s s e d  The  i n some d e t a i l by Shemdin and Hsu (1966), who  found t h a t by measuring p r e s s u r e s  over waves w i t h a probe f i x e d i n  space  above the c r i t i c a l h e i g h t z w h e r e U = c , (as was the probe used by Wiegel e  and Cross) they o b t a i n e d r e s u l t s  which were s i m i l a r to those o b t a i n e d by  Wiegel and Cross but which d i f f e r e d s h a r p l y from r e s u l t s t h e i r probe stayed beneath of M i l e s '  z  c  .  They took t h i s  to mean that  (1957) theory could not be checked w i t h p r e s s u r e  made above  z  c  the v a l i d i t y  measurements  .  The second cause of u n c e r t a i n t y i s to measure p r e s s u r e s of  o b t a i n e d when  t h e i r use of a p i t o t - s t a t i c  i n c l o s e p r o x i m i t y to the^waves.  As the  tube  wavelength  the waves used i n the wind-wave tunnel decreased w i t h i n c r e a s i n g f r e -  quency,  the p r e s s u r e measured at the s t a t i c  p o r t s of  the tube would con-  t a i n i n c r e a s i n g f r a c t i o n s of the s t a g n a t i o n p r e s s u r e L p u U , where U i s the mean f r e e - s t r e a m a i r speed and u' i s speed f l u c t u a t i o n s which i s  the p o r t i o n of  coherent w i t h the waves (the  b e s e t s the buoy measurement system used i n the p r e s e n t its  effects  on p. 91  ff  are d i s c u s s e d ).  the t o t a l  air  same d e f e c t  experiments,  and  i n d e t a i l i n "Data A n a l y s i s and I n t e r p r e t a t i o n "  The i n f l u e n c e of  this  dynamic p r e s s u r e contamination on  the  33 results ment  cannot be estimated w i t h o u t  a d e t a i l e d knowledge of  measure-  system.  3.3.3b  Shemdin and Hsu (1967)  Shemdin and Hsu (1967) s t u d i e d  the p r e s s u r e 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, using a s t a t i c  probe which moved v e r t i c a l l y w i t h the water s u r f a c e a s h o r t above i t .  the a x i s of to yaw,  pressure distance  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 c o n t a i n i n g the wind d i r e c t i o n . the d i s c .  P r e s s u r e was measured  A l t h o u g h such d e v i c e s are n o t o r i o u s l y  available  which might cause l a r g e e r r o r s  the same experiment were attempted i n the f i e l d .  In o r d e r to  t h a t the c r i t i c a l l a y e r f o r the waves was s u f f i c i e n t l y probe to remain w i t h i n i t ,  i t was n e c e s s a r y  over the water a r t i f i c a l l y u s i n g roughness  insure  thick for  the  to t h i c k e n the boundary l a y e r elements l o c a t e d a t a t r a n s -  i t i o n p l a t e where the a i r f i r s t met the water.  In t h i s way they were  a b l e to o b t a i n the r e q u i r e d t h i c k n e s s e s f o r wind speeds up to 600 F o r comparison of  their results  measured v e r t i c a l p r o f i l e s wind speed;  (1957)  theory,  cm/sec.  carefully  of mean wind speed were made a t each mean  a t i c d e v i a t i o n from a l e a s t - s q u a r e s i g n o r e d f o r the c a l c u l a t i o n of  l o g a r i t h m i c , although a system-  f i t was f o u n d .  T h i s d e v i a t i o n was  the p r o f i l e p a r a m e t e r / ! .  r e q u i r e d f o r comparison w i t h M i l e s ' To o b t a i n the phase d i f f e r e n c e  recordings,  with Miles'  they were found to be c l o s e l y  imposed c h a r t r e c o r d i n g s of  at  sensitive  the s m a l l t u r b u l e n c e l e v e l s and a c c u r a t e l y a l i g n e d flow  i n the wind t u n n e l e l i m i n a t e d yaw e f f e c t s , if  their  (Equation  theory. between p r e s s u r e and waves,  they  the observed p r e s s u r e on simultaneous  and found l e a s t - s q u a r e  2.12)  fit  sinusoids  super-  wave  f o r each wind speed.  34 The phase i n f o r m a t i o n o b t a i n e d i n each case i s  effectively  that which would be c o n t a i n e d at one frequency i n a  equivalent  to  cross-spectral  analysis. The f i n a l r e s u l t s theoretical  were p r e s e n t e d  ( p r e d i c t e d from M i l e s '  and t h e o r e t i c a l v a l u e s 0.6  of  i n a table  1957  theory)  showing measured and phase s h i f t s ,  the p e r t u r b a t i o n p r e s s u r e amplitude f o r 0.4  Hz waves over a range of wind speeds from 120 - 1160  Miles'  i n v i s c i d theory was found by the authors  v e r i f i e d by t h e i r observed phase s h i f t s . so s a n g u i n e - - h e  finds  that  agreement w i t h ,  a l t h o u g h somewhat  extent)  than,  3.3.3c  B o l e and Hsu (1969)  to be  Miles himself  t h e i r phase measurements  the t h e o r e t i c a l  l a r g e r (to  predictions  ful  observations  of  the a c t i o n of  same wind-water  cm. not  to 1.4  thickness  of  (as was  t h a t of  not  significant  to M i l e s '  They made c a r e -  waves.  V e r t i c a l wind  both w i t h the m e c h a n i c a l l y -  They used a m o d i f i e d v e r s i o n of  cm/sec.  the  T h e i r waves  Hz; the water depth was about  Wind speeds used were 350 - 1350 thickened  (1967) i s  relevant  tunnel used by Shemdin and Hsu (1967).  v a r i e d i n f r e q u e n c y from 0.9  satisfactorily  the growth w i t h f e t c h under  speed p r o f i l e s were taken a t numerous l o c a t i o n s , and absent.  The  the i n v i s c i d laminar m o d e l . . . " .  the wind of m e c h a n i c a l l y - g e n e r a t e d  generated waves p r e s e n t  and  " . . . are i n f a i r  t h a t of Bole and Hsu (1969).  w i t h c a p a c i t a n c e probes of  cm/sec ^ .  a statistically  The l a t e s t p u b l i s h e d account of o b s e r v a t i o n s i n v i s c i d laminar theory i s  and measured  100  T h e i r boundary l a y e r was  Shemdin and Hsu) and as a r e s u l t ,  the  the c r i t i c a l l a y e r was s m a l l , r a n g i n g between .013 and  .34  cm. They doubted the e x i s t e n c e of a laminar s u b l a y e r ,  noting that at  the  35 wind speeds they used the s t a n d a r d d e v i a t i o n s  of s u r f a c e roughness  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  estimated  thicknesses  layer.  They a l s o ,  of both the laminar s u b l a y e r and of the  therefore,  motion below the c r i t i c a l to e l i m i n a t e  layer.  These two f a c t s ,  critical  taken t o g e t h e r ,  their results  laminar model or the v i s c o u s model of M i l e s .  to e i t h e r  appear  the i n v i s c i d  They n e v e r t h e l e s s  applied  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 the water s u r f a c e - - t h i s generated waves.  to be l o g a r i t h m i c c l o s e  to  r e s u l t b e i n g o b t a i n e d i n the absence of m e c h a n i c a l l y  When these waves were p r e s e n t  the p r o f i l e s were s i m i l a r ,  but c o u l d no l o n g e r be extended as c l o s e to the water s u r f a c e . evidence  the  doubted the e x i s t e n c e of o r g a n i z e d v o r t e x  the d i r e c t a p p l i c a t i o n of  their results  of  They found  t h a t r i p p l e s on the m e c h a n i c a l l y - g e n e r a t e d waves were steep and  sharp-crested,  and appeared to " r i d e " the wave  crests.  They computed s p a t i a l wave growth i n a stepwise manner, s i n c e observed t h a t  the boundary l a y e r e v o l v e d w i t h f e t c h .  growth curves they o b t a i n e d measured v a l u e s ( E q u a t i o n 2 . 8 , p.  13),  and found t h a t  From t h e i r  they  spatial  f o r the growth parameter  the r a t i o s  of  these to v a l u e s  com-  puted from M i l e s i n v i s c i d laminar model u s i n g measured p r o f i l e  parameters  v a r i e d from 1 - 1 0 ,  evidence  of  w i t h a mean o f about 3.  They took t h i s  the t r u t h o f M i l e s (1957) statement t h a t h i s  only give surface that  i n v i s c i d model would  q u a l i t a t i v e agreement w i t h r e a l i t y when the flow over the water  c o u l d be taken as a e r o d y n a m i c a l l y r o u g h .  They go on to  the presence of s h a r p - c r e s t e d r i p p l e s on the c r e s t s of  indicates  as  the p o s s i b i l i t y  suggest  the waves  t h a t s e p a r a t i o n may indeed occur over  their  waves, and t h a t t h i s may account f o r the l a r g e r a t i o s of observed  to  36 predicted  <f .  Because o f the s m a l l c r i t i c a l  l a y e r t h i c k n e s s e s e x i s t i n g i n the  measurement, i t i s p r o b l e m a t i c a l whether the r e s u l t s shown s h o u l d be applied  to the i n v i s c i d o r the v i s c o u s laminar model.  I t should be  p o s s i b l e w i t h the d a t a they o b t a i n e d f o r Bole and Hsu to make the l a t t e r comparison;  3.4  the r e s u l t s of such a comparison would be e d i f y i n g .  Summary o f O b s e r v a t i o n s In  summary, there a r e a v a i l a b l e i n the l i t e r a t u r e  o n l y f o u r observa-  t i o n s o f f l u c t u a t i o n s i n s t a t i c p r e s s u r e over wind-generated waves--two, those o f Longuet-Higgins  et  al  and K o l e s n i k o v and Efimov--  b e i n g f i e l d measurements and two, those of Wiegel and Hsu, b e i n g made i n wind-water t u n n e l s . suspected o f b e i n g s t r o n g l y contaminated  gravity  and Cross and Shemdin  Both f i e l d measurements a r e  by dynamic p r e s s u r e  the l a t t e r p r o b a b l y somewhat more than the former.  fluctuations,  The measurements of  phase made by the E n g l i s h buoy a r e not a c c u r a t e enough to permit the computation Of  o f a pressure-wave quadrature  the two wind-tunnel  spectrum.  measurements o n l y those of Shemdin and Hsu  remain above s u s p i c i o n , although s t r o n g r e s e r v a t i o n s about t h e i r a n a l y s i s a r e expressed  i n Bole and Hsu ( 1 9 6 9 ) .  e q u i v o c a l when compared w i t h M i l e s ' i n v i s c i d s h i f t s r e p o r t e d i n Shemdin and Hsu ( 1 9 6 7 )  T h e i r r e s u l t s are  laminar theory; phase  a r e o n l y s l i g h t l y g r e a t e r than  those p r e d i c t e d by theory, w h i l e some r e s u l t s from published l a t e r  (Shemdin 1968)  data  the same  experiment  a p p a r e n t l y i n d i c a t e growth r a t e s l a r g e r  by a f a c t o r o f about two than those p r e d i c t e d by M i l e s ' 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 B a r n e t t and W i l k e r s o n (1967). They both reached the same c o n c l u s i o n :  observed r a t e s of wave growth  exceed those p r e d i c t e d by the i n v i s c i d laminar model by about one o r d e r of magnitude.  In f a c t ,  than a f a c t o r of  the growth r a t e s  two than the l a r g e s t  they observe  are l a r g e r by more  ones r e p o r t e d by Shemdin (1968).  Since r e a l waves b r e a k , energy f l u x e s  computed from f i e l d  measure-  ments of s p a t i a l growth s h o u l d be s m a l l e r than those o b t a i n e d from measurements tunnel.  It  of p r e s s u r e f l u c t u a t i o n s can thus be seen t h a t  u n d e r s t a n d i n g of  the mechanism of  waves i n some c o n f u s i o n .  over s i n u s o i d a l waves i n a wind  the experiments  d e s c r i b e d l e a v e our  energy t r a n s f e r from the wind to  sea  SECTION 4 :  EXPERIMENT  4.1  Rationale  38  4.2  Design C r i t e r i a  39  4.2.1  The P r e s s u r e Measurement System  39  4.2.2  The Wave Measurement System  .  44  The Measurement S i t e  4.4  The Measurement of Wind Speed  45  4.5  The Wave Sensor  46  4.5.1  Calibration  46  4.6  The P r e s s u r e Sensor  47  4.7  The P r e s s u r e Measurement System  51  4.7.1  The W a t e r p r o o f i n g Diaphragm  '51  4.7.2  The Backup Volume  52  4.9  4.10  4.11  .  44  4.3  4.8  .  .  Electronics  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  C a l i b r a t i o n of P r e s s u r e Sensor 4.9.1  Laboratory Calibrations  4.9.2  Field Calibrations  .  .  .  57 58  .  59  The Buoy  60  4.10.1  Underwater Shape  60  4.10.2  Upper S u r f a c e :  Aerodynamics  62  Wind Tunnel T e s t s  64  4.11.1  65  The Wind Tunnel  SECTION 4:  EXPERIMENT ( c o n t i n u e d )  4.11.2  E x p e r i m e n t a l Procedure  4.11.3  The Dynamic P r e s s u r e  4.11.4  The A r t i f i c i a l T u r b u l e n t Boundary L a y e r . . .  .  .  .  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 A t t a c k Angle V a r i a t i o n  Rejection  65 Ring  . . . . . .  .  66  70  SECTION 4: The and  EXPERIMENT  p r e c e d i n g s e c t i o n has s e t  e x p e r i m e n t a l evidence  the stage f o r t h i s one;  the  theoretical  a v a i l a b l e up to mid-1965 formed the b a s i s  for  the  d e s i g n c r i t e r i a and e x p e r i m e n t a l procedures d e s c r i b e d immediately below. These d e s c r i p t i o n s  are f o l l o w e d by d e t a i l e d accounts  of apparatus and  calibrations.  4.1  Rationale As mentioned i n the i n t r o d u c t i o n , the f i r s t d e c i s i o n which had to  be made i n the d e s i g n of  the experiment was whether  the buoy should be  allowed to d r i f t more or l e s s f r e e l y on the waves ( L a g r a n g i a n measurement), or be c o n s t r a i n e d to move o n l y 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 s p e c t r a between p r e s s u r e and wave e l e v a t i o n were r e q u i r e d , as  as w e l l  the power s p e c t r a of each v a r i a b l e . If  the waves are measured w i t h accelerometers  on the buoy,  then  r e l a t i v e phase of p r e s s u r e and waves can be r e c o v e r e d c o r r e c t l y .  the  The  true frequency to which a g i v e n frequency i n the r e c o r d e d i n f o r m a t i o n i s r e l a t e d cannot be e x t r a c t e d , waves moving i t For  u n l e s s motions of  are s m a l l w i t h r e s p e c t  this reason,  and s i n c e  the buoy r e l a t i v e  to  the  tor.the wave m o t i o n s .  two attempts  a t measuring p r e s s u r e  fluctua-  t i o n s w i t h f r e e l y moving a c c e l e r o m e t e r - e q u i p p e d buoys had a l r e a d y been made (see  p.20ff),  i t was f e l t  that the next a t t e m p t - - t h e  one  technique used was to measure the p r e s s u r e a t  the  under d i s c u s s i o n h e r e - - b e The  experimental  necessary  quasi-Eulerian.  s u r f a c e of a d i s c - s h a p e d styrofoam f l o a t w i t h a gymbal-suspended b e a r i n g 38  39 on i t s v e r t i c a l a x i s , which moved up and down on the wave sensor, a t e f l o n - s h e a t h e d b r a s s rod clamped v e r t i c a l l y  4.2  Design  4.2.1  The The  i n the water.  Criteria P r e s s u r e Measurement System  e n t i r e p r e s s u r e measurement apparatus was  designed  so t h a t as  p r e c i s e a measurement as p o s s i b l e c o u l d be made of the phase r e l a t i o n between s u r f a c e e l e v a t i o n and p r e s s u r e .  Rather  severe compromises were  found n e c e s s a r y w i t h amplitude measurements, as w i l l become c l e a r i n subsequent  paragraphs,  but no such compromises were p e r m i t t e d i n the phase  measurements. That The  the phase measurement i s c r u c i a l  can be seen i n T a b l e  4.1.  t a b l e shows t h e o r e t i c a l l y p r e d i c t e d phase d i f f e r e n c e s between p r e s s u r e  and wave e l e v a t i o n f o r r e p r e s e n t a t i v e wind speeds as computed by M i l e s (1959a) and used  6  =  to o b t a i n h i s F i g u r e 6:  tan  ' [ - p d l . / c f / d  -  a(U,/cf)}  4.1  ( t h i s i s a m o d i f i c a t i o n of M i l e s ' c a l c u l a t i o n made by Longuet-Higgins al,  1963;  Q  i s the phase l e a d of p r e s s u r e over wave e l e v a t i o n 1J , C  phase speed, L)  t  -  K  i s the wave  2.5  where U j , i s the f r i c t i o n v e l o c i t y and  ) .  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 g i v e n by  U, - = . 0 W K )  1964)  p Cj  i t i n c l u d e s the e f f e c t of the s t a t i c p r e s s u r e term  et  4.2, (see, f o r i n s t a n c e , Lumley and  i s von Karman's constant^  a.  and p  are numerical  c a l c u l a t e d by M i l e s (1959a) i n h i s theory of wave g e n e r a t i o n .  Panofsky, constants  TABLE  4.1  P r e d i c t e d Phase Lag of P r e s s u r e Behind Wave E l e v a t i o n over Wind-Generated Waves ( C a l c u l a t e d from M i l e s , 1959; the s t a t i c p r e s s u r e termp g»£ has been i n c l u d e d i n the p r e s s u r e , as i n Longuet - H i g g i n s et a l , 1963) fc  Radian  Frequency  U.  -a  C  r  .%  |3  -p(U,/C)  | - a(U,/c)  Q  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.  -  the t a b l e  Wind p r o f i l e v a l u e s used i n 20  cm/sec  U, =  2.5  a. =  Zo =  7 x  10"  U »  =  XI =  2  gz / U, = 1  e  given  50 U-*  0  cm/sec 1  3 x  Z , U,  are:  3..5 x 10"  = 10"  3  (fl  values i s  3  cm  c a l c u l a t e d from 2.8 x  10 ) - 3  41 U s i n g the estimates f o r the l o g a r i t h m i c p r o f i l e f o r "moderate wind over wavy water" i n M i l e s n  ^  3 x 10- ),  a  for values  rad/sec.  (u* & 20 cm/sec, z a 7 * 10"  ,  c  and|3 can be c a l c u l a t e d , and the t a b l e d i s p l a y s  J  results  (1959a)  of  the r a d i a n wave-frequency CJ  Expected phase s h i f t s  between 0.5  the  and 10  are t h e r e f o r e s m a l l , r e q u i r i n g an  a c c u r a t e phase measurement. A further calculation w i l l of  the p r e s s u r e f l u c t u a t i o n s .  g i v e some i d e a of If  the expected amplitude  the water s u r f a c e e l e v a t i o n i s  given  by  r[ = a sm (ftx - cot) , then the wave s l o p e has amplitude  dx I The momentum f l u x to the waves from p', the p a r t of field  i n phase w i t h the wave s l o p e ,  the t o t a l  pressure  is  dx where the overbar denotes a time average.  Root mean square s l o p e s on  wind d r i v e n waves have been measured (Cox and Munk, 1954a, and  are  (kaf If  b; Cox, 1958),  it  for  is  Then  - 0.1 .  assumed that h a l f  a 5 meter/sec  T=  |  f C U a  D  of  s i z e of  then  wind, l  -  U.«|0" « l.5"|0" « Z5«\0  ~  0.5" dyne c m .  5  jp'| ~ ( Y / E ) ( l / k a j The  the t o t a l wind s t r e s s goes i n t o waves,  3  4.3.  - 4  ~ 1.5 dyne cm'  Z  4.4.  the expected p r e s s u r e amplitudes becomes even more  important when the e f f e c t of  the buoy shape on the a i r flow over i t  is  42 considered. the  The mean f l o w s t r e a m l i n e s w i l l be d i s t o r t e d by the buoy,  d i s t o r t i o n being r e l a t e d  buoy above the w a t e r l i n e .  to the shape chosen f o r the p a r t of the  This d i s t o r t i o n w i l l  cause the presence of  dynamic p r e s s u r e s on the buoy s u r f a c e , 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 b e i n g a f u n c t i o n of p o s i -  2  t i o n on the buoy s u r f a c e and of the shape of the buoy.  I f a turbulent  wind of speed U + u', where u i s the f l u c t u a t i n g p a r t of the t o t a l wind, 1  blows over the buoy, then the f l u c t u a t i o n s i n dynamic p r e s s u r e a t the sensor would be P =  F e  d  a  U  U  4.5;  t y p i c a l v a l u e s o b t a i n e d from wind tunnel t e s t s f o r the shape of the buoys used and a 500 cm/sec wind g i v e , assuming u to be g i v e n by(a*j f o r a situation similar  o  to those observed a t the measurement  =  0.2 x 1.2 x 1 0 "  ~  7 dynes /  3  site,  x 60 x 500  4.6.  cm^.  T h i s means t h a t to a c h i e v e the modest s i g n a l to n o i s e r a t i o of t e n i t became n e c e s s a r y to d e v i s e means f o r r e j e c t i n g a t l e a s t 96% of the dynamic p r e s s u r e  fluctuations.  The s i z e of the buoy was of g r e a t importance.  I t had to be l a r g e  enough to c a r r y the r e q u i r e d e l e c t r o n i c s and y e t s m a l l enough to respond to h i g h frequency waves beyond the peak of the spectrum. One of the major f a c t o r s which had to be c o n s i d e r e d i s the a b i l i t y of  the buoy to keep water o f f i t s s u r f a c e , 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 p r e s s u r e s i g n a l has  been mentioned ( p. 3 ) .  I t remains to note t h a t water, once on the buoy  43 surface,  could very e a s i l y  location.  Also surface  waves a t the f r o n t of  then be blown by the wind to the  sensor  c u r r e n t s i n the wind d i r e c t i o n produced bow  the buoy which were then blown over the  sensor.  T h i s c o n s i d e r a t i o n demanded buoy shapes d i a m e t r i c a l l y opposed the shapes suggested as good f o r r e j e c t i n g dynamic p r e s s u r e s ; a compromise had to be reached which s a t i s f i e d each of  the two  as c l o s e l y  to  therefore,  as i s  feasible  criteria.  The frequency response  of  the p r e s s u r e system was much h a r d e r to  c o n t r o l a t low than a t h i g h f r e q u e n c i e s .  The h i g h e s t  frequency a t which  waves c o u l d i n p r a c t i c e be r e s o l v e d by the wave probe used i n the e x p e r i ment was not more than 5 H z , and most p r e s s u r e measurement d e v i c e s  are  q u i t e capable of r e s p o n d i n g to such f r e q u e n c i e s - - i n d e e d . ; many, s u c h . a s microphones, Therefore, one more of digital filtered  are not u s u a l l y designed  the c h o i c e of convenience  sampling r a t e .  to respond to such low  the h i g h - f r e q u e n c y c u t o f f of than a n y t h i n g e l s e .  frequencies'.  the system was  One c o n s i d e r a t i o n was  F r e q u e n c i e s above h a l f  this  the  frequency had to be  out to a v o i d a l i a s i n g .  The low-frequency c u t o f f spectrum i s  at i t s  l a r g e r i n the  largest  (lower)  presented more problems.  there;  in fact,  The p r e s s u r e  i t becomes thousands  of  times  frequency range a s s o c i a t e d w i t h the passage of  storms and f r o n t a l systems.  The lowest f r e q u e n c i e s  a t which w i n d - d r i v e n  waves were encountered a t the e x p e r i m e n t a l s i t e were about 0.2 H z , and the u s u a l frequency a t which the peak of Hz.  the spectrum o c c u r r e d was  0.5  Because a c c u r a t e measurement of phase i n the r e g i o n of wave genera-  t i o n was so i m p o r t a n t , i t became n e c e s s a r y a t l e a s t a decade below t h i s ,  a t about 0.05  to s e t  the c u t o f f  H z , thus keeping  frequency phase  44 corrections  less  than 1 0 ° at 0.5  low i n t r o d u c e d two problems. ponents  Both of  Setting  the c u t o f f  the l a r g e ,  introduced d r i f t s ;  second,  this  internal  low-frequency  these h i d e i n f o r m a t i o n a t h i g h e r f r e q u e n c i e s  tempera-  noise.  because of  windows  analysis.  The Wave Measurement System Demands on the wave probe were l e s s s t r i n g e n t  p r e s s u r e measurement system. about 100 cm, and the e n t i r e  3.0 H z .  The probe i t s e l f  by the e x t r a drag of  frequency band of  were  i n t e r e s t w i t h i n which a l l  i n amplitude were found was 0.1  had to be s t r o n g enough so t h a t f l e x u r e  the buoy was kept s m a l l ,  t o r t e d o n l y the observed shape of the r e l a t i v e  than those on the  Expected maximum wave e x c u r s i o n s  waves more than one or two m i l l i m e t e r s  of  frequency  n a t u r a l low-frequency com-  i n the sensor i n t r o d u c e d s p u r i o u s  used i n the s p e c t r a l  4.2.2  First  i n the p r e s s u r e s i g n a l  ture v a r i a t i o n s  Hz.  the waves;  although t h i s it  phase between p r e s s u r e and waves.  caused  flexure  d i d not a f f e c t  -  dis-  the a c c u r a c y  A b s o l u t e water  depth  a c c u r a c y was unimportant; o n l y the r e s o l u t i o n m a t t e r e d . The frequency response  of  the wave probe used was e s s e n t i a l l y  from DC to a f r e q u e n c y somewhere between 3 and 5 H z , c o v e r i n g the range of  4.3  interest  expected  a t the e x p e r i m e n t a l  The Measurement  Site  The l o c a t i o n of  the measurement  a tidal flat (distance  site is  from which the wind blows  is  shown i n F i g u r e  e a s t are apartment b l o c k s r a n g i n g i n h e i g h t  2.  1.  It  The a v a i l a b l e  as a f u n c t i o n of  shown i n F i g u r e  completely  site.  a d j o i n i n g P o i n t G r e y , Vancouver, B. C.  from shore to o b s e r v a t i o n p o i n t )  flat  the  is fetch  direction  On the l a n d to the  above sea  level  from 20  southto  45 80 m e t e r s .  Water depth a t the s i t e and over most of  from 0 to 4 meters depending on the t i d e . circular masts  tidal  c u r r e n t i n E n g l i s h Bay.  themselves  ranges  a weak ( < lm/sec)  F i g u r e 3 shows the measurement  and the p l a t f o r m on which the d a t a were r e c o r d e d .  d i s t a n c e from p l a t f o r m to mast i s  4.4  There i s  the sand f l a t  The  60 meters i n a N o r t h e r l y d i r e c t i o n .  The Measurement of Wind Speed Wind speed was measured i n two d i f f e r e n t ways d u r i n g the course of  the experiment, u s i n g cup anemometers, F o r the f i r s t f o u r of  and u s i n g a s o n i c anemometer.  the s i x runs p r e s e n t e d , mean wind speed and d i r e c t i o n  were measured w i t h three Thornthwaite s e n s i t i v e at heights  of 3, 4,  cup anemometers  and 5 meters above the water s u r f a c e  were deployed a t lower l e v e l s ,  (anemometers  but were d e s t r o y e d j u s t p r i o r  p e r i o d d u r i n g which the f i n a l l y - u s e d runs were made).  located  to  Of these  the three  anemometers the one a t the 4 meter h e i g h t read low c o n s i s t e n t l y , d a t a from i t was not u s e d .  and  S i n c e o n l y two r e l i a b l e d e t e r m i n a t i o n s of  wind speed were t h e r e f o r e a v a i l a b l e , no attempt was made to e s t i m a t e  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 p r e s e n t e d no cup anemometers were a v a i l a b l e ,  and wind speed was measured w i t h 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 Denki 3 - d i m e n s i o n a l S o n i c Anemometer ( M i t s u t a ^ t a l ,  1967).  was used to o b t a i n the mean and f l u c t u a t i n g components of a i r v e l o c i t y U.  From t h i s  Kaijo-  This  device  the downwind  q u a n t i t y the mean wind speed tJ, the mean d i r e c -  t i o n 8 , and the wind s t r e s s  T  = -  were e x t r a c t e d .  P^UW  =  ^"1  The mean wind speed U5 at 5 meters height;.-' was e x t r a -  p o l a t e d from the v a l u e of U a t the h e i g h t of the s o n i c anemometer  (about  46 1.5 meters) assuming a l o g a r i t h m i c wind p r o f i l e as d e f i n e d  i n Equation  2.11. 4.5  The Wave Sensor The sensor used to measure waves was of  similar stock  to t h a t used by G i l c h r i s t  (1965).  the  It  capacitance  consisted  type,  of a \  inch  (0.635 cm O . D . ) b r a s s rod 180 cm i n l e n g t h covered w i t h a  dielectric sealed  i n the form of s p a g h e t t i  t u b i n g (0.719 cm O . D . , 0.681  a t the bottom and w i t h an e l e c t r i c a l  a t the t o p .  cm I . D . ) ,  c o n n e c t i o n to the b r a s s rod  T h i s 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  determing network of a b l o c k i n g o s c i l l a t o r Burling  teflon  (1955),  and Kinsman (1960).  v a r i e d w i t h changes i n water  level  to the  frequency-  s i m i l a r to that used by  The frequency of the o s c i l l a t o r on the probe.  thus  This frequency-modulated  (FM) s i g n a l was r e t u r n e d from the sensor l o c a t i o n to the r e c o r d i n g p l a t form v i a c o a x i a l c a b l e . magnetic  tape,  I t was  then e i t h e r  r e c o r d e d d i r e c t l y on analog  or f o r m o n i t o r i n g purposes was f i r s t  demodulated and then ii  recorded. three  The demodulator used f o r t h i s  ii  purpose was a V e t t e r Model 3  channel FM r e c o r d e r a d a p t o r .  4.5.1  Calibration The f r e q u e n c y - d e t e r m i n g e q u a t i o n of  the o s c i l l a t o r  was  ? = K/(C mH)  4.  p+  where f i s  frequency,  K is  a constant  the f r e q u e n c y - d e t e r m i n i n g network, H is  including resistive  Cp i s  a f i x e d "padding  the water depth to which the probe i s  r a t e of  that i s ,  in  capacitance",  immersed, and m i s  change of probe c a p a c i t a n c e w i t h immersion d e p t h .  were s t a t i c ;  elements  7 >  the  (constant)  Calibrations  they were made by measuring the frequency of  the  47 oscillator -  a t v a r i o u s f i x e d immersion depths i n s a l t water  (salinity 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 q u i t e n o n l i n e a r ; a t y p i c a l  example i s  shown i n F i g u r e 4.  shown i n F i g u r e 5; from t h i s probe i s  The r e s u l t of p l o t t i n g H against t = 1/f i s it  can be seen t h a t the s i g n a l from the  amenable to l i n e a r i z a t i o n d u r i n g d i g i t a l a n a l y s i s ,  i n f a c t done (see  "Data A n a l y s i s and I n t e r p r e t a t i o n " , p.  and t h i s was  84).  The e q u a t i o n f o r H , from 4.7, i s H  The  =  K / m f -  second term i n 4.8  C p / m  4  S i n c e , however,  differences  these d r i f t s were not i m p o r t a n t .  which were r e q u i r e d ,  Changes i n K/m were important, and w i t h t h i s  and  i t was o n l y h e i g h t  i n mind,  a s e r i e s of  six  c a l i b r a t i o n s of the probe was made which extended over three days,  f o r which the s a l i n i t y i n the t e s t i n g  range of  2 - 5.5%».  range of 2 - 1 3 ° C . slopes  _  i n the system used had to be a d j u s t e d by  as much as 307„ from day to day.  static  8  A i r temperature was u n c o n t r o l l e d but v a r i e d over a H was p l o t t e d a g a i n s t ' ^ f o r  (values of K/m) f o r a l l the l i n e s  On the b a s i s  of  tank was v a r i e d over a s a l i n i t y  this  a l l the t e s t s ,  are g i v e n i n T a b l e  and the  4.2.  t e s t n e i t h e r a i r temperature nor water  caused the minor' changes o f s l o p e which o c c u r r e d , l i m i t s f o r wave h e i g h t d i f f e r e n c e s  salinity  and the 957» confidence  observed i n the experiments were +37o  p r o v i d i n g t h a t 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  analysis. 4.6  The P r e s s u r e Sensor The p r e s s u r e sensor was a m o d i f i e d commercial c a p a c i t a n c e m i c r o p h o n e - -  TABLE  4.2  Values of K/m f o r S i x Wave Probe C a l i b r a t i o n s  Date (Jan 1968)  Time of Day  Salinity (°/oo)  Notes:  A i r Temperature (°C)  lCT^K/m (cm/sec)  (If  (2)  22  aft  2.0  13.3  1133  23  morn  4.0  10.0  1126  24  aft  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  is  t h a t measured a t Vancouver  International  Airport. 2.  Mean K x 10" /m 3  =  1122  cm/sec; S t d . D e v i a t i o n  =  15.8  cm/sec.  49 a schematic sensing  diagram ( F i g u r e 6)  shows i t s  diaphragm was a g o l d - c o a t e d  about 0.08  cm t h i c k .  glass disc  1.27  g o l d c o a t i n g made e l e c t r i c a l  its  distance  steel  its of  considerable  time and  the manufacturer ( A l t e c L a n s i n g ,  the microphone diaphragm i n s i d e i n the p r e s e n t  the s t a i n l e s s s t e e l experiment so t h a t  the microphone c o u l d be c o n t r o l l e d .  seconds f o r  to the  casing.  diaphragm s e a l e d  The time constant was  i n t o a volume of  of about 20 seconds was r e q u i r e d .  The  the time con-  the microphone as r e c e i v e d from the m a n u f a c t u r e r , w i t h  space behind i t s  effort  Inc.)  a s m a l l l e a k which allowed a i r to pass from the f r o n t to  s e a l was deemed n e c e s s a r y s t a n t of  the plane of  accelerations.  were spent i n t r y i n g to get  r e a r of  capaciacross  thus m i n i m i z i n g the e f f e c t on the s i g n a l  When the microphone was f i r s t o b t a i n e d ,  off  caused  the p r e s s u r e d i f f e r e n c e  The microphone was mounted i n the buoy so t h a t  seal  and  Variations in  i n s u l a t e d from the s h e l l  tance v a r i a t i o n s which were a measure of  vertical  casing,  c o n t a c t w i t h the s h e l l .  from a b a c k i n g p l a t e  diaphragm was v e r t i c a l ,  The p r e s s u r e -  cm i n diameter and  I t was clamped to a s t a i n l e s s  its  it.  construction.  2 cm , a time  six the  constant  The attempt had to be abandoned when  the manufacturer was u n s u c c e s s f u l  i n sealing  each one e n t a i l i n g about a two-month d e l a y . had to be lengthened  by c o n n e c t i n g  to a l a r g e r volume.  T h i s volume i s  the leak a f t e r  three  T h e r e f o r e the time  the space at  the r e a r of  r e f e r r e d to h e r e a f t e r  as  tries,  constant  the diaphragm the "backup"  volume. The p r e s s u r e measurement p o r t on the s u r f a c e of be l e f t open. of  the buoy c o u l d not  Whenever water was shipped by the buoy i t  the p o r t and clogged  even 0.5  f l o o d e d the  area  cm diameter h o l e s f o r p e r i o d s up to -r-  50  ten seconds " a f t e r  the f l o o d " .  A number of d i f f e r e n t  f o r r e d u c i n g s u r f a c e t e n s i o n were used i n an attempt of  the s m a l l e s t  ful.  "uncloggable" h o l e ,  chemical agents to reduce the  size  but none were p a r t i c u l a r l y s u c c e s s -  T h e r e f o r e , the h o l e was s e a l e d w i t h a t h i n rubber diaphragm,  sealing off  a s m a l l forevolume  (about 2 cnP) i n f r o n t of  the microphone  diaphragm. Because l a r g e ambient low-frequency p r e s s u r e s weather  systems e x i s t e d  i t became necessary  e q u a l i z i n g the i n t e r n a l p r e s s u r e of w i t h ambient p r e s s u r e s .  of  with  to p r o v i d e some means  the microphone-backup volume  for system  T h i s was achieved by v e n t i n g the backup volume  to atmosphere w i t h a slow a l s o designed  associated  ( p e r i o d about 5 0 seconds)  to e l i m i n a t e as much as p o s s i b l e  leak.  T h i s leak was  the e f f e c t s  of  variations  ambient temperature on the p r e s s u r e 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 f a r the most i m p o r t a n t , and was the source of the major component of n o i s e c a l c u l a t i o n shows i t s by an amount  a t low (  effect.  If  0 . 1 Hz) f r e q u e n c i e s .  the backup volume changes  S T , then the a s s o c i a t e d  p r e s s u r e change  (if  A simple temperature  the backup  volume were s e a l e d ) would be  4.9,  where P i s Q  ambient atmospheric p r e s s u r e and T Q i s ambient temperature.  P u t t i n g 1 0 0 0 dyne cm" f o r p , 2  Q  0.3 If cm  290° K for T  Q  , and 0 . 1 ° K f o r  dyne c m  t h i s &T occurs i n 1 0 seconds p r e s s u r e d r i f t s of per minute can be expected;  pressure signals  T gives  the o r d e r of 2 dyne  these are a s m a l l f r a c t i o n of  a t these f r e q u e n c i e s ,  the  observed  but may be important i n cases  where the a i r - s e a temperature d i f f e r e n c e was l a r g e and changing r a p i d l y .  51 4.7  The P r e s s u r e Measurement System A schematic diagram of  the p r e s s u r e system i s  The e n c l o s e d space i n f r o n t of  shown i n F i g u r e  7.  the diaphragm had a r e s o n a n t frequency of  between 150 and 250 Hz depending on the t e n s i o n of  the rubber w a t e r p r o o f -  i n g diaphragm.  4.7.1  The W a t e r p r o o f i n g Diaphragm Many methods of keeping water out of  and some were t r i e d .  The o n l y s u c c e s s f u l  the microphone were c o n s i d e r e d , one was to s e a l  behind the aforementioned rubber diaphragm, experiment. advantages;  Along with i t s  and t h i s was used i n the  obvious advantages  came l e s s obvious d i s -  because i t was mounted h o r i z o n t a l l y , f l u s h w i t h the measure-  ment s u r f a c e , accelerations.  it  caused the sensor to become s e n s i t i v e Being t h i n n a t u r a l rubber i t  hence the p r e s s u r e s e n s i t i v i t y of the system, Further,  the microphone  treatments  as to not change i t s  to make i t  shed water  to v e r t i c a l  changed i t s  t e n s i o n , and  as i t aged w i t h u s e .  (seebelow) had to be chosen so  tension.  Because t h i s diaphragm s e a l e d the system except f o r a slow v e n t atmosphere v i a the backup volume, any b u i l d u p of p r e s s u r e i n s i d e  to  the  system was b a l a n c e d by an i n c r e a s e i n diaphragm t e n s i o n , w i t h a concomit a n t decrease  i n amplitude s e n s i t i v i t y of  b u i l d u p s o c c u r r e d a l l too e a s i l y .  the i n s t r u m e n t .  Such p r e s s u r e  Changes i n ambient temperature  (see  p. 50) were t h e i r main source* another was the i n t e r m i t t e n t passage of water over the diaphragm.  This could,  if  i t happened f r e q u e n t l y ,  cause  the mean p r e s s u r e on the diaphragm to d i f f e r from ambient enough to change the diaphragm t e n s i o n . Because i n p r a c t i s e there was always water on the buoy s u r f a c e ,  it  52 was n e c e s s a r y  to prevent the f o r m a t i o n of water drops which "nested"  the diaphragm and prevented p r e s s u r e measurement. p e r t y of  the water  i n fact  microphone s e n s i t i v i t y To prevent t h i s  and i t s  by a f a c t o r of  "nesting",  2 -  oil.  the sys tern.  tension  (which i n c r e a s e d  agent,  basic  3.  its  acceleration  i t was  sodium s i l i c a t e  sensitivity  treated with a l i g h t  The area immediately around i t was,  coated w i t h a w e t t i n g  This  and a l s o to prevent the diaphragm from  t e n s i o n by an unknowable amount),  silicone  of  and e r r o r to be t h a t which reduced the  b e i n g wetted by the water  pro-  determined the minimum p r a c t i c a b l e diaphragm  t e n s i o n and hence the maximum s e n s i t i v i t y was determined by t r i a l  T h i s "nesting"  in  except f o r s m a l l  or water g l a s s .  areas,  This  treatment remained e f f e c t i v e f o r p e r i o d s of up to two hours i n  conditions  where the diaphragm was b e i n g inundated once every 10 - 15 seconds on the average.  4.7.2  The Backup Volume The backup volume was an aluminum c y l i n d e r 4 cm long w i t h a 3.5 cm  i n s i d e diameter. by 10 cm of  I t was connected  to  1.5 mm I . D . p o l y e t h y l e n e  the r e a r of tubing.  I t was vented  by 4 cm of 0.1 mm I . D . s t a i n l e s s s t e e l  tubing.  always wet,  into  the vent  itself  was s e a l e d  p r o v i d e d v i a 10 cm of s t a i n l e s s s t e e l vertically,so  that i t s  the microphone diaphragm  Because  the buoy and an a i r passage  tubing (1 mm I . D . )  the backup volume to the microphone c a s i n g was from f l e x i n g  which was  fixed  The t u b i n g from  cemented f i r m l y i n p l a c e  as the buoy moved on the  The backup volume was embedded i n the styrofoam buoy. i t was not t h e r m a l l y i n s u l a t e d .  atmosphere  the buoy was  top d i d not o f t e n become b l o c k e d .  i n the buoy to p r e v e n t i t  to  water.  Other than  A l t h o u g h t h i s meant t h a t  this,  temperature-  53 induced p r e s s u r e d r i f t s i n t r o d u c e d some n o i s e a t f r e q u e n c i e s 0.1  Hz, space and weight  design.  less  than  c o n s i d e r a t i o n p r e c l u d e d any more e l a b o r a t e  A d e s s i c a n t was s e a l e d  i n t o the backup volume to prevent con-  d e n s a t i o n from o c c u r r i n g e i t h e r on the microphone diaphragm or i n the vent  to atmosphere.  E i t h e r would p r e v e n t the system from f u n c t i o n i n g 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 d u r i n g the f i e l d  4.8  experiments.  Electronics The e l e c t r o n i c s  recorder,  system,  two components  (tape  FM tuner) had to be designed s p e c i f i c a l l y f o r the experiment.  The reasons  f o r choosing the p a r t i c u l a r system used are f i r s t o u t l i n e d ;  a more d e t a i l e d account of  4.8.1  w i t h the e x c e p t i o n of  the p a r t s of  the system  follows.  The Design C r i t e r i a The d e s i g n 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  follows: 1.  Buoy e l e c t r o n i c s  2.  Pressure s e n s i t i v i t y  change i n p r e s s u r e .  small, light, greater  E l e c t r o n i c s noise  quency range o f i n t e r e s t  (set  waterproof;  than 2 m i l l i v o l t s per dyne cm"*  2  l e s s than 0.2 mv p-p i n the  by the tape r e c o r d e r n o i s e  fre-  level);  3.  Frequency and phase response n e a r l y f l a t  4.  P r e s s u r e s i g n a l f r e e from c o n t a m i n a t i o n caused by the waves.  The requirements of No. 1 above are d i s c u s s e d (p. 6 0 f f ) .  S t a t e d b r i e f l y they a r e :  had to be l e s s interest--about  than o n e - h a l f 50 cm i n t h i s  i n the range 0.1  later i n this  the p h y s i c a l dimensions of  the wavelength of experiment;  - 5.0 Hz;  chapter  the buoy  the s h o r t e s t waves of  the d i s p l a c e m e n t had to be  54  minimized to reduce the drag of  the water on the buoy, which produces  instability  the buoy i n t o  i n the alignment of  waterproofing  l e d to the i n c l u s i o n of  the wind.  The need f o r  the b a t t e r y f o r the e l e c t r o n i c s  the buoy r a t h e r than b r i n g i n g i n DC power v i a the s i g n a l  in  cable.  The l i g h t e s t t r a n s d u c e r w i t h the r e q u i r e d p r e s s u r e s e n s i t i v i t y  was  a c a p a c i t a n c e microphone.  I n o r d e r to get  response,  to use frequency m o d u l a t i o n r a t h e r than the  i t was necessary  u s u a l DC s i g n a l c o n d i t i o n i n g system.  the r e q u i r e d low frequency  S i n c e the c a p a c i t a n c e and the modu-  l a t i o n of c o m m e r c i a l l y a v a i l a b l e microphones was extremely s m a l l  (typically  5 pf w i t h T-0.02% m o d u l a t i o n per m i c r o b a r p r e s s u r e d i f f e r e n c e  across  diaphragm)  The FM band  i t was n e c e s s a r y  (88 - 108 megahertz) were e a s i l y bought  to use a h i g h frequency c a r r i e r .  was chosen,  c h i e f l y because components  the  f o r t h i s band  locally.  The u n d e s i r a b l e c o n t a m i n a t i o n of the output s i g n a l by wave motions proved of major importance i n the d e s i g n i n two separate  areas:  the c h o i c e of methods of d a t a t r a n s m i s s i o n from the buoy; and the d e s i g n of s h i e l d i n g f o r the  first,in second,  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 the buoy to the h u t . of  After  considerable e f f o r t  this  idea,  t r a n s m i t t i n g to the instrument mast had to be g i v e n up.  on the b a t t e r y and s i z e l i m i t a t i o n on the antennae,  v a r i e d w i t h wave m o t i o n s . limited,  the water s u r f a c e  As a r e s u l t  fluctuations  that  The power limitations  v a r i e d over enormous  (the  ground plane)  the r e c e i v e r was not always  and wave-induced modulations of the amplitude of  caused s i g n i f i c a n t  from  and even  r e c e i v e d from the buoy, s m a l l to b e g i n w i t h because of weight  ranges as the c o n f i g u r a t i o n of  in  the  fully  carrier  i n the demodulated tuner o u t p u t .  55 An e f f e c t w i t h a s i m i l a r r e s u l t o c c u r r e d i n the buoy o s c i l l a t o r . The movement of  the ground plane caused l a r g e changes i n l o a d i n g on the  carrier oscillator, limits  which could not adequately be removed ( w i t h i n  on power consumption s e t  The problem remained a f t e r  by the b a t t e r y )  enclosed  the  4.8.2  System  D e s c r i p t i o n of  A b l o c k diagram of into  A ground to  but not used when i t was found to have no  on the performance of  effect  oscillator.  the e l e c t r o n i c s  three p a r t s :  the  system i s  shown i n F i g u r e 8.  transducer e l e c t r o n i c s  buoy; an RF a m p l i f i e r on the instrument mast; ing a m p l i f i e r ,  cable to the s h i e l d of  (RF) a m p l i f i e r on the instrument mast.  the sea was c o n s i d e r e d ,  to  a box m i l l e d from a b l o c k of aluminum,  grounded o n l y v i a the s i g n a l  the r a d i o frequency  can be s e p a r a t e d  i t was s o l v e d by c a r e f u l a t t e n t i o n  The o s c i l l a t o r used i n the experiment was  i n (and grounded to)  which was i t s e l f  amplifier.  i t was d e c i d e d to t r a n s m i t from buoy to  mast v i a a c o a x i a l c a b l e ; however, s h i e l d i n g and g r o u n d i n g .  with a buffer  the  in  the  and an FM r e c e i v e r ,  and tape r e c o r d e r i n the hut (60 meters  from the  It  match-  mast-  see F i g u r e 3 ) . Sealed i n t o the  the buoy were a b a t t e r y  transducer e l e c t r o n i c s ,  to the 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  to be switched on or o f f  t e r m i n a l s on the buoy, electronics  (15 v o l t s ,  carbon-zinc)  r e l a y e n a b l i n g the power  by a p p l y i n g a DC v o l t a g e  and the t r a n s d u c e r e l e c t r o n i c s  were p o t t e d w i t h beeswax i n t o  which the microphone was a t t a c h e d .  aluminum box,  partments, were the 110 mHz (Clapp) c a r r i e r o s c i l l a t o r stage c o n s i s t i n g  themselves.  the s m a l l aluminum box  In t h i s  to power  to  The to  i n s h i e l d e d com-  and a s m a l l  of a low -Q tuned a m p l i f i e r which matched to the  buffer signal  56 c a b l e j w h i c h was a seven meter l e n g t h of 50 ohm c o a x i a l c a b l e (Amphenol RG -  174/u). C i r c u i t diagrams f o r the o s c i l l a t o r  F i g u r e 9. as  C o n s i d e r a b l e care was  l a r g e as p o s s i b l e .  and a m p l i f i e r are shown i n  taken to make the Q of  The tank c i r c u i t was  the  oscillator  temperature-compensated  through the use of n e g a t i v e temperature c o e f f i c i e n t  capacitors;  the  coil  was hand wound on a ceramic form ( M i l l e r 4300 - Y) and the c o i l w i r e was a s t r i p of  copper 1 mm wide and 0.05 mm t h i c k .  f u r t h e r reduced temperature e f f e c t s ing for  the o s c i l l a t o r  transistors less  The beeswax p o t t i n g  and p r o v i d e d an e x c e l l e n t  and b u f f e r when i t  d i d s h i p some water.  of both c i r c u i t s were b i a s e d f o r low power d r a i n ,  than 10 ma from the 15 v b a t t e r y .  thus  improving i t s  increasing its  buffering  a total  of in  input-to-  capability.  The RF a m p l i f i e r on the instrument mast was designed 100 mHz c a r r i e r ,  The  The l o w - c u r r e n t b i a s i n g used  the b u f f e r a m p l i f i e r had the added advantage of output impedence r a t i o ,  waterproof-  to r e c e i v e  the  a m p l i f y i t by 25 db, and d r i v e a 75 meter l e n g t h of  s h i e l d e d 300 ohm t w i n - l e a d  cable.  I t was powered by two Mercury  cells  and used a commercially a v a i l a b l e  integrated  as a low - Q tuned a m p l i f i e r .  i n p u t was matched to the 50 ohm c o a x i a l  c a b l e from the buoy and i t s  Its  (balanced)  c i r c u i t ( M o t o r o l a MC 1110)  output matched to the 300 ohm  twin-lead. The t w i n - l e a d c a b l e r a n a l o n g the sand from the instrument mast  to  the h u t , where i t was connected  d i r e c t l y to the antenna t e r m i n a l s of a  c o m m e r c i a l l y a v a i l a b l e FM tuner  (EICO ST - 97).  demodulated and the v o l t a g e amplifier  analog of  to an FM r e c o r d channel of  There the c a r r i e r was  the p r e s s u r e fed v i a an o p e r a t i o n a l the tape r e c o r d e r , which was a 14-  57 channel Ampex "CP - 100", or"FR - 1300".  The tuner was m o d i f i e d i n two  ways to i n c r e a s e the s t a b i l i t y o f i t s l o c a l o s c i l l a t o r ;  the B  power  +  s u p p l y was r e p l a c e d by a b e t t e r - r e g u l a t e d one (Dressen - Barnes Model 32 - 101), and the ganged a i r - d i e l e c t r i c  t u n i n g c a p a c i t o r s were r e p l a c e d  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 m o d i f i c a t i o n s and the s p e c i a l care taken w i t h the c a r r i e r oscillator less  i n the buoy reduced  the reproduced  n o i s e o f the system to  than 1 m i l l i v o l t p-p over a frequency range o f 0.01 - 10 Hz.  n o i s e t e s t was of n e c e s s i t y c a r r i e d o u t w i t h a f i x e d  This  5 pf c a p a c i t o r  r e p l a c i n g the microphone, to remove ambient p r e s s u r e n o i s e . The dynamic range o f the e n t i r e system was l i m i t e d by the bandwidth of the I n t e r m e d i a t e Frequency a m p l i f i e r was  carefully  tuned  a g a i n s t i n p u t frequency  amplifier  In the paragraphs  The  (from a r a t i o d e t e c t o r demodulator) i s p l o t t e d  to the I F a m p l i f i e r  C a l i b r a t i o n of P r e s s u r e  i n F i g u r e 1.0..  Sensor  t h a t f o l l o w , d e s c r i p t i o n s w i l l be g i v e n o f the  c a l i b r a t i o n techniques used the p r e s s u r e sensor  This  to g i v e maximum bandwidth and good l i n e a r i t y .  output v o l t a g e o f the tuner  4.9  i n the FM tuner/.  i n the d e t e r m i n a t i o n of the s e n s i t i v i t y of  (the term " p r e s s u r e sensor" w i l l h e r e a f t e r r e f e r to  the complete p r e s s u r e measurement system). c a l i b r a t i o n s were done.  The f i e l d  Both f i e l d  t e s t s determined  and l a b o r a t o r y  amplitude  o n l y , and were n e c e s s a r y because o f day-to-day v a r i a t i o n s  sensitivity  i n sensitivity  caused by the rubber w a t e r p r o o f i n g diaphragm c o v e r i n g the p r e s s u r e p o r t (see the d e s c r i p t i o n of the p r e s s u r e measurement system on p.  5 1).  l a b o r a t o r y c a l i a b r a t i o n s were more p r e c i s e and more e l a b o r a t e , g i v i n g phase response  as w e l l as amplitude  response.  The  58 4.9.1  Laboratory C a l i b r a t i o n s The setup f o r the c a l i b r a t i o n s i s  shown i n F i g u r e 11.  The e n t i r e  buoy was p l a c e d i n s i d e a "clean" t e n - g a l l o n o i l drum a t one end of which was a back p l a t e ,  through which were l e d the c a b l e a t t a c h e d to the buoy  and a s h o r t l e n g t h of 2.5 mm ID s t e e l fitting  t u b i n g , which was l e d v i a an 0 - r i n g  i n t o one p r e s s u r e p o r t of a " B a r o c e l " p r e s s u r e s e n s o r .  o t h e r p o r t of  the B a r o c e l was at ambient atmospheric p r e s s u r e .  p r e s s u r e tube to the B a r o c e l and the buoy c a b l e were s e a l e d the back p l a t e of  carefully  the drum was c l o s e d w i t h a t h i n ,  (horizontal)  to  t a u t rubber mem-  B e a r i n g on t h i s membrane was a c i r c u l a r aluminum or perspex  about 15 c e n t i m e t e r s its  Both the  the drum.  The o t h e r end of brane.  The  i n diameter.  T h i s d i s c was made to o s c i l l a t e  a x i s by a l i n e a r d r i v e r  disc along  ( L i n g A l t e c v47/3 V i b r a t i o n  G e n e r a t o r ) , which was c o n t r o l l e d v i a a low-frequency power a m p l i f i e r by a s i n e wave g e n e r a t o r .  By s u i t a b l e  by the membrane on the d i s c and of signal  to the d r i v e r ,  500 dyne c m  - 2  adjustment of the p r e s s u r e  the amplitude and frequency of  (as measured w i t h the B a r o c e l ) over a range of i n the drum.  c a l i b r a t i o n runs were 30 - 100 dyne cm  these l a r g e amplitudes i s B a r o c e l sensor  is  the  s i n u s o i d a l p r e s s u r e s w i t h amplitudes from 10 to  from 0.003 to 10 Hz c o u l d be generated actual  exerted  better  cm ^ f o r the 30 dyne cm  Amplitudes used i n  (the reason f o r  d i s c u s s e d on p.  59);  than * 1% of f u l l  scale,  amplitude and - 1.3  frequencies  the a c c u r a c y of  choosing the  which m e a n t * 0.4  dyne cm  dyne  f o r the 100 dyne  cm"2 a m p l i t u d e , over a frequency range 0 - 10 Hz.  The analog outputs on a two-channel  from the buoy system and the B a r o c e l were r e c o r d e d  chart recorder (for frequencies  below 0.1 H z ) .  They were  59 a l s o observed w i t h an o s c i l l o s c o p e , phases b e i n g measured by the L i s s a j o u s f i g u r e technique. than 1 Hz.  This method proved  A t f r e q u e n c i e s below 1 Hz  d i r e c t l y from  the s i n u s o i d a l  impractical at frequencies less the phase d i f f e r e n c e s were measured  t r a c e s on the c h a r t r e c o r d i n g s .  The p r i n c i p a l source of e r r o r i n making the c a l i b r a t i o n s was v a r i a b i l i t y of ambient atmospheric apparatus was wind speed  s e t upV.  T h i s was  o u t s i d e , and was  bars) to p r e c l u d e any  p r e s s u r e i n the room where the  presumably due  to f l u c t u a t i o n s i n the  on windy days l a r g e enough (up to 100  c a l i b r a t i o n s at a l l .  T h i s upper amplitude  buoy system, which was  l i m i t was  from ambient p r e s s u r e was b r a t i o n s decreased  107o  Field  Hz, w i t h amplitude  than d i d phase accuracy.  d i f f e r e n c e a t the two  a s c r i b e d to a n o n - l i n e a r response  4.9.2  S i n c e most of the n o i s e  r a p i d l y below about 0.05  c a l i b r a t i o n of the p r e s s u r e system used The  s e t by the dynamic range of the  i n the f i e l d  amplitudes  of  accuracy  The l a b o r a t o r y  i s shown i n F i g u r e  of the rubber w a t e r p r o o f i n g diaphragm.  Calibrations  S i n c e i t was  n a t u r a l rubber and was  o i l b e f o r e the s t a r t of each run, i t was  field  cali-  i n the s e n s i t i v i t y i s  These were made n e c e s s a r y by the presence diaphragm.  arising  a t low f r e q u e n c i e s the a c c u r a c y of the  f a l l i n g o f f considerably faster  12.  as p o s s i b l e i n  i n t u r n s e t by the bandwidth of the tuner IF  (See " E l e c t r o n i c s " , p. 5 7 ) .  amplifier  micro-  Hence the c a l i b r a t i o n s were  done on calm days and w i t h as l a r g e a p r e s s u r e amplitude the drum.  the  c a l i b r a t i o n s , and  of the w a t e r p r o o f i n g treated with a t h i n  f e l t necessary  to c a r r y out  these were done b e f o r e and p r e f e r a b l y a f t e r  run. The method used was  simple:  the buoy was  lifted  from  film  the water  every  60 (or from the bench,  if  the c a l i b r a t i o n was made b e f o r e  moved v e r t i c a l l y a measured d i s t a n c e , The v o l t a g e  the r u n ) , and  u s u a l l y about 100  change r e s u l t i n g from t h i s  centimeters.  p r e s s u r e change was r e c o r d e d on  the c h a r t of an o s c i l l o g r a p h and t h i s  d i v i d e d by f 3 ^ h > where  air density,  the h e i g h t  vity.  g is  and &V\ i s  change,  gave the  U s u a l l y the buoy was moved up and down a number of  average v o l t a g e of  gravity,  change taken.  t h i s method was  ^  f t  The a c c u r a c y (about  107 i n o  is sensiti-  times and the sensitivity)  c o n s i d e r a b l y l e s s than the l a b o r a t o r y c a l i b r a t i o n .  I n a c c u r a c i e s were caused by the i n v a r i a b l e presence pressure f l u c t u a t i o n s  and by the measurement of  of l a r g e  the s h o r t  ambient  distances  over which the buoy c o u l d be r a i s e d and lowered.  4.10  The Buoy  4.10.1  Underwater Shape  The buoy was made from styrofoam; lathe  and a l l f u r t h e r work was done by hand.  cutting,  the e n t i r e  is  A f t e r f i n a l shaping and  s u r f a c e was coated w i t h epoxy r e s i n .  a diameter of about 23 c e n t i m e t e r s It  the b a s i c shape was turned on a  shown i n F i g u r e 13.  The buoy had  and was about 3 c e n t i m e t e r s  Note i n the f i g u r e  c o n t a i n i n g the backup volume was a t t a c h e d  that  thick.  the p a r t of  to the f r o n t  the buoy  section.  The f i n a l underwater shape was a r r i v e d a t p r i m a r i l y by t r i a l error. of  and  Slow-motion movies were taken of many " t r i a l buoys" i n a v a r i e t y  conditions,  and most of  o b t a i n e d from these movies. characteristics  of  the i n f o r m a t i o n c o n t a i n e d i n t h i s I t was n e c e s s a r y  to make the  seakeeping  the buoy such t h a t i t would conform to the  s u r f a c e f o r as l a r g e a percentage  of  the  time as  possible.  section  water  was  The most s t r i n g e n t  e x p e r i m e n t a l requirement f o r the buoy was  i t be c o n s t r a i n e d to move o n l y v e r t i c a l l y . t i l t w i t h the waves, b e i n g a t t a c h e d gymbals,  i t was t h e r e f o r e u n s t a b l e  currents;  that i s ,  amplified.  Since the buoy was f r e e  of  to drag f o r c e s to  as  that  10 cm below the s u r f a c e of  e x e r t e d on i t by  is  at  by making the e n t r y of underwater s u r f a c e ,  from t h i s  instability.  to the c r e d i t of  the  water.  the bow of the bow i n t o  the buoy.  by p u t t i n g as much  T h i s was a i d e d d y n a m i c a l l y  the water v e r t i c a l , by " f a i r i n g " the  and by p l a c i n g the gymbals as low as  I t was d i s c o v e r e d e a r l y t h a t i n s i t u a t i o n s  possible.  of a c t i v e wave g e n e r a t i o n ,  the s u r f a c e water c u r r e n t formed by the wind commonly doubles c r e s t v e l o c i t i e s of  the waves.  taken w i t h the downwind s i d e of critical  the  t h e i r product s u r v i v e d "divesV of as much  The d i v i n g p r o p e n s i t y was minimized s t a t i c a l l y buoyancy as p o s s i b l e  surface  the water s u r f a c e were  the buoy s u f f e r e d  Not even the p r e s e n t model was immune, and i t microphone manufacturers  to  to the b e a r i n g on the wave probe w i t h  small t i l t s with respect  Most e a r l y models  that  Thus whereas l i t t l e the buoy,  the o r b i t a l  care needed  the shaping of i  to be  the bow was of  importance.  The e f f e c t on the buoy of a l r e a d y mentioned,  this  c u r r e n t , b e s i d e s the d i v i n g p r o p e n s i t y  was the f o r m a t i o n of a "bow wave" ahead of i t which  could w i t h the a i d of  the wind be swept up and over  r e a c h i n g the s e n s o r .  When t h i s happened a l a r g e "spike" appeared i n  pressure  the buoy,  usually the  signal.  Because of on the wave.  the f a i r i n g of  the bow the c u r r e n t caused i t  T h i s e f f e c t was c o u n t e r b a l a n c e d i n two ways:  centre of g r a v i t y of  to r i d e up first,  the  the buoy was kept w e l l forward by j u d i c i o u s a r r a n g e -  62 ment of i t s  "payload"; second,  some of f l o t a t i o n of  the buoy was l o c a t e d behind the gymbals, waves s h o r t e r than the r a d i u s of  the f r o n t h a l f  so t h a t t i l t s  associated  the buoy were r e s i s t e d .  separation,  thus s t a l l i n g  the flow beneath the buoy and enhancing the l i f t  bow f a i r i n g .  In t h i s way the c o n f i g u r a t i o n was designed  the buoy i n a s t a t e of s t a t i c  with  The f l o t a t i o n  under the bow was ended a b r u p t l y a f t e r 4 cm to cause flow  of  of  of  the  to keep the bow  and dynamic b a l a n c e over a wide range  of water m o t i o n s . The r e a r s e c t i o n of  the buoy was hinged to a l l o w the buoy to respond  to waves w i t h wave-lengths flotation 2 0 ° of  comparable w i t h i t s  radius.  to support a h i g h - d r a g f i n , which kept the buoy p o i n t e d w i t h i n  the wind d i r e c t i o n .  buoy b e s i d e s  The s o l e f u n c t i o n of the r e a r s e c t i o n o f  those mentioned, was to make the buoy s y m m e t r i c a l ,  a v o i d i n g the p o s s i b i l i t y of r e v e r s a l of where the wave o r b i t a l v e l o c i t i e s wind-generated s u r f a c e c u r r e n t . significant  i n the troughs were g r e a t e r  allowing i t  T h i s c o n t r i b u t e d g r e a t l y to i t s 4.10.2Upper  Surface:  the t i l t i n g p a r t s o f  switching off  to f l e x  i n F i g u r e 13)  the  on e i t h e r  as much as T 3 0 ° from the h o r i z o n t a l .  seakeeping  abilities.  the buoy was a cover beneath which were  and i t s  electronics,  which powered the e l e c t r o n i c s , of  than the  Aerodynamics  The upper s u r f a c e of the p r e s s u r e sensor  thus  Because s h o r t wavelength waves have  amplitudes i n a wind-generated sea  the h i n g e s ,  the  the buoy d i r e c t i o n i n cases  buoy had to be cut away ( i n d i c a t e d by d o t t e d l i n e s s i d e of  I t had enough  the backup volume,  sealed  a battery  and a s m a l l r e l a y which p r o v i d e d a means  the b a t t e r y c u r r e n t when the instrument was not i n use.  Pressures were t r a n s m i t t e d to the sensor  through the t h i n rubber  63 diaphragm, which was g l u e d over a 0.8 plate  (8.2  and i t s  cm diameter h o l e i n the  cm x 3.9  cm x 0.3  cm--see F i g u r e 13) which s e a l e d  electronics  into its  compartment.  perspex the  sensor  The b a t t e r y and the r e l a y were h e l d i n p l a c e w i t h a s i l i c o n e sealant. of  The upwind s u r f a c e of  the buoy was waterproofed w i t h a sheet  t h i n p o l y e t h y l e n e which covered a l l j o i n t s .  i n place with a f a s t - s e t t i n g final  and i n p a r t i c u l a r  securely As a  the buoy except f o r the diaphragm  to keep r a d i a t i v e heat t r a n s f e r to the buoy,  to the backup v o l u m e , t o a minimum.  P r i o r to the d e s i g n phase o f the d i s t o r t i o n of  T h i s was h e l d  r u b b e r - b a s e cement and masking tape.  p r o t e c t i o n the e n t i r e s u r f a c e of  was s p r a y - p a i n t e d w h i t e ,  rubber  the experiment,  the mean s t r e a m l i n e s of  cause s p u r i o u s p r e s s u r e f l u c t u a t i o n s  i t was r e a l i z e d  that  the a i r flow by the buoy c o u l d  at the p r e s s u r e s e n s o r ,  not r e j e c t e d i n some way c o u l d s e r i o u s l y a f f e c t  which i f  the v a l i d i t y of  the  results. To see how g r e a t t h i s instance,  Lamb (1932),  e f f e c t would be the s o l u t i o n (see,  103)  for  f o r the p o t e n t i a l flow over a p l a n e t a r y  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 p r e s s u r e d i s t r i b u t i o n i s From the f i g u r e , a s y m p t o t i c to - 0 . 2  shown i n F i g u r e  the dynamic p r e s s u r e s  times  14.  e x e r t e d on t h i s  the s t a g n a t i o n p r e s s u r e i P U" . 2. *  shape are Although  the  v  shape'used has a bow ( l e f t  s i d e of  the f i g u r e ) which i s much l e s s  bluff  .1 than t h a t used i n the a c t u a l buoy, w i t h i n 507o of buoy.  If  the f i g u r e of - 0 . 1 p^U  i s probably  the a c t u a l f i g u r e a t the p r e s s u r e measurement p o r t on the  -0.1 P U  is  c a l c u l a t e d f o r a 500 cm/sec wind i t g i v e s a  v a l u e f o r the dynamic p r e s s u r e which must be r e j e c t e d a t the p r e s s u r e  64 p o r t of about 7 dynes/cm  2  Further calculations  (see  p.  4 2 ).  show t h a t  by i n c r e a s i n g the e c c e n t r i c i t y of m i n i m i z e shape e f f e c t s s l i m as It is  is  this  dynamic p r e s s u r e  the e l l i p s e .  to make the p a r t of  is  easy to see  (vertical)  The b l u f f i n water  bow i s  point  the b e s t shape would have a  to  changes  level.  bow p r e s s u r e measurements  I t was n e c e s s a r y  of wind tunnel  4.11  the p r e s s u r e s e n s i n g  a l s o demanded to i n s u r e f a s t buoy response  and t h a t l a r g e dynamic p r e s s u r e s  to f i n d  as  bow w i t h a s m a l l downward s l o p e i n the upwind d i r e c t i o n .  the l a t t e r was dominant.  on the water  T h i s meant t h a t the buoy d i s t o r t e d  point.  the buoy above water  on the other hand t h a t i f  Of the two opposing requirements a bluff  Thus the b e s t way to  possible.  to be kept r e l a t i v e l y f r e e from water,  bluff  can be reduced  s u r f a c e were  to r e j e c t  impossible.  the a i r flow over i t  c o u l d be expected  at the  these p r e s s u r e s ;  Without  considerably,  measurement  therefore,  a series  t e s t s d e s c r i b e d i n the f o l l o w i n g s e c t i o n was c a r r i e d  the b e s t method f o r doing  Wind Tunnel  this.  Tests  Three s e r i e s of  t e s t s were performed, a l l of  them i n the wind  tunnel of  the Department of M e c h a n i c a l E n g i n e e r i n g of  The f i r s t  two s e r i e s were developmental.  the U n i v e r s i t y .  The f i r s t was concerned w i t h  the d e t e r m i n i n g an optimum aerodynamic shape f o r the buoy w i t h i n limitations  s t a t e d i n the l a s t s e c t i o n ;  the second was  s e a r c h f o r an adequate method f o r r e j e c t i n g by the bow of  out  the buoy.  the  taken up w i t h  the dynamic p r e s s u r e  The t h i r d s e r i e s of measurements  the  caused  were used  to  65 " c a l i b r a t e " the buoy a e r o d y n a m i c a l l y .  4.11.1  The Wind Tunnel  The wind t u n n e l used f o r the t e s t s was of It  the c l o s e d - r e t u r n  generated wind speeds between 1 and 45 m e t e r s / s e c , w i t h low  turbulence l e v e l s , section  and s p a t i a l v a r i a t i o n s  l e s s than 0.257».  i n mean v e l o c i t y  i n the  I t was p r o v i d e d w i t h t a p e r i n g  which p a r t l y compensated f o r boundary l a y e r growth a l o n g i t s  difference The speed density  i n the t e s t s e c t i o n was measured i n terms of  along a 7:1  (0.17„) test  The t e s t s e c t i o n was r e c t a n g u l a r and was  meters wide and 0.7 meters h i g h .  A i r speed  type.  c o n t r a c t i o n immediately upstream of  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  caused by ambient temperature and p r e s s u r e .  0.9  fillets,  length. the  the  pressure section.  i n mean a i r  The p r e s s u r e  decrease  a l o n g the c o n t r a c t i o n was measured w i t h a micromanometer to an a c c u r a c y of  T20 dynes/cm.  a c c u r a t e to  4.11.2  T h i s p r o v i d e d values f o r the f r e e  T17 a t 1 m e t e r / s e c and G  T0.57<> a t 5 m e t e r s / s e c .  E x p e r i m e n t a l Procedure  Wind t u n n e l t e s t i n g The g e n e r a l setup i s f l o o r of water;  stream wind speed  was done d i r e c t l y on the buoy used i n the  shown i n F i g u r e 15.  the t e s t s e c t i o n  a t the same l e v e l  i t was p l a c e d a t the center of  tunnel a x i s .  The bow of  The buoy was r e c e s s e d i n a t which i t  the t e s t s e c t i o n  the buoy c o u l d be s e t  f l o o r on  at v a r i o u s p i t c h  angles by simple adjustments.  openings  the buoy and around i t s  the t u n n e l f l o o r ,  were c a r e f u l l y s e a l e d w i t h tape.  would e x i s t i n the f i e l d measurements  the  f l o a t e d on the  angles and yaw ( o r i e n t a t i o n ) i n the s t r u c t u r e of  field.  the (attack)  In a l l  cases,  edges, where i t  met  The wave probe which  was r e p l a c e d w i t h a l a b o r a t o r y stand  66 of the same diameter,  the base of which supported  the buoy.  P r e s s u r e s along the upstream h o r i z o n t a l a x i s of the buoy were measured a t f i v e c a r e f u l l y prepared p r e s s u r e taps were d r i l l e d  (0.75  mm  i n a s l a b of perspex which c l o s e l y resembled  holes).  These  the one which  s e a l e d the microphone and e l e c t r o n i c s i n t o the buoy (see F i g u r e  13).  They were l o c a t e d a t 0.53,  (slant  d i s t a n c e ) from  the bow  0.57,  0.63,  0.66,  and 0.69  buoy r a d i i  of the buoy, and are shown i n F i g u r e  16.  P r e s s u r e s were measured r e l a t i v e to the s t a t i c p r e s s u r e r i n g of tunnel w i t h a " B a r o c e l " d i f f e r e n t i a l p r e s s u r e t r a n s d u c e r . from  the p r e s s u r e taps and  cally filtered  found  to remove h i g h - f r e q u e n c y  pressure f l u c t u a t i o n s present i n  I n s p i t e of t h i s ,  (amplitudes of about 10  to be s m a l l e r on calm days and  of wind-induced  considerable  T h e r e f o r e the wind tunnel work was  low-frequency  dynes/cm) remained; these were so were presumed to be the  dynamic p r e s s u r e s on the b u i l d i n g housing  result  the t u n n e l .  done whenever p o s s i b l e on days when  the winds were l e s s than a few k n o t s . ments were averaged  lines  the s t a t i c r i n g to the B a r o c e l were pneumati-  the t u r b u l e n t boundary l a y e r . fluctuations  The  To reduce  these e r r o r s a l l measure-  over a p e r i o d (one minute) exceeding  the range of  p e r i o d s c o n t a i n i n g most of the ambient p r e s s u r e f l u c t u a t i o n energy 20  the  (0  to  seconds).  4.11.3  The Dynamic P r e s s u r e R e j e c t i o n Ring  During  the e a r l y d e s i g n stages of the experiment  considered f o r c a n c e l l i n g  the dynamic p r e s s u r e , which i n t h i s case  a n e g a t i v e ( s u c t i o n ) p r e s s u r e caused by the f l o w over method used ring  many methods were  i n the a c t u a l experiment  (see p l a n view of F i g u r e 13)  4.2  was  the boy.  by f a r the b e s t .  cm I.D.  and 0.27  was  The  A thin half-  cm h i g h was  placed  67 on the s u r f a c e of  the buoy a s h o r t d i s t a n c e  pressure port, with i t s it  j u s t upstream of  pressure at  (1.9  a x i s normal to the buoy s u r f a c e and  the p o r t .  the p o r t which c a n c e l l e d w i t h i n "tO.l dyne c m  v i t y of  its  symmetry,  4.11.4  r o t a t i o n of  The A r t i f i c i a l  boundary l a y e r i s  h i g h wind speeds.  tunnel was l a m i n a r , w h i l e  F i g u r e 15).  extending  turbulent.  S i n c e the  so t h i c k ( t y p i c a l l y 100 - 300 m e t e r s ) ,  its  The t r i p c o n s i s t e d w i d t h of  t r a v e r s i n g mechanism which i s  U/u* where u . i s  its  Reynolds extremely  of  a 2.5  upstream of  cm square l e n g t h of wood  separa-  tube mounted on a r e m o t e l y - c o n t r o l l e d  p a r t of  the wind t u n n e l equipment. the "Law of  The  the W a l l " :  f(zu*/Vo)=f(y*),  the f r i c t i o n v e l o c i t y  they e x h i b i t a s t r a i g h t  a range 2 £ l o g y* 4 3 . 6 .  to 2 cm, a t  i n the t u r b u l e n t boundary l a y e r  produced i n t h i s way f o l l o w e d  =  the buoy  the tunnel f l o o r y . i n t o which were cut  Velocity profiles  were measured w i t h a s m a l l P i t o t  mean wind p r o f i l e s  atmospheric  except a t  cm deep w i t h widths v a r y i n g from 0.5  1 - 2 cm.  in  T h e r e f o r e an a r t i f i c i a l t u r b u l e n t boundary l a y e r was  over the f u l l  s l o t s about 1.5  airj  sensiti-  t h a t i n the f i e l d ,  c r e a t e d i n the wind tunnel u s i n g a t r i p 1.7 meters  y<  tunnel.  the buoy about  number cannot be matched i n l a b o r a t o r y wind tunnels  of  i n the  suction  T u r b u l e n t Boundary Layer  the atmospheric boundary l a y e r , was  tions  the  - 2  axis).  The a i r flow i n the  (see  positive  the r i n g removed almost c o m p l e t e l y any  the system to yaw ( t h a t i s ,  vertical  level  the  intersecting  T h i s o b s t r u c t i o n produced a  p r e s s u r e from the bow when the buoy was s i t t i n g Because of  cm) downstream of  and Da. the k i n e m a t i c v i s c o s i t y  l i n e on a p l o t of U / u  Because  the  Vf  versus  log  of (y )  t r a n s i t i o n to t u r b u l e n c e was  Vr  the over artifi-  68 cially  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 l a r g e r and hence  the l i n e  below that found e x p e r i m e n t a l l y f o r the s m o o t h - t r a n s i t i o n Law of Wall  (see,  4.11.5  for instance,  values  the p r e s s u r e a t a l l f i v e  of t h r e e d i f f e r e n t parameters:  Of these t h r e e , of  the  Figure 3).  The Aerodynamic C a l i b r a t i o n of the Buoy  Measurements of of  W i l l m a r t h and Wooldridge 1962,  lay  taps were made over  wind speed,  ranges  p i t c h , and yaw.  wind speed and p i t c h were the most important; symmetry  the r i n g was e f f e c t i v e  f o r measurements  up to  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  1^45°.  Three wind speeds:  3.5,  4.3,  to yaw  and 6.0  i meter sec  , and f i v e  p i t c h angles:  used throughout the t e s t s .  -5,  The r e s u l t s ,  0,  5,  10,  and 20 d e g r e e s ,  which c o n s t i t u t e  c a l i b r a t i o n of the buoy, are shown i n F i g u r e s 17,  18,  were  the aerodynamic  and 19.  These  show the p r e s s u r e d i s t r i b u t i o n over the buoy under the most severe cond i t i o n s expected  i n the  field.  F i g u r e 17 shows the d i f f e r e n c e between ambient p r e s s u r e and t h a t ' measured along the buoy s u r f a c e as a f u n c t i o n of d i s t a n c e from the bow i n buoy r a d i i ;  the p r e s s u r e d i f f e r e n c e  p r e s s u r e so t h a t a b e t t e r dynamic p r e s s u r e s  existing  i s not n o r m a l i z e d to the  " f e e l " may be o b t a i n e d f o r the s i z e of on the s u r f a c e of a.  second the s t a g n a t i o n p r e s s u r e \ £ U a  p r e s s u r e v a r i a t i o n along the buoy i s  is  the buoy (at 6 meters o  about 220 dyne c m " ) . z  if  the bow of  upwards.  A t a t t a c k angles l e s s than - 5 ° the bow of  merged i f  i t were i n the water.  per  The  the buoy i s  tilted  the buoy would be sub-  The arrow between tap p o s i t i o n s  the p o s i t i o n chosen f o r the i n l e t  used i n the f i e l d experiments.  the  shown f o r v a r i o u s a t t a c k angles of  the buoy, which are taken to be p o s i t i v e  3 indicates  stagnation  to the p r e s s u r e  2 and  sensor  I t might be thought that the arrow should  69 be f u r t h e r from the r i n g than shown i n view of the buoy to n e g a t i v e a t t a c k a n g l e s . however,  by the response  angles occur more o f t e n attack  of  This l a r g e r s e n s i t i v i t y  is  (and are l a r g e r on the average)  of  offset, attack  than do n e g a t i v e  angles.  orientations  T30°  large s e n s i t i v i t y  the buoy to a c t u a l waves; p o s i t i v e  F i g u r e 18 shows "worst-case" for  the  of  v a r i a t i o n of p r e s s u r e along the buoy  the buoy w i t h r e s p e c t  f o r an a t t a c k angle of  +  to the wind ("yaw" angles) of  5 ° i n a 6 meter sec"^ w i n d .  The t o t a l  p r e s s u r e v a r i a t i o n a t the buoy s u r f a c e f o r a 6 0 ° change i n yaw angle less  than 2 dyne c m ,  or 17o of  the s t a g n a t i o n  F i g u r e 19 shows the v a r i a t i o n of sensor i n l e t It  can be seen immediately i n view of to l e s s than 0.07  more than + 1 0 ° are u n a c c e p t a b l e ;  957o of p j i s  rejected  if  the buoy's  U  the p r e s s u r e at the "optimum"  l o c a t i o n v e r s u s wind speed f o r v a r i o u s angles of  pressure r e j e c t i o n of  p r e s s u r e pd = \  attack.  the d e s i g n c r i t e r i a (dynamic p^; see  it  p. 4 2)  that attack  angles  can a l s o be seen t h a t more than  a t t a c k angle  is  l e s s than + 1 0 ° and  more than - 5 ° i n the wind speeds as h i g h 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 figure i t in  this  appears  run).  A t a t t a c k angles g r e a t e r  than 1 0 ° , flow s e p a r a t i o n  for a + 2 0 ° attack angle.  sequent a n a l y s i s ,  pro-  the buoy a t wind speeds l e s s than 6 m sec"'';  t e s t s s e p a r a t i o n appeared to be p r e s e n t  above 3 m sec  the  t h a t about 6% dynamic p r e s s u r e c o n t a m i n a t i o n o c c u r r e d  b a b l y occurs a t the bow of d u r i n g the  is  a t a l l wind speeds  When s e l e c t i n g  a l l runs where the a t t a c k angle o f t e n  runs f o r  sub-  exceeded 10  degrees or where the o r i e n t a t i o n angle exceeded 30 degrees were  rejected.  70 4.11.6  Consequences of A t t a c k Angle V a r i a t i o n  F i g u r e 20 shows a schematic r e p r e s e n t a t i o n of observed p r e s s u r e of  l i k e l y v a r i a t i o n s i n the angle of a t t a c k of  buoy as i t r i d e s over a wave on an e x i s t i n g w i t h the wave e l e v a t i o n . its  amplitude, i s  The p r i n c i p a l  the e f f e c t on the  p r e s s u r e s i g n a l i n quadrature  A l t h o u g h the shape of  changed c o n s i d e r a b l y , i t s  the s i g n a l ,  phase i s  and hence  almost u n a l t e r e d .  e f f e c t of angle of a t t a c k v a r i a t i o n s appears to be  i n t r o d u c t i o n i n t o the observed p r e s s u r e s i g n a l of harmonics of frequency,  i n p a r t i c u l a r of  the  the f i r s t harmonic.  the  the wave  SECTION 5:  5.1  DATA ANALYSIS and INTERPRETATION  Analysis 5.1.1  5.1.2  5.1.3  .  .  71  Analog P r e c o n d i t i o n i n g  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  Digitization  75  5.1.2a  75  Hand D i g i t i z a t i o n  The IOUBC F a s t F o u r i e r Transform Package 5.1.3a  FTOR:  79  The F a s t F o u r i e r T r a n s f o r m a t i o n  Program 5.1.3b 5.1.4  5.2  5.2.2  The S p e c t r a l A n a l y s i s Program SCOR .  .  .  81 84  5.1.4a  The Wave H e i g h t S i g n a l  84  5.1.4b  The P r e s s u r e  85  5.1.4c  The S o n i c Anemometer S i g n a l s  86  5.1.4d  S p e c i a l Programming i n SCOR  87  of the S p e c t r a  88  Distortions  .  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 S i z e of the Buoy  91  5.2.1c  Backscattering  94  Distortions  Summary  Signal  I n t r o d u c e d by the Sensors Themselves  I n t r o d u c e d d u r i n g P r e p a r a t i o n and  Analysis  5.3  .  S p e c i a l Programming  Interpretation 5.2.1  80  of Data  98  5.2.2a  F i l t e r i n g of the Wave S i g n a l  98  5.2.2b  The P r e s s u r e  99  Signals  104  SECTION 5:  5.1  Analysis The  of  DATA ANALYSIS AND INTERPRETATION  analysis  what i s  of  the d a t a w i l l be c o n s i d e r e d from the p o i n t of  done to the s i g n a l s  between the time they o r i g i n a t e at  sensors and the time when the f i n a l computer p r i n t o u t of s p e c t r a , spectra, and  etc.  is  subsequent  generated.  For t h i s  reason,  the cross-  analog p r e - c o n d i t i o n i n g  d i g i t i z a t i o n w i l l be d i s c u s s e d  f o l l o w e d by a b r i e f resume of  view  first.  T h i s w i l l be  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  members, which i n c o r p o r a t e the F a s t F o u r i e r Transform technique  its of  Cooley and Tukey (1965) i n t o a p r a c t i c a l scheme f o r o b t a i n i n g d i g i t a l s p e c t r a from d i g i t i z e d analog v o l t a g e specifically  designed  for  this  the author) w i l l be d i s c u s s e d sentation in  the  project  Next,  the programming  (and which i s hence the work of  i n some d e t a i l .  the s t a t i s t i c s of  the i n t e r p r e t a t i o n of  5.1.1  Finally,  methods of p r e -  the s p e c t r a w i l l be mentioned,  as an a i d  spectra.  Analog P r e c o n d i t i o n i n g  5.1.1a The of  of  signals.  Pressure p r e s s u r e r e c o r d i n g system i t s e l f  the s i g n a l .  The amplitude of  was reduced by about a f a c t o r of diaphragm;  the frequency response  filtering  the s i g n a l "seen" by the microphone two by the presence of  the w a t e r p r o o f i n g  up to 10 Hz and was p r o b a b l y f l a t  the n a t u r a l resonance of  w a t e r p r o o f i n g diaphragm and t h a t of  of  the m i c r o p h o n e - w a t e r p r o o f i n g diaphragm  combination i s known to have been f l a t a t 100 H z , s i n c e  performed the f i r s t  the volume e n c l o s e d  by the  the microphone occurs a t 300 H z .  72 The combination of  the l e a k around the microphone diaphragm and  backup volume acted as a h i g h - p a s s -fwas a t 50 _10  frequency cutoff All  of  pneumatic f i l t e r .  Its  the  measured  low-  seconds.  these b u i l t - i n f i l t e r s  caused  the p r e s s u r e measurement  system  to have an amplitude and phase response which v a r i e d w i t h f r e q u e n c y . responses were measured d u r i n g c a l i b r a t i o n (see and were c o r r e c t e d f o r i f n e c e s s a r y i n the The FM p r e s s u r e measurement detector  f o r demodulation of  the r a t i o d e t e c t o r  determined amplitude. flat  set  analysis.  system employed a p h a s e - s e n s i t i v e  The s l o p e of  the v o l t a g e s e n s i t i v i t y  The frequency response of  demodulator was  of  the system and  a c t e d as a b u f f e r  interest.  The s i g n a l from the  through a s i n g l e o p e r a t i o n a l a m p l i f i e r ,  stage between the demodulator and the analog  r e c o r d e r and p r o v i d e d a g a i n which was v a r i a b l e from 0 - 2 . 5 . amplifier  allowed optimum use  tape r e c o r d e r ; keeping  its  the p r e s s u r e  recorder  to be made of  g a i n was s e t  a t the maximum l e v e l  s i g n a l w i t h i n the  limits  (tl.5  This the  consistent with v o l t s ) of  the  tape  system.  The FM r e c o r d mode was u s e d ,  inches per second signal  which  tape  the dynamic range of  Two tape r e c o r d e r s were used; Ampex models 1300".  pressure  the demodulator was found to be  frequency range of  then fed  ratio  the frequency response curve  the v o l t a g e output from the FM tuner f o r a g i v e n i n p u t  over the e n t i r e  12),  the modulated 100 mc frequency which  o r i g i n a t e d a t the microphone. of  Experiment, F i g u r e  These  (ips).  at  "CP -  tape speeds of  Both r e c o r d e r s were s e t  as r e c o r d e d - - n o a m p l i f i c a t i o n was  100" and"FR or 3  3/4  up to reproduce  the  attempted.  1 7/8  73 5.1.1b  Waves  The wave s i g n a l was r e c o r d e d as d e s c r i b e d under Experiment (p. i n one of  two ways:  either  the FM output of  the wave probe b l o c k i n g  o s c i l l a t o r was r e c o r d e d i n the D i r e c t mode as an analog s i g n a l was f i r s t  demodulated w i t h a s e p a r a t e  the FM mode.  5.1.1c  the r e s u l t i n g  of Data to a c h a r t r e c o r d e r ;  c h a r t r e c o r d s were compared w i t h monitor c h a r t r e c o r d s  produced d u r i n g the r u n .  It  is  on the b a s i s of  t h a t p o r t i o n s of d a t a were chosen as s u i t a b l e  right",  it  instrument and then r e c o r d e d i n  The s i g n a l s were p l a y e d back a t the I n s t i t u t e  subjective,  or  I n both cases i t was p l a y e d back i n the FM mode.  Playback and S e l e c t i o n  Because  46)  these c h a r t  for further  recordings  analysis.  the s e l e c t i o n c r i t e r i a f o r the runs were to some e x t e n t b e i n g sometimes the r e s u l t  they cannot be s p e l l e d  of " f e e l i n g s  out c o m p l e t e l y .  that  the r u n looked  The b e s t way seems to be  to enumerate reasons why a c t u a l runs were thrown out: 1.  One of  the f a u l t s  of  the p r o t e c t i v e  diaphragm was that  times "bulged out" or became "sucked i n " under abnormal t e n s i o n f a i l u r e of pressure  the p r e s s u r e e q u a l i s a t i o n  signal,  to p r e s s u r e s  system.  The r e s u l t was a  it  some-  due  to  rectified  s i n c e the diaphragm i n t h i s c o n d i t i o n could respond o n l y  which tended  to decrease the p r e s s u r e d i f f e r e n c e  across  it.  Any runs i n which t h 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. pressure  If  a drop of water  "nested" i n the diaphragm, a  s i g n a l a t the bobbing frequency  Runs where  t h i s happened were e a s i l y  (2 Hz) of  recognized  characteristic  the buoy  resulted.  and thrown o u t .  74 3.  Because of  the u n a v o i d a b l e presence of  large pressure  caused by water s p l a s h i n g on or over the diaphragm, calm-water case had to be exmined c l o s e l y . p r e s s u r e "spikes" were a n a l y s e d . some months w h i l e a (vain) grounds t h a t some of  In f a c t ,  spikes  a l l runs except  Only those w i t h the  one  fewest  a n a l y s i s was postponed f o r  attempt was made to gather more d a t a , on the  the b e s t runs p r e s e n t e d i n t h i s  t h e s i s were c o n s i d e r e d  by the author (not h i s s u p e r v i s o r ) as not worth a n a l y s i n g !  E v e n t u a l l y the  o n l y runs thrown out because of s p i k e problems were those where water remained on the diaphragm long enough to change the mean p r e s s u r e i n the backup volume,  l e a d i n g to the presence of  l a r g e steps i n the p r e s s u r e  records. 4. system.  The l a s t  l a r g e source of "bad runs" was d r i f t  U s u a l l y the g a i n of  i n the p r e s s u r e  the b u f f e r a m p l i f e r at the demodulator o u t -  put was kept as h i g h as p o s s i b l e ,  to maximize the s i g n a l to n o i s e  The p r a c t i c a l upper l i m i t to the g a i n was determined by d r i f t , the g a i n was too h i g h caused the s i g n a l to exceed r e c o r d e r FM system.  Thus runs were r e j e c t e d i f  ratio.  which i f  the l i m i t s of  the  tape  the monitored p r e s s u r e  s i g n a l s were even p a r t i a l l y b u r i e d i n n o i s e or i f v o l t a g e  l i m i t s were  exceeded d u r i n g p a r t o f a ten minute p e r i o d . The r e m a i n i n g runs ( t h e r e were s i x a n a l y s e d , of which two were a n a l y s e d i n two p a r t s ; see  "Results") were then p r e p a r e d f o r d i g i t i z a t i o n .  One r u n which was h a n d - d i g i t i z e d w i l l be d i s c u s s e d s e p a r a t e l y  5.1.Id  75  ff.).  Rerecording  The runs were r e r e c o r d e d at twice noise  (p.  and time used f o r a n a l y s i s )  a m p l i f i c a t i o n and h i g h - p a s s  the tape speed  (thus r e d u c i n g  a f t e r p r e c o n d i t i o n i n g c o n s i s t i n g of  filtering.  The low c u t o f f  frequency (0.02 Hz)  75 of  the f i l t e r was such t h a t n e g l i g i b l e  lowest f r e q u e n c i e s  (0.05  Hz) of  interest.  l o s t due to r i n g i n g i n the f i l t e r s s i g n a l v a l u e near the s t a r t of  ( T l ° ) phase s h i f t  occurs at  the  To minimize the amount of  a DC v o l t a g e e q u i v a l e n t  data  to the mean  the r u n was a p p l i e d to each f i l t e r ;  then  when the r u n began the DC v o l t a g e s were r e p l a c e d by the s i g n a l s  them-  selves.  components  T h i s method e f f e c t i v e l y  from both the wave and p r e s s u r e to make optimum use of  5.1.2  removed s p u r i o u s signals.  low frequency  The a m p l i f i e r gains were  the dynamic range of  set  the tape r e c o r d e r .  Digitization All  s i g n a 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) c o n v e r t e r , of d i g i t i z i n g as many as  ten channels  f a c e d w i t h a s m a l l computer forms p a r t of  of d a t a .  which i s  T h i s instrument  capable is  inter-  ( C o n t r o l Data 8092 Teleprogrammer) which  the computing f a c i l i t y of  8092 was programmed to w r i t e d i g i t a l  the U . B . C .  Computing C e n t r e .  (A/D) tapes f o r subsequent  The  process-  i n g on the main computer a t the Computing C e n t r e , which was an IBM 7044 a t 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 r e c o r d e r were  passed through low-pass l i n e a r phase s h i f t quency of 6 Hz p r i o r to d i g i t i z a t i o n . mately  50 Hz was used f o r a l l r u n s .  composite  n o i s e of  the p r e a m p l i f i e r ,  filters  with a cutoff  fre-  A sampling frequency of a p p r o x i In most cases a s h o r t s e c t i o n of filters,  and tape r e c o r d e r s was  the also  digitized. 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 c h a r t paper r e c o r d i n g  76 of  the p r e s s u r e and wave s i g n a l s  made on October 30,  1967.  two-channel r e c o r d e r .  The r e c o r d e r used was a Sanborn Model  on a "Thomson E l e c t r o n i c s " Model PF-10 P e n c i l  i n the Canadian Oceanographic Data Centre i n Ottawa.  Two separate  p i e c e s of data were a n a l y s e d .  (run 4a) was s p l i t stationarity. difference  into  curves  S i n c e there proved to be no s t a t i s t i c a l l y  these for  signficant  between the p a r t s they were subsequently merged, and are p r e In both r u n s ,  the s p i k e s  i n the  pressure  were smoothed by hand b e f o r e d i g i t i z a t i o n by drawing smooth j o i n i n g the "good" data on e i t h e r  Two sources of those which a f f e c t first  The f i r s t of  two p a r t s i n o r d e r that i t might be analysed  sented t o g e t h e r as r u n 4 a . signals  320  The d i g i t i z i n g was done by M r . J . R. W i l s o n , a  f e l l o w graduate s t u d e n t , Follower  which was taken d u r i n g the a c t u a l r u n  them.  e r r o r i n the measurement of time must be both channels  type of e r r o r i n c l u d e s ,  the d i g i t i z e r .  s i d e of  considered:  e q u a l l y and those which do n o t .  besides chart v a r i a t i o n s ,  The r e c o r d e r c h a r t d r i v e speed of  The  inaccuracies  5 mm/sec v a r i e s  in  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 T 0 . 1 mm, which corresponds  to TO. 02 seconds or T27=; so the  t o t a l expected a b s o l u t e  time  r e s o l u t i o n i s T4.57o. The second  type of e r r o r i s  the more s e r i o u s  of  the two;  it  involves  processes which produce time s h i f t s between the data d i g i t i z e d from different errors:  channels  at d i f f e r e n t  those a s s o c i a t e d  it  taut;  heated  There are two sources of  such  w i t h the c h a r t r e c o r d e r and those a r i s i n g from  the d i g i t i z a t i o n p r o c e d u r e . so t h a t the paper i s  times.  The c h a r t d r i v e of  the r e c o r d e r i s  p u l l e d over a r i d g e w i t h s u f f i c i e n t  s t y l i b e a r i n g on the r i d g e produce the  arranged  s t r a i n to keep  trace.  The  77 s p e c i f i e d maximum misalignment i s TO.25 mm a c r o s s cm wide;  this  causes p o s s i b l e misalignments between  v a l e n t to T.025 seconds. e r r o r s due to t h i s differences  since  source.  the r e c o r d e r s p e c i f i c a t i o n s  be s m a l l .  the channels e q u i -  Another source of e r r o r may a r i s e  S i n c e the wave s i g n a l i s  the a m p l i f i e r responses  the two s t y l u s  essentially  from driver  zero by 10 H z , and  i n d i c a t e t h a t when p r o p e r l y compensated  are o n l y down 3 db a t 100 H z , t h i s  e r r o r should  However a m p l i f i e r compensation had not been a d j u s t e d f o r some  time p r e v i o u s to the r u n ; a l s o  the p r e s s u r e of  waves i n c r e a s e d markedly towards the end of  the s t y l u s  r e c o r d i n g the  the a c t u a l r e c o r d i n g (which  extended 40 minutes beyond the end of r u n 4b)  c a u s i n g an a t t e n u a t i o n  its  as low as 1 - 2 Hz.  response which was v i s i b l e a t f r e q u e n c i e s  facts  5  T h i s s e t s an upper l i m i t of t l 8 ° a t 2 Hz f o r  i n the h i g h frequency response of  amplifiers.  the paper which i s  l e d to a more c a r e f u l c o n s i d e r a t i o n of  response of  the c h a r t r e c o r d e r .  the e f f e c t s  The wave s i g n a l s  of  in  These  the frequency  a t e a r l i e r p a r t s of  c h a r t r e c o r d were v i s u a l l y compared w i t h those a t times  l a t e r than the  two d a t a s e c t i o n s  since  (runs 4a and 4b)  frequency p o r t i o n of and a f t e r  the s p e c t r a l shapes  i n u n t i l w e l l beyond the end of r u n 4b.  A r e a l i s t i c estimate pressure is  filter,  of  stylus  Moreover  computed l a t e r f o r these runs were i n no obvious way  d i f f e r e n t at high frequencies  excess s t y l u s  the h i g h -  the s i g n a l seemed i d e n t i c a l i n appearance b o t h b e f o r e  the p i e c e s chosen i t was assumed t h a t the e f f e c t e d  p r e s s u r e d i d not s e t  low-pass  chosen f o r a n a l y s i s ;  the  of  from those f o r o t h e r r u n s . the phase e r r o r i n t r o d u c e d because  d i f f i c u l t to make.  Its  a c t i o n was t h a t of a  and i t v i s i b l y a t t e n u a t e d f r e q u e n c i e s  beyond the end of r u n 4b; i t had no apparent e f f e c t , beyond the end of r u n 4b.  T h i s means t h a t i t s  of  cutoff  of 1 Hz 40 minutes  however,  20 minutes  frequency was  78 d e c r e a s i n g w i t h time.  An e x t r a p o l a t i o n assuming the e f f e c t began a t  b e g i n n i n g of run 4a g i v e s 75 Hz f o r the c u t o f f run 4b.  If  the f i l t e r  associated  assumed to have a t r a n s f e r pass f i l t e r ,  w i t h excess s t y l u s  p r e s s u r e tan be  then the phase l a g produced by such a f i l t e r  and hence w i l l  the time a x i s by the o p e r a t o r ;  for this  is  i n t h i s work (0.05  be i g n o r e d as a source of phase  The d i g i t i z a t i o n procedure i n t r o d u c e s  reason,  great  c o v e r i n g the range of  three  interest  taken sine  was  twice s i d e by s i d e by a Calcomp  plotter  to resemble  signals  were then h a n d - d i g i t i z e d w i t h the same care and analysed i n  same way as  the d a t a .  computed from the Table  the d a t a as i t  - 5 Hz)  care was  waves of d i f f e r e n t  on a computer and p l o t t e d  than 1 °  e r r o r s through misalignments  As a f u r t h e r check a sum of  frequencies  less  error.  to i n s u r e a c c u r a t e a l i g n m e n t .  generated  the end of  f u n c t i o n s i m i l a r to t h a t of a simple RC low-  over the whole frequency range of i n t e r e s t  of  frequency at  the  appeared on the c h a r t paper.  The phase between channels  two h a n d - d i g i t i z e d t e s t s i g n a l s  from one  These the  cross-spectrum  are g i v e n below  in  5.1. TABLE  5.1  Summary of I n f o r m a t i o n from T e s t H a n d - D i g i t i z e d Data Frequency of S p e c t r a l Peak (Hz)  *Note:  Spectral Estimate at peak (dyne c m " ) Hz" 2  2  R e l a t i v e Phase between Channels  0.3  36.0  -2.0°  0.59  23.0  -2.6°  1.27  32.5  -0.8°  2.54  10.5  4.2' o  The s i g n a 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 f a c t o r used  the p r e s s u r e  i n runs 4a and 4b.  for  79 This is  considered  digitization.  It  the b e s t e s t i m a t e of indicates  that i t  is  the phase e r r o r s caused by the c o n s i d e r a b l y l e s s 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 The of  about  e r r o r s from a l l sources (18  2  + 6  + 4 )^,  2  2  checked i n the f i e l d  and on n e i t h e r  noted t h a t w h i l e  swamp t h i s 5.1.3  than those  associated  of  the r e c o r d e r has  channel were e r r o r s g r e a t e r  than  i n the waves and p r e s s u r e s i g n a l .  the e r r o r i s  should be  about equal to c a l i b r a t i o n e r r o r s f o r  the l a r g e u n c e r t a i n t i e s  e r r o r and i t  It  can s a f e l y  (*207 ) i n the p r e s s u r e o  be i g n o r e d .  The IOUBC F a s t F o u r i e r Transform Package  the I n s t i t u t e .  some i n assembler  It  is  language which i n s t r u c t s by the 8092,  store  the IBM 7044 computer to read  tapes generated  and  perform a F a s t F o u r i e r Transform (FFT) on them;  c i e n t s r e s u l t i n g from the FFT process channels  the d a t a i n the  computer memory,  the F o u r i e r  and c r o s s - s p e c t r a between s e l e c t e d p a i r s of  on a Calcomp p l o t t e r ,  t i o n of d i g i t i z e d v a l u e s  coeffi-  are used to compute s p e c t r a  I n c l u d e d i n the package are r o u t i n e s signals  students  a s e r i e s of programs, some i n FORTRAN I V , and  the  all  the  calibration  T h i s package was w r i t t e n by J . F . G a r r e t t and J . R. W i l s o n , at  been  then the expected maximum e r r o r 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  waves s i g n a l ,  decreases  c a l i b r a t i o n are caused by e r r o r s i n the g a i n of  The g a i n of both channels  This is  error  decreased.  the c h a r t r e c o r d e r .  *27o.  phase  at 2 Hz; the e r r o r  Amplitude e r r o r s i n the wave s i g n a l other w i t h the probe and i t s  digitizer.  produce a maximum expected  or about 3 2 0 °  l i n e a r l y as the frequency i s  f o r the  for  channels.  f o r p l o t t i n g the  digitized  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 -  over the range of  the A / D c o n v e r t e r f o r  all  80 channels is  and the f i r s t  f o u r moments of  run r o u t i n e l y on a l l d i g i t i z e d d a t a and i s  detecting  more s p e c i a l i s e d also  ( t h i s program  extremely u s e f u l  for  d i g i t a l e r r o r s ) , and f o r p l o t t i n g out the d i s t r i b u t i o n of  Fourier coefficient  amplitudes versus  frequency f o r a l l c h a n n e l s .  programs, not used i n the p r e s e n t  investigation,  Other are  available.  5.1.3a  FTOR:  The F a s t F o u r i e r T r a n s f o r m a t i o n Program  T h i s program c a l l s d a t a tape generated of  these d i s t r i b u t i o n s  an assembler  by the A / D c o n v e r t e r  data p o i n t s which can be read i n t o  10,240.  language program which causes to be r e a d .  The maximum number  the computer memory at a time  S i n c e the F F T subprogram PKFORT (Cooley and Tukey,  b l o c k s of  data i n i n t e g r a l powers of  the  two,  1966)  the maximum number of  is  accepts  data  13 points of  per d a t a b l o c k f o r a s i n g l e  data points  8192 or 2  ; if  up to a maximum of  ten.  f o r each channel to be analysed a group of  c a l l e d a "data b l o c k " , where j i s  set  the computer memory and that p o i n t s which i s  a m u l t i p l e of  2^ d a t a p o i n t s  by the u s e r .  hereafter  The o n l y  restrictions stored  coefficients  Each c o e f f i c i e n t contains  two.  the d a t a p o i n t s  called.  i n memory w i t h 2^  which are t h e - b u i l d i n g b l o c k s f o r the  complex spectra.  c o n s i s t s of a r e a l and an imaginary p a r t , and hence  amplitude and phase i n f o r m a t i o n p r e s e n t  frequency.  in  the d a t a b l o c k s must c o n t a i n a number of  When one d a t a b l o c k has' been s t o r e d i n memory PKFORT i s This subroutine replaces  can be  Thus the program reads  are the maximum number of p o i n t s which can be s i m u l t a n e o u s l y  Fourier  the number  per channel i s made l e s s than t h i s more channels  analysed simultaneously, in  channel i s  The h i g h e s t frequency i s  i n the s i g n a l at a g i v e n  the N y q u i s t frequency f / 2 , w h e r e g  f  s  81 is 1/T  the A/D  sampling  frequency f o r the d a t a , and  the lowest frequency i s  > where T i s the b l o c k l e n g t h i n seconds and equals  2^/f . s  These F o u r i e r c o e f f i c i e n t s are then w r i t t e n on a " c o e f f i c i e n t "  tape  a l o n g w i t h p e r t i n e n t i n f o r m a t i o n such as an i d e n t i f i c a t i o n number, b l o c k number, sampling i n from  the A/D  frequency, tape;  etc.  F o l l o w i n g t h i s the next b l o c k i s read  the sequence i s repeated f o r as many b l o c k s as  r e q u e s t e d on the i n p u t d a t a cards or u n t i l encountered  on the A/D  5.1.3b  S p e c t r a l A n a l y s i s Program SCOR  The  T h i s program reads  an e n d - o f - f i l e mark i s  tape.  the c o e f f i c i e n t  tape generated  cards p r o v i d e i n f o r m a t i o n on number of channels  by FTOR.  Data  and number of b l o c k s to  be done, b l o c k number a t which the a n a l y s i s i s to b e g i n , channels a n a l y s e d and which are to have c r o s s s p e c t r a , whether l i n e a r or mic  ( a p p r o x i m a t e l y o n e - h a l f o c t a v e ^ bandwidths are to be used,  corrections, etc. block at a  The program then reads  to be  logarithphase  i n Fourier coefficients a  time.  P r o v i s i o n i s made i n SCOR f o r smoothing the F o u r i e r c o e f f i c i e n t s b e f o r e the s p e c t r a are computed; smoothing i s c a r r i e d out a t t h i s p o i n t by a method (see, f o r example, Bingham and Tukey, 1967) to as' "hanning";  often referred  a t h r e e - p o i n t running average w i t h weights  -% i s a p p l i e d to the F o u r i e r c o e f f i c i e n t s .  -\, \,  and  T h i s i s e q u i v a l e n t to  c o n v o l v i n g the s p e c t r a w i t h a s p e c t r a l window 32TT /(3[4TT 4  where f  2  -  (2 TT f ) T ] 2  2  2  )  5.1,  i s the d u r a t i o n of the data b l o c k i n seconds and f i s frequency.  S i n c e the s p e c t r a are a l r e a d y convolved w i t h a s p e c t r a l window a s s o c i a t e d  82 w i t h the f i n i t e 6"f i s 1,  l e n g t h of  the d a t a b l o c k which v a r i e s  as  &£~ , where  the frequency i n t e r v a l from the c e n t r e frequency (see Appendix  Equation A l . l l ) ,  hanning of the c o e f f i c i e n t s  has  ducing a composite s p e c t r a l window which f a l l s  off  the e f f e c t of as & f ^ .  It  pro-  also  has the e f f e c t  of s p r e a d i n g the energy of sharp peaks i n the unhanned  s p e c t r a and of  c o n f i n i n g to the two lowest f r e q u e n c i e s  s p e c t r a any energy a s s o c i a t e d The F o u r i e r  w i t h l a r g e mean v a l u e s  coefficients  of a l l  experiment have been hanned. large f a l l o f f are r e a l , of  rates  and t h a t  (  the  smoothed  i n the d a t a .  the s p e c t r a a n a l y s e d i n  this  T h i s procedure should i n s u r e t h a t  the  S f " " ^ a p p r o x i m a t e l y ) observed i n the wave  spectra  the energy i n t r o d u c e d i n t o the low-frequency  regions  the s p e c t r a by l a r g e mean v a l u e s  c o n f i n e d to the f i r s t  or d r i f t s  i n the o r i g i n a l d a t a are  t h r e e of f o u r s p e c t r a l e s t i m a t e s .  w i l l be seen when the r e s u l t s d a t a are p r e s e n t e d  of  of  the s p e c t r a l a n a l y s i s  that the f o u r - l o w e s t - f r e q u e n c y  In f a c t of  spectral  the  it  field  estimates  are u n r e l i a b l e . If  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  + i l ^ and R 2 seconds  + i l 2 > where i = ( - 1 )  (and equals  coefficients)  l / A f where  2  and f  Af is  =T(R +  f  the b l o c k l e n g t h  in  the bandwidth between F o u r i e r  R  i )  2  2  2  $22  =T(  ( f )  r 2  +  + I2 1 1 2 Af 2 1  if) 2  2 the cross  equals  then the power s p e c t r a are g i v e n by  <J>n< )  and  f o r two channels are R-^  5.2;  Af  s p e c t r a are  C o  12(  f )  =  T ( 1 2.+ 2 R  R  I  1 2) I  R  1 2 + 1 2 2 Af R  I  I  5.3.  83  Qu (f) 1 2  X  =  ( 2 1 R  "  X  R  i 2 I  =  )  ( 2 1 R  X  2 The f a c t o r of  I  5.3.  )  2 Af  frequencies  Phase c o r r e c t i o n s are made a t t h i s / Qu(f) A , /  correcting this  I Co(f)  -  R  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  = tan"-*  " 2 1  equal to the s i g n a l v a r i a n c e .  p o i n t by c a l c u l a t i n g the phase angle f o r i n s t r u m e n t a l  0(f)  responses,  and then r e c a l c u l a t i n g Co and Qu from Co(f) 0  -  -Vc  2  = V C o ( f ) + Qu (f)  cosO  e  , 5.4  and  where B  c  Qu(f) = V c o ( f ) 2  is  + Qu (f) 2  sin8  t  ,  the c o r r e c t e d phase and the u n c o r r e c t e d e s t i m a t e s are under  the square r o o t After a l l  sign. the r e q u i r e d power and cross  spectra for a given block  and o t h e r r e q u i r e d i n f o r m a t i o n on the b l o c k s t a t i s t i c s have been the sequence i s  stored,  repeated u n t i l a l l the b l o c k s asked f o r have been p r o -  cessed or a n i i e n d - o f - f i l e mark i s  encountered on the c o e f f i c i e n t  tape.  Then a t each frequency the program averages  the s p e c t r a l estimates over  the number of b l o c k s processes and computes  the standard e r r o r of  mean and the average mean i s  trend over the b l o c k s .  The standard e r r o r of  each the  computed from  I  N " « N-l  ( S  I for N r e a l i z a t i o n s  (blocks)  of  ' *  5.5  the q u a n t i t y  A l s o computed f o r  each  frequency are _ [ Co (f) = " L $11 2  Coherence  +,,Qu (f)1 /ft" ' ($22 1 2  N  2  5.6  84 Phase  and The  Qu(f)/Co(f)]  program a l s o computes  (the means of is  =  the  5.1.4  the v a r i a n c e of  i n d i v i d u a l blocks)  then p r i n t e d out i n t a b u l a r  5.7.  and t h e i r  the zero trend.  harmonics This information  form.  S p e c i a l Programming The  programming which was designed  can c o n v e n i e n t l y  be grouped i n t o  two s e c t o r s :  computation of F o u r i e r c o e f f i c i e n t s spectra The  specifically that  for  this  project  concerned w i t h  (FTOR), and that  concerned w i t h  (SCOR). first  sector is  c o n d i t i o n i n g of  by f a r the l a r g e s t ;  the s i g n a l .  it  instead  d e s c r i b e what the programs d i d to the s i g n a l s . d a t a were brought i n t o  the  performed a l l  In the d e s c r i p t i o n of  computational d e t a i l s are not d i s c u s s e d ;  the d e s c r i p t i o n  an attempt  the  that  digital  the programs  Since  computer i n b l o c k s ,  done on one b l o c k a t a time;  the  c o n d i t i o n i n g was follows  is  of  sensors--wave h e i g h t ,  and s o n i c anemometer--will  described.  what  s i g n a l s from three be  The s u b r o u t i n e i n which t h i s c o n d i t i o n i n g was c a r r i e d out  named FIDDLE.  5.1.4a  The Wave H e i g h t S i g n a l  The wave s i g n a l on t r a n s f e r  from FTOR c o n s i s t e d  tape r e c o r d e r output v o l t a g e about some mean v o l t a g e . a p p l i e d to i t It  to  digitized  C o n d i t i o n i n g of  buoy p r e s s u r e ,  actual  i s made  was done to a s i n g l e b l o c k of d a t a .  is  the  of v a r i a t i o n s  in  In FIDDLE i t had  the n o n l i n e a r wave probe c a l i b r a t i o n g i v e n i n E q u a t i o n 4 . 8 .  emerged from t h i s  c a l i b r a t i o n as d e v i a t i o n s  i n c e n t i m e t e r s about  the  known immersion depth o f the wave probe.  Two separate wave s i g n a l s  were r e t u r n e d from FIDDLE to FTOR; the f i r s t was the wave s i g n a l as d e s c r i b e d above and the second was the same s i g n a l m u l t i p l i e d by a "Spike F u n c t i o n " which was one except d u r i n g the times  that spikes  o c c u r r e d i n the p r e s s u r e s i g n a l , i n which case i t was s e t to zero..The reasons  f o r a n a l y s i n g these two wave s i g n a l s a r e o u t l i n e d i n  Appendix 1.  5.1.4b  The P r e s s u r e S i g n a l  In  FTOR, a c a l i b r a t i o n f a c t o r which i n c l u d e s a l l analog g a i n s ,  i n v e r s i o n s and instrument amplitude  c a l i b r a t i o n s was a p p l i e d to the  p r e s s u r e s i g n a l which was then s t o r e d as v a r i a t i o n s i n p r e s s u r e i n u n i t s of dyne cm  .  FTOR generated  i f r e q u i r e d an e x t r a channel  S j f o r Spike F u n c t i o n , a l l the data p o i n t s o f which were a t t h i s set  called stage  equal to 1 . 0 . The p r e s s u r e s i g n a l was next  w i t h the S channel.  t r a n s f e r r e d from FTOR to FIDDLE along  In FIDDLE the f i r s t  o p e r a t i o n performed  on the  p r e s s u r e s i g n a l was to add to each o f i t s data p o i n t s an amount R-f 3**,i a  where R i s a f a c t o r which c o u l d be p r e s e t to any number i n c l u d i n g zero, is  p  i s a i r d e n s i t y , g i s the a c c e l e r a t i o n due to g r a v i t y , and  \  the v a l u e of the wave s i g n a l a t the time of zero, one or two samples  i n advance of the p r e s s u r e s i g n a l . preceded  The number of samples by which 7£  the p r e s s u r e s i g n a l c o u l d be p r e s e t .  The reason f o r choosing  an e a r l i e r wave s i g n a l i s that i n the a c t u a l measurements the wave probe was downwind o f the p r e s s u r e sensor;  the advance i s an a p p r o x i -  mation to the phase c o r r e c t i o n r e q u i r e d by the s p a t i a l s e p a r a t i o n of the sensors.  T h i s c o r r e c t i o n i s d i s c u s s e d i n some d e t a i l i n "Data  86 A n a l y s i s and I n t e r p r e t a t i o n " , The p r e s s u r e s i g n a l ,  if  on p. 102  ff.  i t had s p i k e s  in i t ,  was next t r a n s f e r r e d  from FIDDLE to the s p i k e removal s u b r o u t i n e SPKSKP; ones was a l s o t r a n s f e r r e d a t t h i s SPKSKP are somewhat Appendix 1, spectra is  the p r e s s u r e s i g n a l ;  the s i g n a l drops below the  time,  deferred pressure  of  spikes  on the b a s i s of r a t e of  and number of p o i n t s  threshold.  f r a c t i o n of  to be skipped  SPKSKP r e p l a c e d the  S channel by zero d u r i n g s p i k e s . the data p o i n t s  SPKSKP caused the " d e f e c t i v e "  b l o c k to be i g n o r e d i n  see Appendix 1,  the s p i k e s  spikes deviations  average.  The Sonic Anemometer  Signals  The s o n i c anemometer produced three s i g n a l s : and two h o r i z o n t a l v e l o c i t i e s  gonometrically velocity  in  p. 178 f f .  removed was averaged and the F o u r i e r t r a n s f o r m performed on  (w)  the  on the  F o l l o w i n g p r o c e s s i n g by SPKSKP the p r e s s u r e s i g n a l w i t h  5.1.4c  If  i n a b l o c k were r e p l a c e d  and was used to i n v e s t i g a t e the e f f e c t of  pressure spectra:  after  pressure  of " c l e a r " b l o c k s , which may however be s e p a r a t e d w i d e l y  from t h i s  to  i n f o r m a t i o n was p r e v i o u s l y read i n t o a  and .the ones i n the  more than a p r e s e t  analysis  of  discussed.  d a t a card on t h r e s h o l d v a l u e s  by z e r o s ,  is  i n which the e f f e c t s of s p i k e removal on the also  data points  The a c t u a l mechanics  complicated and t h e i r d i s c u s s i o n  SPKSKP d e t e c t e d the presence change of  time.  the S channel of  to u and v ,  in directions  (A and B ) .  the f l u c t u a t i n g  vertical  velocity  These are r e l a t e d t r i -  components  of  the wind  a l o n g and at r i g h t angles to U, the mean wind  87 v e c t o r f o r the r u n (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  S p e c i a l Programming i n SCOR  The  c r o s s - s p e c t r a between the p r e s s u r e and the wave e l e v a t i o n s  were  used as d e s c r i b e d i n the f o l l o w i n g paragraphs to compute the mean f l u x of  energy  ations this  E  and of momentum  i n pressure.  It  is  from the a i r to the waves v i a f l u c t u -  these c a l c u l a t i o n s which were s p e c i a l  program, and the m o d i f i c a t i o n of  to  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  g i v e n by the  covariance ~~T~  E  =  SCORF f i r s t under t h i s  p ( , t ; a ^ u ; / -a t  5.8.  computed the spectrum of. energy f l u x E ( f ) to get E .  Because  and then  integrated  the p r e s s u r e and wave s i g n a l s were  almost e n t i r e l y of n o i s e above 3 Hz (see i n t e g r a t i o n was t r u n c a t e d at t h i s  p.104 of  frequency.  this  section)  composed  the  The computation done i n  SCORF was 3.0 <E>  3.0  = Y~ ( i):Af, f = 105 E  i  f  = X " 2lt f .05  where the lower frequency l i m i t of 0.05 therelis  one complete  spectral estimates,  Hz i s  c y c l e i n a 1024-point  i n t e g r a t e d energy f l u x f o r one b l o c k , and Q u p j " ^ (^±)  i Q u  '  i e  ) Af.  5.9,  the frequency f o r which is  the bandwidth of  the the  quadrature spectrum of  s p i k e - c o n t a m i n a t e d p r e s s u r e and wave s i g n a l s . E q u a t i o n 5.9 were computed s e p a r a t e l y  (f  data b l o c k ; < E }  A f ^ is t  ,  The q u a n t i t i e s  f o r each data b l o c k .  the  shown i n  The mean  88 energy f l u x E over the number of b l o c k s i n the run and the standard e r r o r i n t h i s mean were then computed. The  spectrum a t a frequency f^ of  to the waves i s  g i v e n by  *^w i ( f  where  )  = g/2TTfi  frequency f^.  -=( i)/ i  =  is  f  =  w a  the i n t e g r a l under E ( f ^ ) . over  5.10,  i  i/si^  w  w  2fff.E(f )/g  =  the phase v e l o c i t y  the i n t e g r a l under * £ * ( f i )  ^'"J** ^  C  of  the wave component of  SCORF computed, f o r each data b l o c k ,  < T w )  of  the momentum f l u x from the wind  4TT f iQu ^ 2  p  s  truncated at  As f o r E ( f ) ,  the b l o c k s ,  2  s  (f )  A f i  ±  5.11;  the same f r e q u e n c i e s  SCORF computed  t  as  , the mean  and the standard e r r o r i n t h i s mean f o r  the  run.  5.2  I n t e r p r e t a t i o n of This section  the  contains  on the s p e c t r a p r e s e n t e d comings,  Spectra f a i r l y detailed  i n "Results"  and a l s o g i v e s b r i e f  discussions  of  of v a r i o u s i n s t r u m e n t a l  accounts  of  effects short-  the f i l t e r i n g p r o c e s s e s p e r -  formed on the p r e s s u r e and wave d a t a p r i o r to and d u r i n g analysis.  the  spectral  The purpose of  the s e c t i o n  is  may i n t r o d u c e s i g n i f i c a n t  distortions  i n the s p e c t r a and g i v e wherever  possible  a quantitative  e s t i m a t e of  to d e s c r i b e processes which  their effect,  may be i n t e r p r e t e d  objectively.  5.2.1  Introduced by the Sensors  Distortions  5.2.1a  The P i t c h i n g A c t i o n of  The f o r c e d response  of  so t h a t  these  spectra  Themselves  the Buoy  the buoy (which i s hinged so t h a t i t  can  89 respond to s t e e p , water s u r f a c e i s analytically.  s h o r t waves"--see F i g u r e 13)  a complex motion which i s not e a s i l y  O b s e r v a t i o n s of  the movements of  w i t h the a i d of slow-motion movies, changes  "pitch'angle") of  water s u r f a c e  a t 2 T 0.2  ( t h i s angle i s  Hz.  It  pressure s i g n a l  is  hinge.  described  the buoy have been made  the buoy makes w i t h r e s p e c t  the  r e f e r r e d to h e r e a f t e r as  the  of  A resonance i n t h i s  thought that the l a r g e s t  the f r o n t s e c t i o n  (bow)  p i t c h i n g mode occurs  d i s t o r t i o n s of  (and to some e x t e n t the wave s i g n a l )  d i r e c t l y a t t r i b u t e d to the motion of  the  to  are a s s o c i a t e d w i t h o s c i l l a t i o n s  the buoy about i t s  the  and these i n d i c a t e t h a t most of  i n the angle t h a t the bow of  instantaneous  to the motions of  the  which can be  the buoy are the r e s u l t of  these  oscillations. The p r e s s u r e s i g n a l i s efficiency  of  a f f e c t e d by the o s c i l l a t i o n s  the  the dynamic p r e s s u r e r e j e c t i o n arrangements v a r i e s w i t h  the p i t c h angle of  the buoy (see  f i g u r e 19);  t i l t e d up) produce s p u r i o u s n e g a t i v e the measurement l o c a t i o n , of  since  p o s i t i v e p i t c h angles  dynamic p r e s s u r e s  (bow  to appear at  so t h a t p i t c h angle and p r e s s u r e are 1 8 0 ° out  phase. Because of  the resonance at 2 Hz the e f f e c t of  the p i t c h i n g of  the  buoy v a r i e s w i t h f r e q u e n c y .  A t resonance  w i t h the waves;  the s p u r i o u s p r e s s u r e s produced s h o u l d be  consequently  i n a n t i p h a s e w i t h the waves and w i l l  the p i t c h i n g w i l l be i n phase  increase  the magnitude of  the p, ^  cospectrum as w e l l as p r o d u c i n g a "bump" i n the p r e s s u r e power spectrum. The e f f e c t  on the wave spectrum should be n e g l i g i b l e .  A t lower f r e q u e n c i e s , expected  the p i t c h f l u c t u a t i o n s  to l a g the waves by 9 0 ° .  This  of the buoy can be  case has been mentioned  90 ("Experiment"; p. 70); such f l u c t u a t i o n s important e f f e c t of  the expected  a r e shown i n F i g u r e 20. is  cause a f a l s e observed  the fundamental frequency of the waves.  at 2 f  spectrum.  can be seen that  , which i s  twice  the frequency of  signal w i l l  l e a d the Stokes'  apparent s h i f t  harmonic and i t s  It w i l l  be  the peak of the wave  T  and is  the s p u r i o u s  pressure  e f f e c t s w i l l be an  i n the observed phase towards - 1 8 0 ° and a decrease (f)  spectra at 2 f  w It  correg  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 E ( f )  the most  coherent w i t h the f i r s t harmonic of the waves.  This spurious pressure s i g n a l w i l l  largest  It  generated by  the g e n e r a t i o n of a l a r g e s p u r i o u s f i r s t harmonic  the p r e s s u r e s i g n a l  l a t i o n a t twice  quadrature p r e s s u r e s  in  . p  possible  to estimate the s i z e of t h i s  effect.  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',' r u n .  For f  P J"s 0.6 H z , kp cs at f  P  1.4  P x 10  -2 -1 i s 40 cm Hz .  If  2  cm  and the wave power spectrum  Cj)^(f)  these waves occupy a bandwidth of 0.1 Hz  they are e q u i v a l e n t to a s i n g l e s i n u s o i d a l wave a t 0.6 Hz of amplitude ^ 2 2 (2 x 40 x 0 . 1 ) ^ 3 cm; t h e r e f o r e ka / 2 ^ 6 x 10 cm. The p r e d i c t i o n 2  f o r the c o n t r i b u t i o n to -2 (6 x 10  cm)  amplitude of of  0 ^ ( 1 . 2 Hz) of the Stokes'  2  -2 /  (2 x 0.1 Hz) s  2 x 10  2 cm  harmonic i s  Hz  The maximum expected  the p r e s s u r e harmonics generated by the p i t c h  the buoy i s  about 2.5 dyne cm ; -2  cm amplitude wave) of t h a t about one h a l f of  2  1 dyne cm this  is  is  if  therefore  -1  fluctuations  a^pressure^'amplitude  assumed and i f  i t is  (for a 3  f u r t h e r assumed  i n quadrature w i t h the Stokes'  first  harmonic of  the waves,  spectrum i n a 0.1  then the contribution to the  Hz bandwidth caused by t h i s e f f e c t  Qu__ (f)  =  { (2 x l O >• \  X  =s  0.2  2  cm^ x 1 Hz 2  dyneA cm /  1  it  the e f f e c t of  expected that  Hz i n r u n 2a i s  at 2 f p .  T h i s may show up i n the Q u p ^ ( f ) near 2 f ;  magnitude of and£" (f) w  ^  as a s l i g h t  port.  the  reduction i n  Since  the  reduced by the e f f e c t ,  the  c o n t a m i n a t i o n of  as  E(f)  and t h i s should be  the buoy.  the s p e c t r a i s  The wavelength  of  a t 2 Hz and 1 a t 3.7  Hz.  caused by  the h i g h - f r e q u e n c y waves  the buoy a t f r e q u e n c i e s  Hz the r a t i o of buoy r a d i u s to wavelength As a r e s u l t ,  and so the wind speed at the bow of  c a n t l y from t h a t a t  ;  pitching  is  0.1;  it  above 1 Hz. reaches  the s c a l e s of v a r i a t i o n s  the wind speed a s s o c i a t e d w i t h the wave motion are comparable w i t h buoy,  - 1  the Buoy  becomes comparable w i t h the s i z e of  0.33  - 1  spectra.  The second source of  At 1.2  cm  t h i s c o n t a m i n a t i o n of  s p e c t r a would a l s o be reduced a t 2 f p ,  s i z e of  3 dyne  i n the p r e s s u r e power s p e c t r a .  The F i n i t e S i z e of  the f i n i t e  the  the quadrature s p e c t r a i s  looked f o r i n these  5.2.1b  p  . l H z )  as mentioned e a r l i e r i t may a l s o appear  p  "hump" a t 2 f  0  t o t a l p,->£ quadrature spectrum  the buoy may be 5 - 10% of  a slight  x  to the Stokes harmonics by the  a c t i o n of  t h e i r magnitudes  2  1  a t 1.2  energy and momentum f l u x e s  (  g i v e n by  dyne cm" H z " .  p  therefore  /  is  quadrature  2  The observed s i z e of Q u ^ ( f ) is  p,?£  the p r e s s u r e  T h i s means t h a t  the buoy may d i f f e r  the s t a g n a t i o n  pressure  the  signifi-  p o r t and at the r i n g downwind of  the f r a c t i o n of  in  the  appearing  at  the p o r t due to the a i r flow over the bow may not be e x a c t l y  by the e f f e c t of If  the r i n g  (see  the f l u c t u a t i o n s  assumed  "Experiment", p.  i n wind v e l o c i t y  to be a p p r o x i m a t e l y those of  particles  at the s u r f a c e of  wave p r o f i l e  -  ^ ?a '  Some f r a c t i o n F of p suction pressure; x direction,this presence varies  of  2 u  o  k  the waves then t h e i r amplitude i s k a c ,  where  and c wave phase speed.  the s t a g n a t i o n  c  s  i  pressure v a r i e s  5.12.  k x  the p r e s s u r e p o r t of  the buoy as a  is  c a n c e l l e d by a p o s i t i v e  p r e s s u r e Fp^ due to  the buoy and i f  If  a wind i s  the p o r t i s  ~  p  -  F  where  ?a. ^o kac ( s i n k (x-d)  F £  a  U  Q  kac ( 6  2  = cos kd - cos ke , and  C  = tan'  the  - F p  sinc(kx-f)  2  ) 5.13,  5.14.  (6/T)  1  for s u f f i c i e n t l y =  - s i n k (x + e)  + V )%  Y  p  then  the  r i n g + Pbow  = s i n kd + s i n ke  P  the  g i v e n a p p r o x i m a t e l y by  h  That g i v e s ,  the  p r e s e n t which  d and e are r e s p e c t i v e l y  d i s t a n c e s from the r i n g and the bow to the p r e s s u r e p o r t ,  =  the  i n an a i r flow over the buoy,which i s u n i f o r m i n  as s i n kx over  Pp  along  ( )  the r i n g downwind from the p o r t .  pressure at  There-  a c c o r d i n g to  n  appears a t  s  w i t h the waves are water  Q  a  associated  the  ( i n the x d i r e c t i o n )  Ps  66).  the o r b i t a l v e l o c i t i e s of  k i s wave number, a i s wave a m p l i t u d e , f o r e i n a mean wind speed U  cancelled  a  U  Q  small values k  3  ac (d + e )  of k, 2  d , and e  cos kx  5.15,  93 i n d i c a t i n g t h a t the s p u r i o u s p r e s s u r e s  caused by the buoys f a i l u r e  to  c o m p l e t e l y c a n c e l out the s t a g n a t i o n p r e s s u r e s w i l l be i n quadrature w i t h the waves,  l a g g i n g them by 9 0 ° .  Run 2a i s  chosen to e s t i m a t e the s i z e of the e f f e c t .  w i l l be made a t two f r e q u e n c i e s : F  ~  0.1  d  2s 2 cm, and  e  — 7.5 cm.  For the r u n , D s Q  (see  3 x 10  0.6 Hz, and 2 Hz.  "Experiment", F i g u r e  2  cm sec  The e s t i m a t e  F o r the buoy  16),  -1  The wave s l o p e ka w i l l be taken to be ka  21  c  = g / w - 2.6 x 10  k  = co / g cs 1.45  0.1;  2  Taking d + e a 10 cm g i v e s , p If  this  is  = -2.0  p  x 10"  2  cm sec  x 10"  , and  cm .  2  - 1  from E q u a t i o n 5.15, dyne c m . - 2  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 e s t i m a t e f o r Pp g i v e s  (t>  Pp (0.6)  =  (2 x 1 0 " )  =i  -2 x 1 0 '  2  3  2  /  (2 x  0.1)  (dyne c m " ) H z " . 2  2  1  To f i n d the observed quadrature p r e s s u r e s p e c t r a l estimate coherent w i t h the waves,  the square of the ppj quadrature spectrum must be d i v i d e d by  the wave s p e c t r a l e s t i m a t e a t 0.6 Hz: 0p  Q  (0.6)  =  Qu .,,  =  (22)2/40  =s  12 (dyne c m " ) H z " .  2  (0.6)  / 0,(0.6)  2  2  1  5.16  Since (J)p of  p  (0.6)  / (J) P  Q  (0.6) <  the buoy can be n e g l e c t e d  2 x 10  - 4  the e f f e c t o f the f i n i t e  size  a t 0.6 H z .  A t 2 Hz n e i t h e r kd nor ke can be assumed s m a l l , and so E q u a t i o n  5.13 must be u s e d . 2s 0.063 cm Hz  A t 2 Hz, Q(ip^(2) 2* -.04 dyne cm"--Hz" , and 1  ; u s i n g E q u a t i o n 5.16 t h i s  Cj)p  o  (2) 2s  -2.5 x 1 0  E q u a t i o n 5.13 g i v e s f o r 0 p  (J)p  p  Thus (J)p  p  p  - 2  (J)^(2)  gives  (dyne c m  - 2  )  2  Hz" . 1  (2)  (2) 2s -6.5 x 1 0  (dyne c m  - 3  ( f ) i s about 307o of (J)p  - 2  )  2  Hz" . 1  (f) a t 2 H z , which i s the upper  Q  frequency l i m i t beyond which e r r o r s i n the measurement o f the phase between p r e s s u r e and waves become  large.  T h e r e f o r e the p r o b a b l e d i s t o r t i o n s of  the buoy a r e s m a l l a t f  spectrum);  i n t r o d u c e d by the f i n i t e  (the frequency a t the peak o f the wave  they i n c r e a s e w i t h frequency and may add as much as 307c, to  the magnitude o f the p,i£ quadrature spectrum by 2 Hz. the waves and hence  the p r e s s u r e s  the p r e s s u r e power s p e c t r a . the same percentage  5.2.1c  t h a t the e f f e c t w i l l  The E andZ"*  as Q u p ^ ( f ) , s i n c e  tend to be too l a r g e a t h i g h  Since,  coherent w i t h them a t t h i s  are q u i t e s m a l l , i t i s n o t expected  will  size  w  however, frequency  show up i n  s p e c t r a w i l l be a f f e c t e d by  they a r e d e r i v e d from i t ;  they  frequencies.  Backscattering  The l a s t  source of d i s t o r t i o n i n the s p e c t r a to be d e a l t w i t h i s  b a c k s c a t t e r i n g o f waves from the mast which supports the wave probe, which i s 10 cm i n d i a m e t e r .  The b a c k s c a t t e r e d waves t r a v e l a g a i n s t the  wind and are hence damped by i t . i n c i d e n t wave i s  The p r e s s u r e d i s t r i b u t i o n over  i n f l u e n c e d by that over the r e f l e c t e d wave;  r e s u l t i n g s p u r i o u s phase s h i f t  the  the  i n the p r e s s u r e can be i n the d i r e c t i o n  of e i t h e r damping or added g e n e r a t i o n depending on the r e l a t i v e phase of  (towards  - 1 8 0 ° or towards - 9 0 ° )  the i n c i d e n t and r e f l e c t e d  T h i s r e l a t i v e phase depends on the l e n g t h  \  of  waves.  the waves b e i n g  scattered,  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  4TT  =  5.17.  g  2TT X  2  The mast which supports  the wave probe i s  the b a c k s c a t t e r i n g from such an o b j e c t flow  theory (Havelock,1940).  can be c a l c u l a t e d from p o t e n t i a l  Assuming t h a t  waves are the o n l y waves p r e s e n t  the i n c i d e n t and  where x i s is  =  A  e  i ( « t  scattered  and t h a t the i n c i d e n t wave i s  wave p r o c e e d i n g i n the - x d i r e c t i o n w i t h the (|).  a v e r t i c a l cylinder;  + kx) kz e  fa  =  e  a plane  potential icot kz e  5  >  1  8  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  two-dimensional  (uniform i n the t h i r d  direction).  The p o t e n t i a l f o r 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 =  e  i  + w  t  e  ^ r  k  In p o l a r c y l i n d r i c a l sum of B e s s e l of  functions  the second k i n d .  (  z  4>i +  4>r)  coordinates  and  § r  The o r i g i n of  a  s  5.19.  (J)i a  n  can be expanded as an i n f i n i t e  infinite  sum of Hankel  the c o o r d i n a t e system i s  functions  taken to be  96 a t the geometric c e n t r e of the s c a t t e r e r . c o n d i t i o n that the v e l o c i t i e s results  I n t r o d u c i n g the boundary  normal to the s c a t t e r e r must v a n i s h  i n an e x p r e s s i o n f o r the r a t i o  (J>i/  (J)  r  in mentioned sums of B e s s e l and Hankel f u n c t i o n s .  terms of the above-  The wave e l e v a t i o n ">£ i s  g i v e n (to the f i r s t o r d e r ) by z = o =  therefore  iw  V. r / V i  ( d>i +  =  <j>r)e  l w t  e  k z  4>r/ <J>i  5.20.  A computer program w r i t t e n by J . F . G a r r e t t , been used to compute of  4>i and <t>r  a f e l l o w s t u d e n t , has  i n terms of the above-mentioned sums  B e s s e l and Hankel f u n c t i o n s f o r v a r i o u s v a l u e s of k , f o r the p a r t i -  c u l a r r a d i u s of the mast used and the ( f i x e d ) the wave probe.  d i s t a n c e from the mast to  The G a r r e t t program t r u n c a t e s  the sums of B e s s e l  f u n c t i o n s a t a p o i n t where the t r u n c a t i o n e r r o r becomes n e g l i g i b l e .  <4>i>  computes three q u a n t i t i e s : The  , <<]>r>  ,  and<4>r>  <4>i)  /  It .  q u a n t i t y <\CL) i s <OL>  =  Max|ae  i  r  |  for0 4 T £ 2 T r  thus the ^ y denote "maximum v a l u e over a f u l l  ;  cycle". <4>i> and  ^4^r^ of  a r e complex and v a r y w i t h the wave number k; the r e l a t i v e  the i n c i d e n t and r e f l e c t e d waves can be determined from them.  phase The  t o t a l amplitude of r e f l e c t e d p l u s b a c k s c a t t e r e d waves can a l s o be computed and compared w i t h the amplitude of the i n c i d e n t wave.  I t must  however be noted t h a t such c a l c u l a t i o n s assume t h a t the i n c i d e n t and  b a c k s c a t t e r e d waves are c o h e r e n t . communication)  I t has been found by G a r r e t t  that at the Spanish Banks s i t e  waves become e s s e n t i a l l y  i n a s o u t h e a s t wind the  incoherent at frequencies  a t downwind wave probe s e p a r a t i o n s  greater  of 2 m (2 m i s  from the wave probe to the s u p p o r t i n g m a s t ) . r e f l e c t e d waves are c o h e r e n t ,  their total  (personal  If  twice  than 1 Hz  the  distance  the i n c i d e n t and  ( i n c i d e n t plus  reflected)  energy must be found by adding them v e c t o r i a l l y b e f o r e s q u a r i n g ; i f are i n c o h e r e n t t h e i r t o t a l energy i s  g i v e n by the sum of  they  the squares of  t h e i r amplitudes. The r e l a t i v e phase of  the i n c i d e n t and r e f l e c t e d waves has  c a l c u l a t e d a t a number of f r e q u e n c i e s ;  the r a t i o of  total  to  been  incident  energy has a l s o been c a l c u l a t e d , on the assumption t h a t i n c i d e n t and r e f l e c t e d waves are coherent a t f r e q u e n c i e s  below 1 Hz and i n c o h e r e n t  above 1 Hz. The energy r a t i o f o r coherent waves R  is  -  C  (<$i>  <$r>) /(<<t>i>) 2  +  displayed for frequencies  from 0.35  to 1.7  2 5  Hz i n T a b l e 5.2;  .  2  1  the  c o r r e s p o n d i n g r a t i o f o r i n c o h e r e n t waves  R =(|<<t>i')| ic  is  2  + |<*r>| ) /  |<<Dl>|  2  d i s p l a y e d i n the same t a b l e f o r f r e q u e n c i e s  From the t a b l e i t  is  2  from 0.85  5.22 to 2.0 H z .  c l e a r t h a t a t f r e q u e n c i e s near 1 Hz s p u r i o u s wave  energy amounting to as much as 157, of p r e s e n t i n the wave spectrum.  the i n c i d e n t wave energy may be  The e f f e c t of  the b a c k s c a t t e r i n g on the  p r e s s u r e power spectrum cannot be e s t i m a t e d without a knowledge of p r e s s u r e d i s t r i b u t i o n over the r e f l e c t e d waves,  the  which are b e i n g damped.  TABLE Expected E f f e c t s  f  R  5.2  of B a c k s c a t t e r i n g on Wave Power Spectrum  100(R -1)  c  C  (Hz)  v  100(R -l)  ic  i c  (%)  (%)  Notes:  (2)  (1)  0.35  .998  0.2  0.5  .982  2.0  0.85  1.06  6.0  1.00  0  1.0  $.91  -9.0  1.00  0  1.2  .872  •14.0  1.01  1  1.4  .923  -8.0  1.02  2  40.0  1.03  3  1.03  3  1.7  1.4  2.0  Notes:  1.  R  is  c  total  computed from E q u a t i o n 5.21;  to i n c i d e n t energy f o r coherent  reflected 2.  R^  c  is  it  the r a t i o of  i n c i d e n t and  waves.  computed from E q u a t i o n 5.22;  ponding energy r a t i o f o r i n c o h e r e n t reflecting  is  waves.  it  is  the  corres-  i n c i d e n t and  A q u a l i t a t i v e e s t i m a t e of  the e f f e c t of  the b a c k s c a t t e r i n g on the  spectrum between p r e s s u r e and waves can be made. = 2TT/X  of  the waves changes by 1 m~\  i n c i d e n t and r e f l e c t e d waves changes  is  As the wave number k  the r e l a t i v e phase  1 radian ( 5 7 . 3 ° ) .  phase a t any frequency can thus be found i f  8 i r of  frequency  The r e l a t i v e phase of  i n c i d e n t and r e f l e c t e d waves was c a l c u l a t e d and p l o t t e d versus number to f i n d 270°;  using  d i s p e r s i o n r e l a t i o n f o r water waves, g i v e n by E q u a t i o n 5.17. e f f e c t on C o ^ ( f ) , p  (f)/Co  (f)  Q u ^ ^ ( f ) , and Phase  ) of a damped r e f l e c t e d wave i s  the f o u r phase angles mentioned above, the r e f l e c t e d wave leads  i t by 9 0 ° .  are d i s p l a y e d i n Table 5.3 frequencies Qu and effect Co,  of  Qu and  dp^(f)  tan'^CQUp^  determined a t each of  assuming t h a t the p r e s s u r e  The r e s u l t s  the phases <9 i r o c c u r .  the b a c k s c a t t e r i n g , and a  i n c r e a s e d by the (thus,  the magnitudes of  © (f)  columns where " - " s i g n s o c c u r ) .  A zero means "not a f f e c t e d " .  above 1 Hz, s i n c e  assumed to be e s s e n t i a l l y 5.2.2  moved towards - 1 8 0 ° i n the  assumed to be unimportant a t  at this  since  Co and Qu  are i n c r e a s e d and the angle of  backscattering effect is  the  A "+" i n the Co,  means a decrease  are commonly n e g a t i v e ,  over  o f the d e t e r m i n a t i o n  columns means the r e l e v a n t spectrum i s  0(f)  =  the  The q u a l i  i n c o n j u n c t i o n w i t h a t a b u l a t i o n of  a t which each of  Q (f)  wave  the wavenumbers where the phase was 0 ° , 9 0 ° , 1 8 0 ° and  these wavenumbers were then converted to f r e q u e n c i e s  tative  the  The r e l a t i v e  the phase a t one  computed from the b a c k s c a t t e r i n g f o r m u l a .  cross  The  frequencies  frequency i n c i d e n t and r e f l e c t e d waves are incoherent.  D i s t o r t i o n s I n t r o d u c e d d u r i n g P r e p a r a t i o n and A n a l y s i s of Data  5.2.2a  F i l t e r i n g of  the Wave S i g n a l  The wave s i g n a l i s  f i l t e r e d i n three ways.  First,  it  is  filtered  TABLE ected E f f e c t s  5.3  of B a c k s c a t t e r i n g on P r e s s u r e -  Waves Cross Spectrum Sir  Co(f)  (Degrees)  A'Vin is  (dyne cm" •'"Hz ) -1  6(f)  (dyne cm"•'"Hz" ) (Degrees) 1  (1)  (1)  (2)  90  +  0  +  180  0  +  270  -  0  0  0  -  +  90  +  0  +  180  0  270  -  0  0  90  +  .  + 0 + 0  the Co or Qu columns i n d i c a t e s  added to  A + i n the  the r e l e v a n t 8  (f)  column i n d i c a t e s  backscattering.  +  that p o s i t i v e  spectrum by the  between p r e s s u r e and waves i s the  Qu(f)  that  energy  backscattering. the phase  angle  i n c r e a s e d by the e f f e c t  of  99 by the buoy i t s e l f  which r i d e s v e r t i c a l l y on the wave probe.  form of d i s t o r t i o n i n t r o d u c e d i s becomes s i g n i f i c a n t l e n g t h of of  not known; i t  above the frequency  the waves i s  (about  is  felt  that i t  the buoy.  1 mm i n observed wave a m p l i t u d e . the wave s i g n a l f o r f r e q u e n c i e s  Typically  earlier,  Second,  above a low-frequency  prior  c e r t a i n l y be n e g l e c t e d  the wave s i g n a l  to d i g i t i z a t i o n ;  is  sample p o i n t s .  The l a t t e r f i l t e r , i s  The wave power spectrum i s  less  as mentioned is  applied the  is  differentiated  f o r the sake of  thus  digitally;  consistency.  the c o n v o l u t i o n of of  it  the o r i g i n a l  the above three  filters,  the r e s u l t i n g unsmoothed spectrum w i t h the hanning window as  g i v e n by E q u a t i o n 5 . 1 ,  5.2.2b  at frequencies  three p o i n t r u n n i n g mean of  wave spectrum w i t h the t r a n s f e r f u n c t i o n s and of  2 Hz  designed p r i n c i p a l l y to reduce h i g h  frequency n o i s e when the wave s i g n a l a p p l i e d to a l l the waves s i g n a l s  of  the b e a r i n g on  in addition a d i g i t a l f i l t e r  by t a k i n g an unweighted  is  of  cutoff  analog f i l t e r e d ,  during analysis  to exceed  the s t a n d a r d d e v i a t i o n of  exceeds 1 mm, so the e r r o r caused by the presence  than 2 H z .  Because  of the b e a r i n g , e r r o r s caused by s u r g i n g of  the water between the b e a r i n g and the probe are not expected  the wave probe can almost  only  2 Hz) where the wave-  comparable w i t h the diameter of  the p h y s i c a l dimensions  The exact  The P r e s s u r e  p.81.  Signals  S i n c e the p r e s s u r e  signals  by E q u a t i o n A l . 2, Appendix 1, plus s t a t i c  head,  p (t)  = p(t)S(t),  s  and  P ( ) t  s  are t r e a t e d d i f f e r e n t l y  they w i l l be d i s c u s s e d  separately.  +  where S ( t )  f ,9 ?.( ) T  0  t  >  t  n  i n some p a r t s of  e  is  defined  pressure the  analysis,  100 As mentioned e a r l i e r , time of  the r u n i s  to d i g i t i z a t i o n .  the "raw" p r e s s u r e  subjected  to some f i l t e r i n g and c o n d i t i o n i n g  During r e r e c o r d i n g i t  f i l t e r w i t h a 50-second time c o n s t a n t , drifts  from the d a t a .  a l i n e a r phase s h i f t off  r a t e of  s i g n a l recorded at  is  passed through a  prior  high-pass  which removes DC o f f s e t s  During d i g i t i z a t i o n  the s i g n a l  is  the  and slow  passed through  low-pass f i l t e r w i t h a 3 db p o i n t a t 6 Hz and a  fall-  12 db per o c t a v e .  During d i g i t a l  conditioning prior  to F o u r i e r  transformation  of d a t a p o i n t s where s p i k e s occur are r e p l a c e d w i t h z e r o , dual mean v a l u e  is  removed from the s i g n a l .  It  sections  and any r e s i -  should be noted  that  in  run 4y, the h a n d - d i g i t i z e d r u n , the s p i k e s were removed by hand by drawi n g smooth curves  from the b e g i n n i n g  After  t r a n s f o r m a t i o n has been performed the  the F o u r i e r  Fourier coefficients valent  after  g  + Za.^  signal  but b e f o r e  the  spikes.  resulting equi-  spectra. is  generated  d u r i n g the d i g i t a l  a l l f i l t e r i n g and a v e r a g i n g on the p r e s s u r e  complete  of  are smoothed by a d i g i t a l f i l t e r which i s  to hanning the  The p  to the end p o i n t s  the F o u r i e r  transform i s  analysis,  and wave s i g n a l s  computed.  ^  is  is  computed  from ^  =  1.29  (223) T  (  P ) 1000  where T i s mean a i r temperature P i s mean sea.  level  i n degrees K e l v i n ,  pressure  T and P are a v a i l a b l e f o r most of are not r e a s o n a b l e  5.23,  estimates are  and  in millibars.  the runs (see  Table 6.1);  where  they  used.  There are two important sources of e r r o r i n the computation of  101 P  s  f»9 \  p  s  and f 3'"£  +  •  <  a  First,  over the range of f r e q u e n c i e s  are comparable any c a l i b r a t i o n e r r o r s i n p  proportionately greater be s i g n i f i c a n t second of  in their difference.  Because of  this  i n time d i g i t a l l y p r i o r  the l a t t e r  i n t r o d u c i n g a phase s h i f t  the waves;  is  the sensor s e p a r a t i o n  this r a t i o varies  effects  of each of  i n the f o l l o w i n g  as  i n the p r e s s u r e  T h e r e f o r e the time  s  +  frequencies, f<")  |p| »  shift  on  either  these two sources of e r r o r w i l l  f g 1*^1 a  be  phasor of a 507o  (top)  the h y p o t h e t i c a l  the " c o r r e c t measured" phasor,  i.e.  sensor l o c a t e d  about 4 cm upwind of  calibration is  correct.  calibration.  meant  s i g n a l may o c c u r .  a  is  m u  is  which  and | p | ~ p^g \v(\  (bottom).  " c o r r e c t " p r e s s u r e mea-  s u r e d 'by an i d e a l sensor at the e x a c t l o c a t i o n of  estimated  is  to the wave-  c a l i b r a t i o n f o r two s i t u a t i o n s  the phasor P i s  P  has  paragraphs.  commonly occur i n the d a t a :  is  the  the frequency squared through  F i g u r e 21 shows the e f f e c t on the p + f^g^  In the f i g u r e  time s h i f t  The phase e r r o r which the time s h i f t  s i d e of which l a r g e phase e r r o r s i n the p  underestimate  separation  i n the wave s i g n a l which i n c r e a s e s  o n l y c o r r e c t the ^ phase over a range of  discussed  The  cm downwind of  The d i g i t a l  the d i s p e r s i o n r e l a t i o n f o r g r a v i t y waves.  The p o s s i b l e  3.8  to  the wave s i g n a l must be advanced  to t r a n s f o r m a t i o n .  to c o r r e c t v a r i e s w i t h the r a t i o of  will  are  found  the p h y s i c a l  a  the phase of  l i n e a r l y with frequency.  l e n g t h of  and v[  This error is  + Pg">j ) i n v o l v e s  s  the p r e s s u r e and wave s e n s o r s ;  the e f f e c t of  s  i n both the power s p e c t r a and the c r o s s s p e c t r a .  source of e r r o r (in p  former.  where the s i z e of  the wave probe.  that measured by the  the wave probe,  P  m  actual  and f o r which  the  the measured p r e s s u r e w i t h the under-  the measured wave phasor and ?£ i s c  the  102 wave phasor c o r r e c t e d f o r the 4 cm sensor proper phase w i t h P ; m  tude of  the p + f^q^  The diagrams show that i f  |p ( c-  shift  of  For this  It  the phase response of  the sensor  the s p a t i a l s e p a r a t i o n of  as d i s c u s s e d on the p r e c e d i n g page,  is  this  r e a s o n the  (linear)  ampli-  e  calibrais  affected  is  which ( F i g u r e 12)  the p r e s s u r e  exhibited i n  curve which must be approximated by the  the ^ s i g n a l p r i o r  the  large.  to the a d d i t i o n of pjj ^  time  to the p r e s s u r e .  phase c o r r e c t i o n curves f o r an advance of  the wave s i g n a l i n time by one and two samples of  by the a  n o n l i n e a r e f f e c t of  and wave sensors F i g u r e 22.  n  phase c o r r e c t i o n curve f o r the p r e s s u r e s e n s o r ,  i n c l u d e s both the e f f e c t of and the f u l l  t  the amplitude of p + P<j ^  but the phase e r r o r i s  The f u l l  \P|»  s i g n a l i s most s t r o n g l y a f f e c t e d  tion error, while i f little,  s e p a r a t i o n so t h a t i t has  50 Hz the advances are 0.02  and 0.04  (at a sampling frequency  seconds)  are i n c l u d e d i n F i g u r e  22. During a n a l y s i s ,  the s i z e of  the advance to be used was  s e p a r a t e l y f o r each run on the b a s i s  chosen  t h a t the phase c o r r e c t i o n be as  s m a l l as p o s s i b l e near the peak i n the spectrum of energy f l u x to  the  waves. The e f f e c t of  the phase e r r o r s i n t r o d u c e d by a p p r o x i m a t i n g the  p r e s s u r e 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 23 f o r f o u r d i f f e r e n t  of  9 0 ° to 1 8 0 ° ) ,  diagrams P and P  for | p | » m  shown i n F i g u r e  s i t u a t i o n s which appear i n the r e s u l t s :  g e n e r a t i o n (p l a g s * ig by  9 0 ° ) and wave damping (p leads P^\^\  and  a r e , as f o r F i g u r e 21,  \p\ & f 3 a  1^1  full  •  wave  by an" angle On a l l  the  the p r e s s u r e s observed by  103 ideal  (at the wave probe) and a c t u a l (3.8  sensors  r£  respectively.  is  cm upwind of wave probe)  the wave phasor c o r r e c t e d u s i n g the  p r e s s u r e phase c o r r e c t i o n curve and  is  that corrected using  l i n e a r a p p r o x i m a t i o n r e s u l t i n g from the simple time advance of wave s i g n a l s .  I t w i l l be seen that a t h i g h f r e q u e n c i e s  phase c o r r e c t i o n underestimates i t a t low f r e q u e n c i e s ,  the a c t u a l one w h i l e i t  the the  the approximate overestimates  as i n d i c a t e d i n F i g u r e 22 (at v e r y low  the a p p r o x i m a t i o n underestimates  full  frequencies  the phase c o r r e c t i o n , but t h i s i s  not  considered here). The f i r s t p o i n t to be n o t i c e d i s quencies  the phase of p +  and i t s  t h a t a t both h i g h and low f r e amplitude are a f f e c t e d  strongly  if  | p | s.P <"||^"|> w h i l e n e i t h e r the amplitude nor the phase of p + P 3*£  are  greatly affected  a  a  if  | P | >) f <3 a  i n a l l cases the e f f e c t of i£  and p  m  + f S^ a i  underestimate  its  the r e q u i r e d phase d i f f e r e n c e )  1P| » f ^ l l l  is  is  that  between to  magnitude.  Since i n g e n e r a l i n the r e s u l t s and  The second p o i n t to-note  the a p p r o x i m a t i o n on the phase angle  (which i s  c  •  a  t  t  n  e  | p | ~ f ^l*jlat 0  higher frequencies,  the lower  frequencies  the o v e r a l l e f f e c 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  is  a t low f r e q u e n c i e s , t o  cause an o v e r e s t i m a t i o n of  its  power spectrum and an underestimate of any damping or g e n e r a t i o n which may o c c u r .  A t h i g h f r e q u e n c i e s n e i t h e r the p +f a g»£  the phase w i t h r e s p e c t to ^ It  are a f f e c t e d  power spectrum nor  g r e a t l y by the a p p r o x i m a t i o n .  should perhaps be noted that the phase angles  g i v e n by the  cross-  s p e c t r a between p and i£ and between p + P 3^ and ^ d i f f e r by the e r r o r s A  104 mentioned on the p r e c e d i n g page, the f u l l  s i n c e the phase of p i s  c o r r e c t i o n curve f o r the c r o s s - s p e c t r u m between p and J£  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 p,>£ phase angles to be p r e s e n t e d from 0.1  of  the  (p +  ?£ ) >  i s ± 5 ° from 0.05  to 0.1  also present.  run.  (these are g i v e n i n T a b l e 6 . 3 ) ,  the  estimate  pressure  * 10  to 2 0 ° i n  f r e q u e n c y range from 0.1  to 1 Hz, and l a r g e r o u t s i d e t h i s range.  e s t i m a t e s do not i n c l u d e  the expected e f f e c t s  5.3  ff).  which v a r y from r u n to  T h e r e f o r e o n l y a rough e s t i m a t e can be g i v e n :  mentioned on p. 88  With  since  A reasonable  the a c c u r a c y a l s o depends on the expected accuracy of  calibrations  .  Hz, i 1 °  Experiment, p. 58  phases i s h a r d e r to d e f i n e ,  caused by low coherences i s  using  the expected accuracy of  to 1 H z , and i" 5 ° from 1 to 2 Hz (see  The a c c u r a c y of scatter  corrected  of  the sensors  the  These  themselves  ff.  Summary The d a t a a n a l y s i s  the i n t e r p r e t a t i o n of course spectly  is  seen to be f a i r l y s t r a i g h t f o r w a r d ; not  the r e s u l t s i n some frequency r a n g e s .  ( t h a t chosen i n d i s c u s s i n g the frequency ranges  0.05  s p e c t r a except perhaps ^ ) ^ ( f ) ,  the r e s u l t s ) to 0.1  If  it  i n these frequency ranges more c l o s e l y , s e c t i o n should be r e f e r r e d to  is  to 2.0 Hz i n a l l  from these "suspect"  ranges  n e c e s s a r y to examine  the  then the  extensively.  The s a f e s t  to t r e a t o n l y c i r c u m -  Hz and 1.0  extracting  only corroborative information.  is  so  "Interpretation"  data  SECTION 6:  RESULTS  6.1  Introduction  105  6.2  Method of P r e s e n t a t i o n  105  6.3  Summary of Runs  106  6.4  The Power S p e c t r a  107  6.5  The C r o s s - S p e c t r a  113  6.5.1  6.5.2  6.5.3  The Pressure-Waves  Cross-Spectra  114  6.5.1a  Coherence  115  6.5.1b  Phase  118  The F l u x e s  of Energy and Momentum to the Waves  6.5.2a  Summary of Mean Values of E and *£*  6.5.2b  The S p e c t r a of Energy and Momentum F l u x  The S p e c t r a o f  £  w  .  .  122  .  122 125 130  SECTION 6:  6.1  RESULTS  Introduction The p r i n c i p a l r e s u l t s  They c o n s i s t  of  the t h e s i s are p r e s e n t e d i n t h i s  section.  of power s p e c t r a of wave e l e v a t i o n and normal p r e s s u r e  the sea s u r f a c e ;  at  they c o n t a i n i n f o r m a t i o n h e r e t o f o r e not a v a i l a b l e on  the p r o c e s s e s o f wave growth and momentum t r a n s f e r from the a i r to  the  sea. The measurements  from which the above-mentioned  s p e c t r a have been  computed were made under c o n d i t i o n s which taxed the d e s i g n and use of the i n s t r u m e n t a t i o n to t h e i r l i m i t s ; o r d e r to ensure confidence s e v e r a l d i f f e r e n t ways,  they cannot be repeated e a s i l y .  i n the r e s u l t s  the d a t a have been a n a l y s e d  and a comparison experiment has been made  the "Boundary Bay" experiment and d i s c u s s e d which the performance of t h a t of o t h e r  turned up no s e r i o u s sensor  6.2  (called  i n Appendix 2)  in  against  d i s c r e p a n c i e s between the r e s u l t s  from the buoy  and those from the sensors a g a i n s t which i t was compared, i n  below (0.1  in  The comparison experiment  frequency range c o n s i d e r e d to be of i n t e r e s t  p.  in detail  the buoy p r e s s u r e sensor was t e s t e d  types of p r e s s u r e s e n s o r s .  In  i n the r e s u l t s  to be  the  presented  to 2 H z , as i n d i c a t e d i n "Data A n a l y s i s and I n t e r p r e t a t i o n " ,  104).  Method of  Presentation  The d a t a w i l l be p r e s e n t e d i n v a r i o u s ways to b r i n g out points.  First,  w i l l be g i v e n .  a summary of  the important f a c t s  different  about each of  the runs  T h i s w i l l be f o l l o w e d by the power s p e c t r a of waves and 105  106 pressure.  Then the cross  phase p l o t s  s p e c t r a w i l l be shown, f i r s t as coherence and  of the c o r r e l a t i o n s of p r e s s u r e versus wave e l e v a t i o n and  p r e s s u r e p l u s s t a t i c head spectra E(f)  and T ( f )  versus wave e l e v a t i o n .  of  w  wind to the waves w i l l  Finally,  computed energy and momentum f l u x e s  be p r e s e n t e d ,  a l o n g w i t h s p e c t r a of  the  from the  the  negative  damping r a t i o £ = E / u E , d e f i n e d by M i l e s (1960) as  the f r a c t i o n a l  increase  the mean momentum  i n wave energy per r a d i a n .  Where p o s s i b l e  f l u x *7j' found from these computations w i l l be compared w i t h t h a t w  o b t a i n e d from measured  uw  covariances.  Where t h i s  i s not  (  X ) -  S  possible,  the mean wind speed w i l l be used to c a l c u l a t e an approximate mean momentum flux  from  T  c  =  f  C  a  ^  where C , the drag c o e f f i c i e n t , D  wind speed a t 5 m e t e r s ,  6.3  6.1  2 5  will  be taken as 1.2  3  U ^ , the mean  was measured .with cup anemometers.  Summary of Runs Most of  the a c c e s s o r y  i n f o r m a t i o n f o r the s i x runs i s  o r d e r of i n c r e a s i n g wind speed i n T a b l e s 6 . 1 , cover a range of wind speeds ( e x t r a p o l a t e d cm s e c  - 1  and a range o f v a l u e s  of  6.2,  -150  to 535 cm s e c  - 1  .  m a i n t a i n e d to b e t t e r  of  in  The runs 150 -  800  (U" - c ) , the d i f f e r e n c e between wind 5  the wave spectrum,  A t l e a s t two and sometimes three cup anemometers  were p r e s e n t f o r the f i r s t f o u r r u n s ; d u r i n g the l a s t anemometer was u s e d .  presented  and 6 . 3 .  to 5 m h e i g h t )  speed a t 5 m and wave phase v e l o c i t y a t the peak of of  x 10" ;  two runs a s o n i c  The a c c u r a c y of wind speed e x t r a p o l a t i o n was than ^57 ; hence 0  the wind speeds a t 5 m e t e r s ,  are e x t r a p o l a t e d from assumed l o g a r i t h m i c p r o f i l e s ,  not which  are o n l y meant to be  107 this  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 f e t c h w i t h azimuth f o r t h i s has been computed by G i l c h r i s t F i g u r e 2.  (1965; h i s F i g u r e 2)  The f e t c h i n the w e s t e r l y  arose from the f a c t  (run 5)  than the water at a depth of  neutrally stable  conditions.  t h a t the buoy  the t i d a l c u r r e n t  s u r f a c e to the  c a u s i n g an u n a c c e p t a b l e number of s p i k e s With one e x c e p t i o n  cooler  If  drift  the s m a l l "bow wave" formed by the buoy  was c o n t i n u a l l y d r i v e n by the wind over i t s diaphragm,  1 - 10 cm below the water  i n these c o n d i t i o n s .  was s l a c k or a i d i n g the wind,  Currents  I n g e n e r a l the observed mean t i d a l  c u r r e n t was opposing the wind; t h i s only operated s u c c e s s f u l l y  reproduced i n  d i r e c t i o n was about 40 km.  were measured by t i m i n g t i s s u e paper f l o a t i n g s u r f a c e over known d i s t a n c e s .  and i s  quadrant  pressure  i n the d a t a .  the a i r was 0.5  - 1 . 5 ° C (3 0 . 4 ° )  5 - 10 cm, i n d i c a t i n g u n s t a b l e or  The runs were taken i n the f a l l  the year  at  the s t a r t of  of  lows r e a c h i n g the c o a s t from the west and d r o p p i n g t h e i r m o i s t u r e  against all  6.4  c o a s t a l B r i t i s h Columbia's "winter  of  the c o a s t a l mountains.  For this  monsoons"--successions  r e a s o n s k i e s were o v e r c a s t  r u n s - - i t was r a i n i n g l i g h t l y d u r i n g r u n 6.  The Power S p e c t r a Three power s p e c t r a a r e p r e s e n t e d f o r each run i n F i g u r e s 24 -  P  s  for  denotes the p r e s s u r e measured by the buoy and contaminated by the wave h e i g h t ,  and p  s  +  f<J^ the p r e s s u r e w i t h f^ ^ 1  a  p r e s s u r e head caused by the v e r t i c a l e x c u r s i o n s  of  the  31.  spikes,  static  the buoy, added to  it.  The l a t t e r p r o v i d e s an i n d i c a t i o n of the spectrum which would be measured by a h y p o t h e t i c a l sensor All  f i x e d a t the mean water  level.  the power s p e c t r a have l o g a r i t h m i c ( r o u g h l y h a l f - o c t a v e )  band-  108 TABLE 6.1  Run No.  Time/Date 1967  tes:  Length of Run F e t c h (min.) (km)  (1)  U  9  w  _ (cm sec  1545/XI/17  12.0  1  1940/X/16  16.3  2b  2030/X/19  15.5  2a  2010/X/19  3  1435/X/20  1.0  ) (  true)  T  a  T w  P Comments  (°C)  (°C)  (5)  (5)  (mb)  (4) (6) Tide slack. Water depth 3 . 0 m .  10.2  8.9  1015.2(f)  9.8  11.2  1027.6(u)  18(E)  9.8  11.2  1023.9(f)  Tide f a l l i n g . Water depth 2.0 m. Dark.  115  18(E)  10.0  11.2  1023.9(f)  Tide f a l l i n g . Water depth 2.1 m. Dark.  80  5(W)  10.4  11.0  1010.7(f)  Tide r i s i n g . Water depth 2.8 m. T rising.  150(s)  135  220(c)  260  2.4  310(c)  115  11.0  2.4  320(c)  16.8  6.7  340(c)  40.  ( _^ sec ) cm  1  (3)  5  U  5(SW)  Tide slack. Water depth 2.4 m. Just a f t e r sunset.  9.  6  1225/XI/23  10.5  1.6  570(s)  120  4b  1200/X/30  3.1  2.4  700(c)  ESE  1150/X/30  4.0  2.4  800(c)  ESE  1015.9(f)  Tide slack. Fine r a i n . Water depth 3.4 m.  30(W)  1020.8(f)  Tide f a l l i n g . Water depth 2.7 m.  30(W)  1020.8(f)  Tide f a l l i n g . Water depth 2.7 m.  8(W)  7.8  8.2  (2) 4a  TABLE 6.1  Notes:  (continued)  1.  Time/Date months i n Roman numerals.  2.  Run 4 wind speed and water c u r r e n t i n f o r m a t i o n from memory; o r i g i n a l d a t a d e s t r o y e d .  3.  U^:  (c)  f o l l o w i n g wind speed means "measured w i t h cup anemometers";  which two were c o n s i d e r e d a c c u r a t e enough to use (s) 4.  U : w  5.  T , T : a w  their  results.  f o l l o w i n g wind speed means "measured w i t h 3-component  observed s u r f a c e d r i f t mean a i r and water  three cups were used of  sonic  anemometer".  and t i d a l c u r r e n t ; d i r e c t i o n i n b r a c k e t s . temperatures, measured to 1*0.2°C w i t h a t h e r m i s t o r bead at  the  r  recording platform. 6.  P:  p r e s s u r e a t time of run as measured at Vancouver I n t e r n a t i o n a l A i r p o r t ,  sea l e v e l ;  trends are (u)  unchanged,  (f)  falling,  and (r)  rising.  c o r r e c t e d to mean  109 TABLE  5 (cm s e c " l )  p (cm s e c ~ l )  U  Run No.  6.2  c  Notes:  (1)  k (Hz) f  (2),  2  (4)  cp (cm)  (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  a t the peak of  the  locally-generated  wave spectrum. 2.  is is  the frequency a t which the phase speed c of  equal to the mean wind speed  the mean water 3.  z  c p  is  at 1/k = X/2K  above  level.  the M i l e s '  c r i t i c a l h e i g h t f o r the peak o f  l o c a l l y - g e n e r a t e d wave spectrum: 4.  the waves  the h e i g h t were c  The mean wind speeds used i n the c a l c u l a t i o n of f are e x t r a p o l a t e d v a l u e s the drag c o e f f i c i e n t ,  the  c  p  = U.  and  z  c p  from l o g a r i t h m i c p r o f i l e s w i t h C ,  set  n  equal to 0.0012.  110  TABLE  6.3  Pressure Calibration Used  Run  Estimated A c c u r a c y of Calibration  (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  5  30  4a  Notes:  -  1.  1  .  7  A l l c a l i b r a t i o n s used field  (Column 2) were made i n  immediately f o l l o w i n g  the run f o r which  the they  are used. 2.  The same w a t e r p r o o f i n g diaphragm was used f o r a l l runs;  its  laboratory c a l i b r a t i o n is  and was 2.4 mv/dyne cm" .  the  shown i n F i g u r e 12,  Ill widths.  The s p e c t r a have been smoothed by hanning so  t h a t they are not  d i s t o r t e d i n r e g i o n s which change l e s s r a p i d l y than f-* or f "'  (the  -  window f a l l s  off  as  5f~^).  analog f i l t e r i n g p r i o r of  these p o i n t s  tion",  pp.  71  I n a d d i t i o n a l i a s i n g has been avoided by  to d i g i t i z a t i o n .  and those f o l l o w i n g ,  F o r more d e t a i l e d  see  "Data A n a l y s i s and I n t e r p r e t a -  ff.  The wave s p e c t r a , marked w i t h an for  the s i t e ;  3.  i  "peaks";  is  the presence  the f i g u r e s ,  these are p a r t i c u l a r l y n o t i c e a b l e  i n deep water of about 8 m s e c  - 1  c e r t a i n l y not l o c a l l y g e n e r a t e d , i n the approaches  observed p a s s i n g  appear normal and - 5 .  i n some of the s p e c t r a of  The low-frequency peaks occur near 0.2  traffic  n  they show h i g h - f r e q u e n c y s l o p e s between - 4 . 5  i n t e r e s t i n g feature separate  discussions  .  One  two  i n runs 2a,  2b, and  Hz, i n d i c a t i n g a phase  speed  These "peaks" near 0.2 Hz were  almost  and were p r o b a b l y caused by marine  to Vancouver h a r b o u r ; many f r e i g h t e r s  were  the e x p e r i m e n t a l s i t e d u r i n g the p e r i o d when the runs  were made. The most s t r i k i n g f e a t u r e o f the p regions  g  spectra is  of excess p r e s s u r e energy a t f r e q u e n c i e s  are l a r g e ;  s  + PJ]^  pletely  where the wave  i n runs where the wave s p e c t r a show two peaks,  show c o r r e s p o n d i n g "humps" a t both f r e q u e n c i e s . p  the presence  spectra;  the removal of  -  P 3^ a  has  the p  spectra s  are the o n l y r e g i o n s  of  p a r t s of  the  the e f f e c t of almost com-  The h i g h e r - f r e q u e n c y "humps" i n the p r e s s u r e s p e c t r a remain, but  The l o w e s t - f r e q u e n c y  spectra  T h i s i s not so i n  e l i m i n a t i n g the low-frequency r e g i o n s of excess p r e s s u r e  c e n t e r of mass i s moved to s l i g h t l y h i g h e r  of  energy. their  frequencies.  the p r e s s u r e s p e c t r a (near l o g (f)  the e n t i r e 0.05  =-1)  - 2.0 Hz bandwidth covered by the  p r e s s u r e s p e c t r a p r e s e n t e d which appear to c o n t a i n i n f o r m a t i o n not d i r e c t l y associated tral  w i t h waves.  The s m a l l number of  low-frequency  e s t i m a t e s which are a v a i l a b l e here suggest t h a t  have a shape near f  , a t v a r i a n c e w i t h the f  s i m i l a r i t y considerations The r e s u l t s  of  (Stewart,  R. W . ;  the Boundary Bay experiment  may i n f a c t be s p u r i o u s ,  since  than f  .  For t h i s  personal indicate  communication). that  the f"^  the low-frequency shapes of  reason the shapes of  and are not d i s c u s s e d  From 0.2  shape  the power  comparison are  the p r e s s u r e  closer  spectra  shown i n F i g u r e s 24 - 31 can be assumed to be a t b e s t h e a v i l y with noise,  spectra  form suggested by  s p e c t r a o b t a i n e d from the two sensors used i n the to f  the p r e s s u r e  spec-  contaminated  further.  Hz to the upper d e s i g n frequency of  the p r e s s u r e  system  a t 3 Hz the p r e s s u r e s p e c t r a appear to be dominated by the waves. e f f e c t on the s p e c t r a of varies  the removal of - P ^ ^ from the p r e s s u r e  d r a m a t i c a l l y with frequency.  In some of  Hz, presumably caused by s w e l l  ponding e x c u r s i o n s  of  This indicates  that  (see  p.111).  the p r e s s u r e s p e c t r a above t h e i r ' b a s e  the w a v e - i n f l u e n c e d  hence wave g e n e r a t i o n or damping must be s m a l l a t damping i s u s u a l l y observed a t low f r e q u e n c i e s ;  It tions  on  level"  are  signal.  the p r e s s u r e i s  i n a n t i p h a s e w i t h the waves and a t most a s m a l l p a r t i s  section  Corres-  from the p r e s s u r e  p a r t of  signal  the wave s p e c t r a a l a r g e  peak occurs a t 0.2  almost c o m p l e t e l y removed by removing - f ^ ^  The  largely  i n quadrature;  these f r e q u e n c i e s  this  is  (some  described i n  the  cross-spectra).  i s worth n o t i n g a t t h i s  important to  d u r i n g f i e l d use of  p o i n t t h a t removal of  the assessment of the s e n s o r .  -  has  the a c c u r a c y of p r e s s u r e  S i n c e the s e n s i t i v i t y  of  implica-  calibrations  the sensor has  113 been known on other o c c a s i o n s during f i e l d use, the s e n s i t i v i t y  to change by as much as a f a c t o r of  every a v a i l a b l e check on the v a l u e and s t a b i l i t y of  d u r i n g each r u n i s made.  Removal of - p^cj v£  reduce the p r e s s u r e spectrum to the l e v e l which i s I t has been noted above t h a t the r e g i o n s power s p e c t r a caused by s w e l l i t w i l l be p o i n t e d out l a t e r r e d u c i n g the pressure-waves values.  This i s  independent of waves.  o f excess energy i n the  t h a t the -f^*^  pressure  coherence if  removal has the e f f e c t of  at the s w e l l  the s e n s i t i v i t y  frequencies of  are based p a r t l y on the above  to  the sensor  The e s t i m a t e s of p r e s s u r e s e n s i t i v i t y  At frequencies  can a t most  are s u b s t a n t i a l l y reduced by removing - f ^ ^ ',  only possible  well-determined. i n T a b l e 6.3  two  different.  is  quite  accuracy given  considerations.  near and above the l o c a l l y - g e n e r a t e d peak i n  wave spectrum the s i t u a t i o n i s  low  After  the  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 p r e s s u r e spectrum, which often  shifted  towards h i g h e r f r e q u e n c i e s .  dominated p a r t of  occurring.  r e g i o n of  follows  t h a t a t these f r e q u e n c i e s  F u r t h e r d i s c u s s i o n on t h i s  the p r e s s u r e spectrum i s  s t u d i e d by r e f e r r a l  postponed,  to the pressure-waves  The s p e c t r a a t f r e q u e n c i e s  above 1.5  2 H z , which are d i s c u s s e d  wave genera-  l a r g e wave-dominated as i t  can be more  fruitfully  cross-spectra. Hz must be viewed w i t h c a u t i o n ;  a number of e x p e r i m e n t a l and a n a l y t i c a l d i f f i c u l t i e s  6.5  t h a t a wave-  the p r e s s u r e s p e c t r a remains which cannot be i n a n t i -  phase w i t h the waves; i t tion is  This indicates  is  a r i s e a t and above  i n d e t a i l i n "Data I n t e r p r e t a t i o n " ,  p. 88  ff.  The C r o s s - S p e c t r a Presented i n this  s e c t i o n are three s e t s of s p e c t r a .  presented are the c r o s s - s p e c t r a between p r e s s u r e and wave  First  to be  elevation.  114 Second,  the s p e c t r a E ( f )  and momentum f l u x e s  and£* (f),  from the wind to the waves,  be d e s c r i b e d are the s p e c t r a energy per r a d i a n (see  p.  13).  The l a t t e r  is  are p r e s e n t e d .  Last  + ("^^  cross-spectra.  and J£ .  the coherence and phase  and waves i£ , and the phase spectrum between  g  The s u b s c r i p t "s" means t h a t the spectrum concerned  convolved w i t h the s p i k e f u n c t i o n .  c r o s s s p e c t r a between p  s  and  been o u t l i n e d i n Appendix 1.  The reasons  for presenting  r a t h e r than t h a t between p  g  s e p a r a t i o n of  A l l the s p e c t r a p r e s e n t e d are c o r r e c t e d  the p r e s s u r e and wave s e n s o r s ,  d e s c r i b e d i n "Data I n t e r p r e t a t i o n " , p.  101  spatial  a c c o r d i n g to the methods ff.  The c r o s s - s p e c t r a are p r e s e n t e d i n F i g u r e s 32 through 39. on each f i g u r e are the f r e q u e n c i e s and f^,  fp of  the peak of  where the phase v e l o c i t y c equals  z where kz = 1,  so z = X/2TI  = g/4Tf  2  the wave spectrum  the wind speed  f .  is  2  Shown  a t the  .0012  i n runs 1,  i n runs 5 and 6.  2,  3,  and 4,  breaks are l e f t  example near 1.2 Hz i n F i g u r e 32). drawn on each f i g u r e a t phase angles  coeffici-  and from s o n i c anemometer measure-  Where l a r g e d e v i a t i o n s  a t t r i b u t e d d i r e c t l y to n o i s e ,  height  e x t r a p o l a t e d from cup  anemometer wind speeds assuming a l o g a r i t h m i c p r o f i l e and a drag ent of  the  and r£ have  f o r phase e r r o r s i n t r o d u c e d by the p r e s s u r e sensor and f o r the  ments  to  The Pressure-Waves C r o s s - S p e c t r a  s p e c t r a between p r e s s u r e p s  energy  three s e t s of s p e c t r a are  F o r each r u n t h r e e s p e c t r a are p r e s e n t e d :  p  the  (f) , the f r a c t i o n a l i n c r e a s e of wave  d e r i v e d from the pressure-waves  6.5.1  which are r e s p e c t i v e l y  w  i n the s p e c t r a can be i n the s p e c t r a l curves  Two h o r i z o n t a l r e f e r e n c e of  1 8 0 ° and - 9 0 ? .  n e g a t i v e phase angles mean the p r e s s u r e l a g s  the waves.  lines  In a l l  (for  are  cases  115 6.5.1a  Coherence  The coherence between p expected,  s  and r^^ behaves  i n a g e n e r a l way as  b e i n g h i g h a t f r e q u e n c i e s where a p p r e c i a b l e wave energy i s  p r e s e n t 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 p r e s s u r e -  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 o r the lengths c o n s i d e r e d here when the coherence drops below 0 . 3 . 4b and 5, the g e n e r a l l y r e g u l a r f a l l increases  above f  p  of r u n  I n two c a s e s ,  runs  i n coherence as the frequency  is temporarily arrested, r i s i n g  to a s u b s i d i a r y  "peak" a t about 1.6 Hz and r e t u r n i n g to low v a l u e s by 1.8 Hz. To a v o i d c l u t t e r e d graphs the coherence s p e c t r a between p and  have been o m i t t e d .  s  + (> 3»2 a  The g e n e r a l e f f e c t o f removing - (^""f^ f  r o m  t  n  e  p r e s s u r e s i g n a l i s to lower the coherences between p and Y) a t a l l s tJ except the h i g h e s t f r e q u e n c i e s .  I n some runs t h e r e i s a low-frequency  p o r t i o n of the t o t a l wave energy p r e s e n t which i s not the r e s u l t of l o c a l wave g e n e r a t i o n ;  this  can l e g i t i m a t e l y be c a l l e d " s w e l l " .  s w e l l i s b e s t observed i n the phase s p e c t r a between p  and 77 ; i n  S  r e g i o n s where i t i s p r e s e n t positive,  w 5  the phase i s n e a r l y 1 8 0 ° ( i f the phase  then wave damping i s o c c u r i n g ) .  the e f f e c t of removing -f g ^  from p  g  d i s p l a y s the percentage of the  It is interesting  is  toscompare  on the low-frequency coherence of  the runs where s w e l l appears to be p r e s e n t : 6.4  The  runs 1, 2a, 3, and 5.  coherence by which the ( p  s  Table +P 3'"£) {t  coherence i s decreased a t the low f r e q u e n c i e s where s w e l l i s assumed to be p r e s e n t ,  and a t the f r e q u e n c i e s  of the wave power s p e c t r a a r i s i n g  from l o c a l l y - g e n e r a t e d waves. Also included are U 5 , Q  (mean wind d i r e c t i o n ) , c^ and c , p  phase speeds a t the s w e l l and l o c a l l y generated s p e c t r a l peaks  the wave respectively  116 and £ ,  the frequency  n  spectra with  . ^  a  r  e  above which the coherences same s i z e to w i t h i n  t n e  of the two  pressure  5%.  A s t r i k i n g f e a t u r e of the t a b l e i s the l a r g e percentage drop a t the s w e l l peak frequency of r u n 1. recalling  T h i s can be understood  =  p  - p  t h i s i s ( E q u a t i o n 13, Appendix 3):  a  n t  1  +  ( 1  - v  c  )  2  ]  6  - 2  and C have the same s i g n the p r e s s u r e w i l l v a r y between - Z P ^ j ^ (U 0 «c)  o  a n d ( U  0  & c ) .  f o r the low frequency waves; i f they are o p p o s i t e i n  s i g n the p r e s s u r e w i l l v a r y between - 2P^Cj 7£ (I U | 4<x) 0  where D >*>  2.  and -D^gi^ (IU I » c ) 0  T h i s means t h a t 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 was  by  the p r e d i c t i o n of the p r e s s u r e a t the water s u r f a c e g i v e n by  p o t e n t i a l f l o w theory;  If U  coherence  |P |  a n <  g  3 f^i^l  w  the case i n run 1, then the  e  r  e  about the same s i z e ,  as  removal should drop the  coherence to low v a l u e s . No  i n f o r m a t i o n 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 i n f e r e n c e s , however, be made (see Appendix 3).  Thus i f the s w e l l were t r a v e l l i n g w i t h  wind the coherence would be h a l v e d i f U value i f U  0  ~  0  «  C and reduced  o t h e r angle  the  to a s m a l l  C; i f on the other hand the s w e l l were moving a g a i n s t  the wind the coherence would a g a i n be h a l v e d i f o n l y be reduced  can  by 207. i f  JU | 0  C.  1007„.  C, but would  Q  I f the s w e l l was  to the wind, then the expected  between the extremes of 207« and  JU | «  running a t some  drop i n coherence would l i e  From the observed  drops i n  coherence a t low f r e q u e n c i e s , the s w e l l could have been t r a v e l l i n g i n almost  any d i r e c t i o n , i n run 2a, and was  runs 3 and  5.  The  probably from the Northwest i n  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 Pressure-Waves  U  5  Run No.  C  n P  c s  p  Cross  ^  ""^"J"^  SE  n  Spectra  7o drop i n swell  (cm s e c ~ l )  i  7„ drop a t l o c a l peak  f -"-n (Hz)  5  150  340  -  1  220  350  -  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  W  40  no peak  1.2  80  no peak  1.7  118 velocity-wave  6.5.1b  phase r e l a t i o n d i s c u s s e d  Phase  The b e h a v i o u r of the energy f l u x sin  i n Appendix 3.  the phase s p e c t r a i s  (pdfl/dt)  (- 0 ) , where 6 i s  signal.  b e s t e x p l a i n e d i n terms of  from the wind to the waves. the phase lead of  As has been mentioned e a r l i e r ,  This v a r i e s  the p r e s s u r e ahead of 6  as  the wave  = - 1 8 0 ° i s p r e d i c t e d by  p o t e n t i a l flow theory and no g e n e r a t i o n occurs i n t h i s  case (except an  i n s i g n f i c a n t amount by the K e l v i n - H e l m h o l t z mechanism).  The most  effici-  ent g e n e r a t i o n occurs when d = - 9 0 ° ( p r e s s u r e l a g g i n g the waves by 9 0 ° ) , and  damping occurs i f + 9 0 ° ^ So f a r l i t t l e  phase angles  6 A  180°.  has been s a i d about the r e l a t i v e importance of  between p  and ^ , and between p  s  wave s u r f a c e e x p e r i e n c e s  g  + p CJ1£  the former, w h i l e a sensor  and  .  the The  f i x e d 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 .  Theoretical predictions  such as those of M i l e s  the phase  p r e s e n t e d here as the  (1960) are made i n terms of (p  s  + f ^^ a  ), 7j" s p e c t r a ; s  d i s c u s s e d below as "the phase angle between The pattern. angles  phase s p e c t r a ( F i g u r e s 32 - 39) At frequencies  below f^  are m a i n l y p o s i t i v e ,  therefore  differences  these w i l l  the p r e s s u r e and the  be  waves".  f o r a l l the runs show a s i m i l a r  (where C = U a t z = A / £ T T )  i n d i c a t i n g wave damping.  the  phase  At frequencies  near or somewhat lower than f^ the phase c r o s s e s 1 8 0 ° , i n d i c a t i n g the onset of wave growth. sudden,  The s h i f t w i t h i n c r e a s i n g frequency i s  usually  the phase changing from 1 8 0 ° to anywhere from - 1 6 0 ° to - 1 1 0 °  a frequency range of 0.2 more s l o w l y , o f t e n as n o t .  to 0.3 Hz.  At higher frequencies  g e n e r a l l y c o n t i n u i n g to s h i f t At s t i l l higher frequencies  it  over  changes  toward - 9 0 ° and r e a c h i n g i t i n most runs there i s  a  as  119 suggestions  (much c l e a r e r i n the p j , ^ phases)  back towards coherences  that  1  180°; it  can o n l y be c a l l e d a s u g g e s t i o n ,  a t the h i g h e r f r e q u e n c i e s  t h a t the phase i s most s p i k e s  the phase may  (greater  not w e l l - d e f i n e d .  i n the p r e s s u r e d a t a ,  Runs 1,  however,  than 1.6 2b, 3,  since  Hz) are so  low  and 6, which had the  d i s p l a y l a r g e r coherences  at  higher frequencies  than those w i t h few or no s p i k e s ,  those runs much of  the phase i n f o r m a t i o n at h i g h f r e q u e n c i e s  spikes  shift  these  i n d i c a t i n g that  themselves and i s hence s p u r i o u s as f a r as t h i s  is  in  from the  discussion  is  concerned. The e r r a t i c behaviour of low f r e q u e n c i e s 0.05  the  (p  i n r u n 1 ( F i g u r e 32)  to 1 Hz the removal of - Pj^^  w e l l below 0 . 3 . nately  +  g  p <j 1£ a  is  ),  phase spectrum a t  caused by the f a c t , r£  has reduced the  A l a r g e n o i s e peak a t  t h a t from  1226 Hz appears  coherences  to have u n f o r t u -  c o i n c i d e d w i t h the frequency a t which the c r o s s o v e r  from damping  to wave growth o c c u r r e d . The p r e s s u r e s i g n a l with spikes, w i t h i n ^107o.  i n r u n 2a ( F i g u r e 33)  and the p r e s s u r e It  show a tendency  is  c a l i b r a t i o n is  is  relatively  uncontaminated  c o n s i d e r e d as known to  the o n l y run where the h i g h - f r e q u e n c y phase does not  to drop back towards  which are c o m p a r a t i v e l y f r e e  1 8 0 ° from - 9 0 ° .  from s p i k e s  and i n which a c t i v e  occurs, so t h a t a comparison can be made w i t h r u n 2a, (which were h a n d - d i g i t i z e d ) ; the p r e s s u r e s i g n a l s  The o n l y other runs generation  are runs 4a and 4b  i n these the s p i k e s were smoothed out of  by hand p r i o r  to d i g i t i z a t i o n .  the h i g h f r e q u e n c i e s .  e r r o r s noted on p.  the h i g h frequency phases i n these runs can be  compared w i t h those of run 2a.  to the  pro-  cess should not a f f e c t 79,  Subject  T h i s smoothing  They both i n d i c a t e a trend of  expected  the  phase  120 at high frequencies  to r e t u r n toward 1 8 0 ° .  be t r e a t e d as the e x c e p t i o n . wave spectrum of  this  Thus i t  is  run 2a which must  A l a r g e 0.2 Hz s w e l l peak i s  run ( F i g u r e 15) which i s  absent  p r e s e n t i n the  i n run 2b taken 20  minutes  later.  The " s w e l l " was thus p r o b a b l y caused by marine t r a f f i c .  In t h i s  run and i n run 2b the sharp phase s h i f t occurs a t a frequency  0.3 Hz below f^; i n any of  this  is  a l a r g e r f^ - f  than e x i s t s  the o t h e r r u n s .  Run 2b';shows phase s h i f t s 90° shifts  frequency d i f f e r e n c e  p  o c c u r r i n g i n run 2a.  of o n l y 6 0 ° from 1 8 0 ° , as opposed to  the  The wind speed had dropped s l i g h t l y f o r t  t h i s r u n f o l l o w i n g run 2a; transfers.  t h i s might account f o r the l e s s e f f i c i e n t  The frequency o f  Hz i n r u n 2a to 0.58  the wave peak has r i s e n s l i g h t l y ,  from  energy 0.56  Hz i n run 2b.  The p r e s s u r e s i g n a l i n run 3 i s  also  quite "spikey".  In t h i s r u n  U5 was s l i g h t l y l a r g e r than i n runs 2a and b and the f e t c h longer Table 6.1).  S w e l l near 0.2 Hz was p r e s e n t ,  direction is  not c l e a r .  i t was b e i n g damped.  (see  but as noted on p.116 i t s  The low-frequency phases  The phase "peak" a t 0.38  ( F i g u r e 27)  Hz i s  a s s o c i a t e d w i t h a low coherence between p ,+ s  •>£  show t h a t  spurious, being and 1£  .  The s h i f t  from damping to g e n e r a t i o n i n run 3 occurs a t a frequency about 0.2 Hz above the peak of behaviour of noise;  the wave spectrum and almost e x a c t l y a t f  the phase between 1.1  and 1.4  the phase a t h i g h e r f r e q u e n c i e s ,  Hz i s  .  The  e f f i c i e n t l y masked by  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 ° (180°  to - 1 1 0 ° )  a t 0.8 H z .  The two h a n d - d i g i t i z e d r u n s , 4a and 4b, were separated by 10 m i n u t e s , 4b being the later.;: of  the two.  During them the wind speed was h i g h e r than  121 for  any of  the o t h e r r u n s .  In both cases the frequency of the peak of  the wave spectrum i s w e l l above f^., from s a t u r a t e d and t h a t stationary.  therefore  The process  i n d i c a t i n g that  d u r i n g n e i t h e r run were wave  of hand d i g i t i z a t i o n i s b e l i e v e d  phase e r r o r s l e s s than 2 0 ° below 1.6 A n a l y s i s and I n t e r p r e t a t i o n " , p. The  the waves were  Hz; see  states  to i n t r o d u c e  the d i s c u s s i o n i n "Data  79.  between p ,i£  phase d i f f e r e n c e  far  %  runs 4a and 4b than i n o t h e r r u n s ,  and ( p + f 3 * 2 ) » ^ <  i  s  a  r  smaller  e  a  i n d i c a t i n g t h a t the p o r t i o n of  c o r r e l a t e d w i t h the waves was much l a r g e r than  |^|  .  p  in  s  The sharp  phase change a s s o c i a t e d w i t h the onset of wave growth occurs at a f r e quency 0.2 Hz above f^ i n r u n 4a and 0.1 Hz above f^ i n run 4b. In run 4a the phase a t f r e q u e n c i e s sudden change behaves -90°  regularly until  to near - 1 8 0 ° w i t h i n 0.1 H z .  w i t h a v e r y low coherence, significant.  1.3  T h i s sharp change of phase i s  i n d i c a t i n g that i t  towards 1 8 0 ° i s  and (p + f o , " " c o h e r e n c e s  is  not  a s s o c i a t e d w i t h a l a r g e peak i n  near 1.6  Hz.  T h i s means t h a t  quencies  b e i n g a s s o c i a t e d w i t h low  The r e a s o n f o r the d i f f e r e n c e i n runs 4a and 4b,  i n the coherence  at  fre-  coherences.  s p e c t r a above 0.8 Hz  separated as they are by o n l y ten m i n u t e s , must f o r  the wind and wave f i e l d s .  quencies  the  the w i l d phase f l u c t u a t i o n s  l a c k of any o t h e r e x p l a n a t i o n be put down to v a r i a b i l i t y i n the of  associated  observed i n run 4b except  whereas  Hz do n o t ,  initial  statistically  change has some s i g n i f i c a n c e , above 1.6  the  Hz, when i t r e t u r n s from near  Roughly the same phase behaviour i s  t h a t the r e t u r n i n phase the p,-^  above the frequency of  The d i f f e r e n c e  above 0.8 H z , and as w i l l  the energy and momentum f l u x e s  of  o n l y becomes l a r g e a t  be seen l a t e r c o n t r i b u t e s either run.  conditions  little  freto  122 Run 5 must be viewed w i t h a knowledge of i t s  time h i s t o r y .  The wind  was f o r the most p a r t weak and v a r i a b l e (about 80 cm/sec) o n l y r i s i n g to 150 - 200 cm/sec f o r the l a s t band from 0.1 (Appendix 3)  -  0.8  t h i r d of  the r u n .  The waves i n the  Hz were not l o c a l l y g e n e r a t e d ,  and as has been shown  they were t r a v e l l i n g a g a i n s t what wind there was.  remains p o s i t i v e  (+150  to 1 8 0 ° ) i n most of  the coherence exceeds 0.3;  hence  this  frequency  The phase  the frequency range f o r which  " s w e l l " was being damped.  v a r i a t i o n s above 1 Hz are c o n s i d e r e d to be random and w i l l  The phase  not be  discussed  further. The p r e s s u r e s i g n a l i n r u n 6 i s of  spikes.  because  contaminated by a c o n s i d e r a b l e number  The wave spectrum f o r the r u n i s v e r y s m a l l , presumably  the f e t c h  i n Table 6.1,  (1.6  km) was l e s s  than f o r the o t h e r r u n s .  a l i g h t r a i n was f a l l i n g d u r i n g the r u n .  frequency phase change occurs a t a frequency 0.2 the peak of  P  - 9 0 ° near f^. and then f a l l s  + f^q,^  180°,  a n  d V£  apparently reaching i t  In g e n e r a l i f this  6.5.2  about t h i s  curve i s  Summary of Mean Values of E and shows, a l o n g w i t h  spectra for a l l runs.  g r a d u a l l y back towards  the coherence drops below  (this  0.4.  and 8  of  q u i t e s m a l l due to the h i g h to 2 Hz.  the Waves  fw , the i n t e g r a l s under the E and  These i n t e g r a l s have been t r u n c a t e d a t 3.0 H z ,  any p r e s s u r e and wave s i g n a l s  to be i n c o h e r e n t  0.4  The phase between  p r e s e n t over the e n t i r e frequency range from 0.2  T a b l e 6.5  since  The sharp low-  drawn through the computed phase p o i n t s  The F l u x e s of Energy and Momentum to  6.5.2a  W  the wave spectrum.  a t 2 Hz b e f o r e  a smooth curve i s  r u n , the s c a t t e r  coherences  T  reaches  noted  Hz lower than f^ and  Hz below the frequency of s  As was  truncation is  which occur above 3 Hz are observed p a r t i c u l a r l y important i n the  'T'w  123 spectra,  2 s i n c e they c o n t a i n an CO i n t h e i r numerator--see  a d d i t i o n the s t r e s s e s computed i n t h i s way ( ( T$ ) measured by the three-component  7J  W  p.88).  In  ) are compared w i t h  s o n i c anemometer,  those  when a v a i l a b l e ,  and w i t h those computed from 7= C  c  =  2 ^> U  0.0012  (jL  A l s o shown are drag c o e f f i c i e n t s  DW  C  A'K!  =  <Tw/  6.3.  5  C^y c a l c u l a t e d from  e„ 5) u  6  A c o r r e c t i o n must be a p p l i e d to the '~C<  V>/  finite  w i d t h 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  r e c o r d s waves from a l l d i r e c t i o n s ; occurs  i n the wind d i r e c t i o n ,  s p e c t r a to account f o r the waves.  s i n c e momentum i n p u t to the waves o n l y  w i t h the wind w i l l v a r y as cos S  G i l c h r i s t (1965) f i n d s  one which v a r i e s  one f o r waves at f r e q u e n c i e s  as cos  ©  t h a t i n E a s t winds a t is  c l o s e to the  h i g h e r than the s p e c t r a l  peak.  correcthe  observed  Therefore  p r o b a b i l i t y of f i n d i n g a g i v e n wave d i r e c t i o n a t an angle between d©  the  The wave probe  A d i r e c t i o n a l d i s t r i b u t i o n f o r the waves must be assumed so the  Spanish Banks s i t e ,  4  the momentum withdrawn from the a i r by a  wave t r a v e l l i n g at an angle ©  t i o n may be c a l c u l a t e d ;  - -  8  the and  to the wind w i l l be g i v e n by  P(6)d0  =  2 cos ed6 2  6 > 5 <  T h i s g i v e s f o r the c o r r e c t e d momentum f l u x X  (f)  =  2  F/2 Ft 2 \  r  w  w  e )cos©  w  -It/2 2C (f)  (f)p(  spectrum  \ -it 12  cos 6de 3  de  124  TABLE  5 cm Run No. s e c "  6.5  e  T  erg cm ^ degrees (true) sec"l  U  1  «1  dyne cm"  5  150  135  -5  7  1  220  260  13  2b  310  115  2a  320  3  c  dyne -2 cm  2  T  C  dyrie cm"  DW  2  -  -.002  .02  .03  17  .11  .11  .07  -  .0018  29  18  .16  .11  .14  -  .0013  80  43  18  .24  .10  .15  -  .0019  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 u n i t s are  2.  6  3.  0~g and <T"r  c. g . s .  i s wind d i r e c t i o n i n degrees ( t r u e )  f 4.  -.03  over the "7* i s  are the standard e r r o r of t o t a l number of data b l o c k s  the means of E and analysed.  the mean momentum f l u x from wind to waves,  w  computed  from the f,^ c o r r e l a t i o n . 5.  "f  c  is  from T 6.  X  s  is  the momentum f l u x from the a i r to the sea, = ?a  c  U D  5'  C  D =  -  0  0  1  2  -  the momentum f l u x from the a i r to the s e a ,  by a s o n i c ^Dy i s  C  U  as measured  anemometer.  the d i m e n s i o n l e s s  ea 5-  computed  drag c o e f f i c i e n t  computed from Cpy =  125  0.85 t h a t i s , 157o  less  T (f)  6.6;  w  than the c a l c u l a t e d v a l u e .  The  l i m i t s of ~t Tf /2  are  chosen s i n c e 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 windd r i v e n waves t r a v e l l i n g a g a i n s t the wind i s n e g l i g i b l e . s o p h i s t i c a t e d and  Taking more  a c c u r a t e d i s t r i b u t i o n s i s not worthwhile i n view of  s i z e of the c o r r e c t i o n and  the expected  accuracy of "7J* ( f ) .  A l l the *7j*  v a l u e s except run 5 have been c o r r e c t e d f o r the assumed d i r e c t i o n a l trum of the waves ( E q u a t i o n 6.4). (512  f o r each data b l o c k analysed 10  or 20  seconds),  and  The v a l u e s of E and  the <P  or 1024  "J*  the w  spec-  were computed  samples, or time i n t e r v a l s of  columns f o r E and 7J* g i v e the r  w  standard  e r r o r of the mean f o r these p o p u l a t i o n s .  6.5.2b  The  The sented  S p e c t r a of Energy and Momentum F l u x  s p e c t r a of Energy and Momentum f l u x from wind to water are pre-  i n F i g u r e s 40  to 47.  - 2 ( E ( f ) ) and dyne cm  They are i n u n i t s of dyne cm  -1 ( T (f)).  Hz  spectrum f o r the run.  w  A l s o noted  I n c l u d e d on each f i g u r e i s the wave  i s f^., the frequency where the wave  phase v e l o c i t y a t the peak of the spectrum equals U^, distance V k  =  ""sec ''Hz  "X /2TT  from the mean water s u r f a c e .  that, as on a l l s p e c t r a presented  the wind speed at a I t should be  i n " R e s u l t s " the l i n e s  s p e c t r a l e s t i m a t e s are meant to be guides  to a i d the eye  joining  noted  the  i n seeing  the  s p 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  7T (f), w  s i n c e both are computed from e x a c t l y the  same data, must c r o s s the zero f l u x l i n e at the same frequency run;  the "guide  f o r a given  l i n e s " r a r e l y do cross t h i s l i n e at e x a c t l y the same  frequency. Run fers.  1,  the o n l y one  The wind speed was  taken i n w e s t e r l y winds, shows low energy t r a n s 220  cm s e c ; - 1  the l o c a l wave g e n e r a t i o n i s weak  126  and has a c o n t r i b u t i o n f o r unknown reasons It  is  considered u n l i k e l y that  at frequencies  below 1.2  Run 2a i s  at frequencies  real.  c o n s i d e r e d as r o u g h l y t y p i c a l of a f e t c h  wind s i t u a t i o n .  A l t h o u g h a l a r g e wave peak i s  or no damping i s  seen to o c c u r .  t h i s may i n d i c a t e ment s i t e . actions, (1961), and  the presence  The peak of  from o t h e r  (f)  the peak of  s p e c t r a approach z e r o , the wave spectrum; F i g u r e 33)  the wave spectrum.  i s near - 9 0 ° a t 1.4 o  quency, but t h i s peaks  in E(f)  frequencies  i n any o t h e r .  (1966),  Phillips  The peaks  i n E(f) than the  and n^. i n t h i s run  Hz, while E(f) by t h i s  i n r u n 2a.  Damping i s  and f ( f )  and f ( f ) w  are i n t h i s  frequency. less  In f a c t  they peak c l o s e r  Hz.  to  The  lower  c o i n c i d e w i t h the peak of  the  to the wave peak i n r u n 2b than  lower than i n r u n 2a,  and 1.2  fre-  significant.  the run s h i f t e d  The wind speed b e i n g lower i n t h i s  are somewhat  well-defined  seen to occur a t one  p a r t of  they almost  have  w  cannot be c o n s i d e r e d as s t a t i s t i c a l l y  o c c u r r i n g between 0.7  inter-  At higher frequencies  + ^ Y £  s  t h e i r peak v a l u e s  than i n r u n 2a;  wave spectrum.  and ' f w ( f )  is  the measure-  than, but l e s s  The l a r g e low-frequency wave s p e c t r a l peak i s i n r u n 2b than i t  at a  b e i n g governed p r i m a r i l y by the r a p i d decrease of  the phase between p  dropped to 15-207 of  higher  little  through n o n l i n e a r  another p o s s i b i l i t y .  occur i n run 2a a t f r e q u e n c i e s  Hz,  s i m i l a r runs  s t r o n g e r winds upstream of  to lower f r e q u e n c i e s  and Hasselman (1963) i s  easterly  the wave spectrum i s  l i g h t of r e s u l t s of  limited,  p r e s e n t a t 0.17  such as those proposed by Benjamin and F e i r  twice that of  (see  i n the  Energy t r a n s f e r  f^.  the apparent energy and momentum t r a n s f e r s  Hz a r e  frequency lower than f^;  l e s s than  p a r t of  the l a r g e s t  the r u n , E ( f ) differences  The two s u b s i d i a r y peaks  in  the  spectra,  a t 1.04  harmonics of of  and 1.62  Hz, are remarkably c l o s e to the f i r s t and second  the frequency a t the peak of the wave spectrum.  Because  the l a r g e 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 e s t i m a t e s E ( f ) and  f  (f),  the two peaks are not c o n s i d e r e d s t a t i s t i c a l l y  significant;  if  they appeared i n the s p e c t r a of every run they would have to be taken seriously,  but they do n o t .  I n run 3, (0.2  some wave damping i s  seen to occur a t the  Hz) s w e l l peak i n the wave spectrum.  present,  however,  Not enough i n f o r m a t i o n i s  to compute the amount of damping.  e f f e c t on E and T , W  I t has  the spectrum was s t i l l growing.  The M i l e s '  frequency of the peak of  (Table  and the f a c t  6.2)  that  the  the wave spectrum i s h i g h e r than i n run 2 a l s o  i s a "younger" spectrum.  i n r u n 2a but g r e a t e r than i n 2b. many s p i k e s  this  The  i n d i c a t i n g that  c r i t i c a l height  cm f o r t h i s r u n and 100 cm f o r run 2;  suggest t h a t i t  negligible  the mean t o t a l energy and momentum f l u x e s .  wind-generated wave peak o c c u r s a t 0.62 Hz, c l o s e to f^,  was 9.7  low-frequency  peak i s  l e s s than  Presumably because of the presence of  i n the p r e s s u r e r e c o r d ,  slowly at high frequencies  E(f) at i t s  than i s  the s p e c t r a converge to zero more the case f o r run 2.  I t may be  this  e f f e c t which masks any evidence of the harmonic peaks so e v i d e n t i n r u n 2b.  On the o t h e r hand,  the p r e s s u r e s i g n a l of t h a t run i s  h e a v i l y contaminated w i t h s p i k e s Runs 4a and 4b, cm/sec r e s p e c t i v e l y ) ,  as i s  as  t h a t of r u n 3.  the two h i g h wind speed runs (U^ = 800 and 700 show the l a r g e s t energy and momentum t r a n s f e r s  (note the l a r g e r s c a l e s i n F i g u r e s 44 and 4 5 ) . damping at low f r e q u e n c i e s , run 4 a .  just  N e i t h e r show s i g n i f i c a n t  i n s p i t e of a s m a l l s w e l l peak at 0.2 Hz i n  In both runs the peaks of E ( f ) and "U (f) w  occur a t  frequencies  128 above t h a t of  the peak of  the wave s p e c t r a .  As i n run 2,  to occur on the h i g h e r - f r e q u e n c y " r e a r face" of occur w e l l below the frequency of the s p e c t r a l e v e l monics. this  off  the wave spectrum;  the f i r s t harmonics.  i n the r e g i o n of  they appear  In b o t h ,  the frequency of  to be r e a l , and i s  the phase s p e c t r a d i s c u s s e d  associated  however,  the f i r s t h a r -  The h i g h - f r e q u e n c y r e g i o n of r u n 4a i s marred by going  i s not c o n s i d e r e d  negative;  w i t h the blowup of  on p. 121.  The comparative b e h a v i o u r of E ( f )  and f ( f )  i n runs 4a and 4b shows  w  some s i m i l a r i t i e s w i t h those of runs 2a and 2b.  I n both r u n 2 and i n  run 4 the wind speed dropped between the two s e c t i o n s of  the r u n s ,  the r e s u l t  at  that  the s p e c t r a show the l a r g e s t  j u s t above t h e i r peaks;  this  is  is  from 0.8  the r u n 4 s p e c t r a a t  strikingly different  d i s p a r i t y can be put down to the f a c t  saturated,  differences  i n runs 4a and 4b t h i s  On the o t h e r hand the behaviour of below t h e i r peaks  advance w i t h i t ,  E(f)  to 1.2 Hz.  frequencies  from t h a t observed i n r u n 2; t h a t the waves i n r u n 2 were  Hz i n run 4a to 0.5 Hz i n 4b.  showing the most s p e c t a c u l a r  E(f)  the water s u r f a c e was u n r u f f l e d f o r two t h i r d s of to see what E ( f )  and ^ ( f )  change.  s c a l e as the o t h e r s  is  the r e s o l u t i o n of  conditions  p l o t t e d on the same  (with the e x c e p t i o n of runs 4a and 4b)  good q u a l i t a t i v e p i c t u r e of the f u l l  It  that  the r u n , and the  would be when p h y s i c a l  p r e c l u d e d the p o s s i b i l i t y of wave g e n e r a t i o n .  the  and TTw(f)  Run 5 was taken as a "noise r u n " ; the wind speed was so low  i n t e n t i o n was  with  frequencies  w h i l e i n run 4 the wave spectrum was growing r a p i d l y ,  peak advancing from 0.6  they  the system.  frequency band where waves e x i s t e d damping i s  and g i v e s a F o r almost  indicated.  The  129 " t i g h t n e s s " of E ( f )  and * C ( f ) a t h i g h f r e q u e n c i e s , w  s u r p r i s i n g h a v i n g seen the phase spectrum of  ( F i g u r e 30)  (Figure 31).  The s m a l l p o s i t i v e  above 0.8  of E ( f )  Hz are c o n s i d e r e d to be n o i s e .  is  the p r e s s u r e and wave  compared w i t h those from, values  first  the r u n ( F i g u r e 38),  e x p l a i n e d by n o t i n g the r e l a t i v e l y s m a l l s i z e of power s p e c t r a  a l t h o u g h at  for instance,  and t ( f )  run 6  occurring  w  Note i n T a b l e 6.5  that  the  s t r e s s d u r i n g run 5 from the s o n i c anemometer agrees c l o s e l y w i t h  that  measured by the buoy. The most s t r i k i n g f e a t u r e spectrum.  E(f)  fetch for this  and T ( f ) , w  the wave The  and l a r g e energy and momentum f l u x e s might  w i t h s m a l l waves i n such a s i t u a t i o n .  j u x t a p o s i t i o n of  l a r g e E and  case r e a s o n a b l e ,  a d d i t i o n a l credence i s  c a l i b r a t i o n of  the s m a l l s i z e of  on the other hand, are f a r from s m a l l .  r u n was s h o r t ,  w e l l be a s s o c i a t e d  of r u n 6 i s  "f  the wave s e n s i n g  w  Since  s p e c t r a w i t h s m a l l waves i s  system,  been c a r e f u l l y s c r u t i n i s e d because of  lent  to  the  in  the a c c u r a c y of  this the  which i n the case of run 6 has  the s m a l l observed wave spectrum.  The s t r e s s computed from the i n t e g r a l under the spectrum shown as f o r runs 4a and b , E q u a t i o n 6.1  somewhat  or i n t h i s  w  hand,  (35% here)  level.  p at  spectrum from runs 1,  exceed those computed from E q u a t i o n 6 . 1 .  2,  and 3,  on the  2,  T h i s evidence  and 3 and i s much g r e a t e r suggests t h a t as  other  These runs d i f f e r the wave phase  the spectrum to the mean wind speed U 5 ; U5 - c  the peak of  zero f o r runs 1,  anemometer  The s t r e s s e s computed from the  p r o m i n e n t l y from runs 4 and 6 i n the r e l a t i o n of C  than t h a t computed from  case from t h a t measured by the s o n i c  1.75 m above the mean water i n t e g r a l under the t ( f )  lower  is,  most  velocity p  is  near  than zero f o r runs 4 and 6.  the wind speed r e l a t i v e  to the waves  130 increases  an i n c r e a s i n g p r o p o r t i o n of the momentum i n p u t to the water  goes not to the waves but to some o t h e r momentum s i n k , C e r t a i n l y from T a b l e 6.5 f r a c t i o n of  (U - c)  6.5.3  The l a s t paragraph i n d i c a t e s  fractional  i n d i c a t e t h a t a dominant  that  the f r a c t i o n decreases  increases.  T  The S p e c t r a of Miles  currents.  the t o t a l wind s t r e s s over water goes i n i t i a l l y d i r e c t l y  i n t o waves. as  the buoy measurements  such as  (1957) d e f i n e s  £  as the n e g a t i v e  damping r a t i o , or  the  growth r a t e i n mean wave energy per r a d i a n :  T  =  - L = CO E  - | E 9  .  6  .  7  t  S i n c e the wave s p e c t r a and E s p e c t r a are a v a i l a b l e ,  C are  s p e c t r a of  computed from f  (  f  )  =  Q rZJ ^. u  6.8  £  where use has been made of E q u a t i o n s 5.3 p o i n t s must be made:  first,  (p.  82)  and 5.9  the presence of s p i k e s  (p.  87).  and o f l a r g e  i n the p r e s s u r e s i g n a l a l l i n t r o d u c e low-frequency n o i s e  i n the  quadrature spectrum, but t h e r e i s no c o r r e s p o n d i n g n o i s e  in  power spectrum of  the wave s i g n a l .  the l o w - f r e q u e n c y  f  for  spectra;  the o t h e r s p e c t r a ,  the  Two drifts  p  ^^(f),  T h i s causes l a r g e f l u c t u a t i o n s  these are not c o n s i d e r e d r e a l .  the in  Second,  If s p e c t r a above 2 Hz (CJ > 13 r a d / s e c )  almost e n t i r e l y composed of n o i s e ,  g  as  are  and are not shown.  A l s o i n c l u d e d on the graphs are  f  s p e c t r a c a l c u l a t e d from  the  formula  (sc  ~  (, -u^W / cc  Cw  -  1) L)  6.9,  where c i s  the wave phase v e l o c i t y  h e i g h t one wavelength  and  is  the mean wind speed at a  above the water s u r f a c e ,  l o g a r i t h m i c p r o f i l e assuming Cp  =  .0012.  e x t r a p o l a t e d from a  T h i s formula i s  one proposed by Snyder and Cox (1967) which f i t s  an e m p i r i c a l  t h e i r wave growth d a t a  and those of B a r n e t t and W l l k e r s o n (1967) moderately w e l l . The s p e c t r a are p r e s e n t e d , The f i r s t  t h i n g to note i s  except f o r r u n 5,  the s i m i l a r i t y of  i n F i g u r e s 48  to  the p r e s e n t r e s u l t s  with  Snyder and Cox's e m p i r i c a l curves i n the frequency range from the of  a c t i v e wave g e n e r a t i o n to u = 10 r a d / s e c .  where they cross f a c t o r of results  two.  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 the  present  f a l l i n g below those computed from the Snyder and Cox r e l a t i o n .  valent regions Wilkerson.  of  a l t h o u g h not n e a r l y so marked,  In the case of  by the buoy w i t h both In runs 3 and 4, frequencies.  shows up i n the e q u i -  the s p e c t r a shown by Snyder and Cox and Barnettand  e x p l a i n e d as the e f f e c t  <  onset  except near  They b e g i n to d i v e r g e a t h i g h e r f r e q u e n c i e s ,  A s i m i l a r tendency,  all  The c u r v e s ,  55.  the p r e s e n t d a t a ,  the f a l l o f f  on the pressure-waves  is probably  coherence of  signals. the d a t a f a l l  below Snyder and Cox's p r e d i c t i o n at  These r u n s , from t h e i r s m a l l c r i t i c a l h e i g h t s  10 cm), were taken w h i l e the wave s p e c t r a were s t i l l  g e n e r a t i o n process was not s t a t i o n a r y or homogeneous. the d a t a f a l l  (all  growing;  The f a c t  the  that  below those of Snyder and Cox, which were taken i n a more  or l e s s s a t u r a t e d wave f i e l d ,  indicates  the p o s s i b i l i t y t h a t the  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 the wave f i e l d approaches Run 6 i s  interference  a special  genera-  (higher E f o r a g i v e n E) as  saturation.  case;  i t has a low c r i t i c a l h e i g h t  (1.2  cm), but  132 t  f a l l s w e l l above  ^  s  km as compared w i t h 6.7 runs.  c  Thus,  although E is  l e a d i n g to the h i g h  £  than f o r the  approached and some f a i r l y  the s h o r e ,  (1.6  other  steep wooded  f u r t h e r r e d u c i n g the e f f e c t i v e  large since U - c is  The o t h e r runs (1, active  case the f e t c h was much l e s s  i n the d i r e c t i o n from which the wind was blow-  the shore i s  30 meters h i g h s h e l t e r  In this  and 2.5 km; see F i g u r e 2)  A l s o , water depths  i n g s h o a l as  -  large,  cliffs fetch.  E is r e l a t i v e l y small,  values. 2) g e n e r a l l y show v a l u e s  g e n e r a t i o n which exceed  these runs the wave spectrum i s  £  s  c  -  of  C  i n the r e g i o n , of  T h i s can be expected.  Since  s a t u r a t e d or even o v e r s a t u r a t e d  in  (U,. < C p ) ,  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 B a r n e t t and W i l k e r s o n .  These i n v e s t i g a t o r s  p o i n t as i n the p r e s e n t  experiment,  measured not energy i n p u t at one but wave growth w i t h i n c r e a s i n g  t h e i r s p e c t r a t h e r e f o r e do not i n c l u d e t h a t p o r t i o n of  the energy i n p u t  which passes from the waves to a l l other sources v i a b r e a k i n g , w h i l e the spectrum i s b e i n g formed.  fetch;  etc.  SECTION 7:  7.1  Introduction  7.2  Recent Attempts  DISCUSSION OF RESULTS  133  7.2.1  Miles  7.2.2  Stewart  to E x p l a i n Wave G e n e r a t i o n  133  (1967)  133  (1967)  135  7.3  The Power S p e c t r a  139  7.4  The Phase S p e c t r a  141  7.5  The Energy and Momentum F l u x S p e c t r a  144  7.5.1  The Mean Energy F l u x  145  7.5.2  The Wave-Supported Wind S t r e s s  146  7.5.3.  The Energy and Momentum F l u x S p e c t r a 7.5.3a  . . . .  149  Observed and P r e d i c t e d T r a n s i t i o n s Fetches  7.5.3b 7.6  150  Energy T r a n s f e r i n the Wave Spectrum  .  152  The f  Spectra  7.6.1  Mean Values of  C  155  7.6.2.  The S p e c t r a of  f  156  7.6.3  A Dimensionless  R e l a t i o n between  Wind Speed  155  { (f)  and 159  SECTION 7:  7.1  DISCUSSION OF RESULTS  Introduction The r e s u l t s  pany.  presented i n the l a s t  s e c t i o n are i n v e r y sparse com-  In f a c t they are the o n l y ( t w o - d i m e n s i o n a l l y ) ' E u l e r i a n p r e s s u r e  measurements made on the s u r f a c e of n a t u r a l l y generated o n l y comparable measurements  are those of Shendim and Hsu (1967),  m e c h a n i c a l l y generated waves i n a wind tunnel tions", in  p. 33 f f . ) .  set  will  enable  models  7.2  relate  theorists  to b e t t e r  over  ( d e s c r i b e d i n "Observa-  s i n u s o i d f o r a wave) more c l o s e l y  out as s i m p l i f i c a t i o n s  the p r e s e n t r e s u l t s  the  Whereas the wind t u n n e l work may i n some ways  the use of the s i n g l e  the c o n d i t i o n s  sea waves;  (i.e.  approximate  i n p a r t i c u l a r i d e a l i z e d models,  d i r e c t l y to r e a l i t y ,  to a p p l y the necessary  and thus i t  is  simplifications  to  hoped, their  effect.  Recent Attempts to E x p l a i n Wave G e n e r a t i o n After  the works of Snyder and Cox (1966) and B a r n e t t and W i l k e r s o n  (1967) appeared and s t r o n g doubts i n the e f f i c a c y  of M i l e s '  inviscid  model were aroused, a l t e r n a t i v e mechanisms were c o n s i d e r e d f o r energy i n t o the waves.  F o r the sake of completeness,  to be r e l e v a n t to the p r e s e n t r e s u l t s  getting  those which appear  w i l l be d i s c u s s e d  i n the  following  paragraphs.  7.2.1  M i l e s (1967) Miles  (1967) reviews  model, and i s of  the evidence f o r and a g a i n s t h i s  inviscid  f o r c e d to r e c o n s i d e r the e f f e c t s on the momentum t r a n s f e r  wave-induced p e r t u r b a t i o n s of 133  the t u r b u l e n t Reynolds s t r e s s  -f uw. a  134 He suggests t h a t importance as  the r o l e of  the Reynolds s t r e s s e s may i n c r e a s e  the time s c a l e of  time s c a l e of a g i v e n s p e c t r a l  the  turbulence increases  relative  component i n the wave f i e l d .  suggests the use of a d i m e n s i o n l e s s  time-scale  in  He  to  the  therefore  parameter  1/kc  7.1,  where the s l o p e of height  z  c  the mean v e l o c i t y  f o r waves of wavenumber k,  and a l o g a r i t h m i c wind p r o f i l e i s Miles of  the  He r e v e r t s  evaluated  the r o l e p l a y e d by these  terms i n the momentum e q u a t i o n s ;  induced p e r t u r b a t i o n s of Equation 4.3.23,  l a t i o n of here).  the f i r s t  He o b t a i n s  is  A l l variables i n t h i s way two  c o n s i s t s of  the v e r t i c a l and the  vorti-  ( c o r r e s p o n d i n g to momentum t r a n s f e r v i a and the second r e p r e s e n t s  the Reynolds s t r e s s a t  which i s  the wave-  the sea s u r f a c e  shown by M i l e s to be e q u i v a l e n t  the momentum e q u a t i o n s ,  The f i r s t of  the  time r e t a i n i n g  the c o v a r i a n c e between the v e r t i c a l v e l o c i t y  the i n v i s c i d laminar mechanism),  1966;  this  t u r b u l e n t Reynolds s t r e s s e s .  c i t y a l o n g a flow s t r e a m l i n e  effects  to h i s o r i g i n a l (1957) f o r m u l a t i o n of  are averaged i n the crosswind d i r e c t i o n .  i n t e g r a l of  the c r i t i c a l  K (en 0.4) iis von Karman's c o n s t a n t ,  e q u a t i o n f o r the momentum f l u x from a i r to waves, the terms c o n t a i n i n g the  at  assumed.  then goes on to e l u c i d a t e  turbulence.  p r o f i l e is  the one under  to h i s  p e r t u r b a t i o n Reynolds s t r e s s e s .  formu-  discussion  these two terms i n the momentum equations  to the second by an e q u a t i o n f o r the a d v e c t i o n of  (Phillips,  is  linked  the v o r t i c i t y by the  M i l e s remarks t h a t the f i n d i n g of  135 Phillips  (1966) t h a t  the second  term can be n e g l e c t e d  and suggests t h a t f u r t h e r t h e o r e t i c a l  progress  is  questionable,  i s b a r r e d u n t i l more  e x p e r i m e n t a l i n f o r m a t i o n becomes a v a i l a b l e on the Reynolds s t r e s s t r i b u t i o n at  7.2.2  the s u r f a c e of  Stewart Stewart  the  water.  (1967)  (1967),  i n a review paper, d i s c u s s e s s e v e r a l  mechanisms which had not up to t h a t The f i r s t  considers  fer v i a Miles'  the e f f e c t s  c u r v a t u r e to the s l o p e of a l l evaluated  equals  the wave speed.  literature.  the t u r b u l e n c e on the momentum t r a n s The amount of momentum  p r o p o r t i o n a l both to the r a t i o of  the wind p r o f i l e ,  and to  at the c r i t i c a l h e i g h t  the  the rms v e r t i c a l  velo-  z^ where the mean wind 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  very nonlinear functions i n z^,  of  nonlinear  time been mentioned i n the  i n v i s c i d laminar mechanism.  t r a n s f e r by t h i s mechanism i s  city,  dis-  of  z^  can cause the i n c r e a s e s  the  t u r b u l e n c e , by c a u s i n g l o c a l  variations  i n momentum t r a n s f e r d u r i n g wind gusts  g r e a t l y exceed the decreases a s s o c i a t e d  with l u l l s ,  are  thus g r e a t l y  to  increas-  i n g the mean momentum t r a n s f e r over t h a t which would be computed by assumi n g a mean z^ which i s He proposed that  averaged over many  there may be d i f f e r e n t  depending on the s i z e of wave h e i g h t . tial  falloff  model of  the r a t i o L of  This idea is of  wavelengths.  the  regimes  of  generation,  c r i t i c a l height  connected w i t h h i s arguments  that  the  s t r e a m l i n e amplitude w i t h h e i g h t used by M i l e s as  the a i r f l o w i s p r o b a b l y u n r e a l i s t i c i n the l i g h t o f  s p e c t r a of  to the rms exponenhis  observed  t u r b u l e n t v e l o c i t i e s above sea waves, which show no obvious  136 wave peaks.  T h i s leads him to propose that  in fact  the s t r e a m l i n e  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 Phillips  (1966; p. 91)  but i n s t e a d ,  w i t h the phase speed of  the waves,  the "cats-eye" a s s o c i a t e d  surface.  i n the troughs  This configuration requires  pressure gradients,  and are thus  pressures  to windward of  low p r e s s u r e s  the wave c r e s t s ,  b u t i o n which generates waves.  If  of normal p r e s s u r e s be measured as w e l l ,  alone;  i n regions  the g r e a t d i f f i c u l t i e s  appears  that  the e x i s t e n c e of  The d i s t r i b u t i o n of  which i s  distri-  operative,  surface.  t h i s mechanism would be h a r d to  In  on the wave as i f  it  detect.  considered,  it  is  the s t r e s s d i s t r i b u t i o n If  the momentum  found t h a t the d i v e r g e n c e  caused by the s t r e s s d i s t r i b u t i o n induces v a r i a t i o n s A  to  the s h e a r i n g s t r e s s a l o n g the wave suggests  l a y e r i n the water.  thickness  indeed  the  i n v o l v e d i n making such measurements,  w i l l be f e l t by a t h i n s u r f a c e  the flow i n i t  exactly  v e r y c l o s e to the water  The e f f e c t s of  layer is  This  i n v i s c i d mechanism by measurement  another mechanism to Stewart.  i n this  streamline  the shear s t r e s s d i s t r i b u t i o n would have  view of  wave.  a l o n g the wave  to leeward and h i g h  t h i s mechanism i s  i t would not be s e p a r a b l e from M i l e s '  its  not o n l y of  low i n the troughs and h i g h on the c r e s t s .  i n t u r n l e a d s to a d i s t r i b u t i o n of  flow  i n s t e a d of  the presence  but a l s o of shear s t r e s s g r a d i e n t s  with  The s t r e s s e s must be arranged to produce the g i v e n  pattern,  balance  (1962) or  i n the c o o r d i n a t e system moving  around the c r i t i c a l h e i g h t may be s i t u a t e d over the c r e s t s .  con-  a l o n g the wave p r o f i l e . they were v a r i a t i o n s  These v a r i a t i o n s  A  in act  A i n the p r e s s u r e on the  With the s t r e s s d i s t r i b u t i o n d e s c r i b e d above t h i s  lags the waves by 9 0 ° , and thus r e p r e s e n t s  in  of  generation.  extra Stewart  pressure calculates  137 t h a t the v a l u e f o r the momentum t r a n s p o r t a s s o c i a t e d w i t h the  latter  mechanism i s k a T o ,  where k i s wave number, a i s wave a m p l i t u d e , and tT  is  the s h e a r i n g s t r e s s ;  the amplitude of  0  by L o n g u e t - H i g g i n s (1969a) t h a t t h i s a f r a c t i o n ka/4  (2:  0.05  t r a n s f e r from a i r to sea; this  i t has s i n c e been p o i n t e d out  should be ka t " o / 2 .  This  f o r t y p i c a l sea waves) of the t o t a l momentum f o r s h o r t wavelength waves w i t h l a r g e  f r a c t i o n might become c o n s i d e r a b l y l a r g e r than the  f i g u r e of 5%, which i s  represents  taken as  slopes,  above-mentioned  that l i k e l y f o r n o r m a l l y observed  slopes  i n a w i n d - d r i v e n sea. Stewart then goes on to suggest a f i n a l be added s e l e c t i v e l y  possibility.  Momentum can  to the c r e s t s of l o n g waves v i a the b r e a k i n g there  of steep s h o r t - w a v e l e n g t h waves, which would presumably pass 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 further its  that neither  of  the long waves.  He notes  t h i s mechanism nor the p r e c e d i n g one depends  a c t i o n on normal p r e s s u r e s ,  measurements  of  their  these  and hence n e i t h e r would be observed by  pressures.  O b s e r v a t i o n s on wave growth which are c o n s i d e r e d to have been i n a regime i n some sense s i m i l a r to that o b t a i n i n g i n the p r e s e n t of measurements have been reviewed e a r l i e r ("Observations", As was p o i n t e d o u t ,  they give  c o n f l i c t i n g evidence;  ments g e n e r a l l y show l a r g e r d i s c r e p a n c i e s than those i n wind t u n n e l s .  Miles,  fer,  17  set ff).  from M i l e s '  i n v i s c i d theory  as mentioned e a r l i e r (p.  134), has  i n the  field  of the t u r b u l e n t Reynolds s t r e s s e s on the momentum t r a n s -  as r e p r e s e n t e d by the largeness  Equation 7.1.  p.  taken  the f i e l d measure-  suggested t h a t t h i s may be due to the i n c r e a s e d e f f e c t s measurements  for  (Stewart  of the parameter TV  g i v e n by  (1967) has suggested i n t u r n t h a t the type of  138 momentum t r a n s f e r regime o p e r a t i n g i n a g i v e n s i t u a t i o n may be governed by the s i z e of  = c (V?)'  L  Phillips of  (1966) d e s c r i b e s  the regimes o f wave g e n e r a t i o n i n terms  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 .  cludes of  7.2.  k  z  t h a t three s e p a r a t e mechanisms are o p e r a t i v e f o r d i f f e r e n t  c/u, ; for c/u* ^ v  5 ( a l t h o u g h he does not s p e c i f i c a l l y say so)  mechanism must become c o n t r o l l e d by v i s c o s i t y model);  20 the i n t e r a c t i o n of  w i t h the a i r flow over the waves i s r a t i o s : J\.  values the  ( M i l e s ' v i s c o u s laminar  f o r 10 < c / u * ^ 20 the i n v i s c i d laminar model i s  w h i l e f o r c/u^y  7.1.  He con-  operative,  the t u r b u l e n t Reynolds s t r e s s e s  assumed to be dominant.  A l l of  the  , L , and c / u ^ , are d i s p l a y e d f o r the p r e s e n t runs i n T a b l e  A l l values  are taken a t the l o c a l l y - g e n e r a t e d peak of  the  relevant  wave spectrum. From the t a b l e ,  it  is  c l e a r t h a t _A-  and L are dominated by z ; c  they are w i t h the e x c e p t i o n of run 6 n e a r l y the same i n s i z e . moreover t h a t the p r e s e n t measurements  cover a l a r g e range of  They show values,  and thus should p r o v i d e i n f o r m a t i o n on more than one wave g e n e r a t i o n regime. viscous likely  The range of v a l u e s laminar,  of c/u^. suggest t h a t a l l three  i n v i s c i d laminar,  to be p r e s e n t i n the r u n s .  and Reynolds s t r e s s - d o m i n a t e d , It  is  range of v a r i a t i o n of C p / u ^ (where Cp i s peak of  the wave spectrum)  regimes:  interesting  are  that a l t h o u g h the  the wave phase v e l o c i t y at  i s v e r y much l e s s  than f o r .A.  and L , the  three parameters, which are a l l d e r i v e d from wave c h a r a c t e r i s t i c s a t s p e c t r a l peak, are a l l arranged i n the same o r d e r .  the  the  TABLE 7.1  u  C 5  Run No. Notes:  U  P  cm/sec (1)  (On racl/ sec  / *  (2)  (2)  "c cm  cm  (2)  A  (3)  (4)  (>10 )  (>10 )  5  150  5.,2  332  (64)  2..95  10  4  3.,4  1  220  7..6  370  49  2.,65  10  4  4. ,2  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  6  570  19..6  270  13.8  3..63  1.2  2..4  4b  700  24..1  305  12.6  3.,21  0.7  4. ,6  4a  800  27..6  265  9.6  3..70  0.2  4, ,8  Notes:  1.  u,  ( C U ) ^ i s c a l c u l a t e d as U / 2 9  2.  S u b s c r i p t "p" means  2  D  5  3  no  3  no  3  3  1.9  1.5 9 x 10"  2  5 x 10"  1  3.7 x 1 0  - 2  1.5 x 10"  1  1.1 x 1 0  - 2  4.2 x 10"  2  (i.e.  C = .0012). D  ' e v a l u a t e d a t the peak of the wave  spectrum'. 3. 4. 5.  A  L  =  0.4  CJ Z /u*. P  c  = z /p?)%. c  In r u n 5, the v a l u e s f o r c / u * , A . realistic,  since swell  wave energy p r e s e n t .  , and L may not be  comprises the major p a r t of the  139 7.3  The Power S p e c t r a The p r e s s u r e power s p e c t r a are b e s t c h a r a c t e r i s e d as r i s i n g mono-  t o n i c a l l y w i t h d e c r e a s i n g f r e q u e n c y , w i t h the p r i n c i p a l 5 Hz range h a v i n g a wave-induced "hump".  T h i s hump i s  p a r t of  the 0.1  -  superimposed on  what w i l l be c a l l e d the "basic" p r e s s u r e spectrum: t h a t found over sand a t Boundary Bay; see F i g u r e s 81 and 82. found to s c a l e as w e l l as available f i e l d results results  c o u l d be expected  always  the t h i c k n e s s  i n accord with r e s u l t s  very w e l l ,  for instance,  of  best  the wind t u n n e l  The p r i n c i p a l  uncertainty  the atmospheric t u r b u l e n t  satisfactory  on the t h i c k n e s s  The monotonic r i s e of  of  and to  to the  the wind t u n n e l boundary l a y e r s  there seems to be l i t t l e  from i n f e r e n c e s  pp.199ff)  (those of P r i e s t l e y 1965)  boundary l a y e r ; w h i l e t h a t of  is  (see  of W i l l m a r t h and Wooldridge (1962).  i n the s c a l i n g i s  known,  The Boundary Bay s p e c t r a were  is  accurately  d a t a b e s i d e s those o b t a i n e d  the c o r r e s p o n d i n g atmospheric l a y e r .  the p r e s s u r e s p e c t r a as  the frequency decreases  from many o t h e r i n v e s t i g a t i o n s ;  it  shows up  i n the p r e s s u r e s p e c t r a g i v e n by Gossard (1960).  T h i s f a c t makes i t hard to s p e c i f y  a value for p ,  the mean square  2  pressure  fluctuations,  as has been done i n the past by P h i l l i p s (1957), Longuet-  H i g g i n s et a l  (1963),  and K o l e s n i k o v and Efimov  (1962).  The power s p e c t r a measured u s i n g the buoy are v e r y n o i s y a t frequencies. when s c a l e d For t h i s  A t these f r e q u e n c i e s to the r e s u l t s  they are h i g h by f a c t o r s  this  p o r t i o n of  i s not c a r r i e d beyond the above simple s c a l i n g .  boundary l a y e r ,  (see  are b e t t e r  of 2 -  10  of P r i e s t l e y or W i l l m a r t h and W o o l d r i d g e .  reason the d i s c u s s i o n of  Boundary Bay r e s u l t s  low  Appendix 2 ) ,  the p r e s s u r e It  is  felt  spectra  that  the  a l t h o u g h taken i n a d i f f e r e n t  representations  of  the l i k e l y form of  the  140 p r e s s u r e spectrum a t f r e q u e n c i e s At higher frequencies, of  the waves,  l e s s than about 0.2 Hz.  the power s p e c t r a are dominated by the  i n the sense t h a t i t  which determines  their slope.  more or l e s s continuous s l o p e ,  is  the wave-induced p a r t of  the present  s p e c t r a d i s p l a y a "hump";  line  p o r t i o n of  to the h i g h - f r e q u e n c y  the spectrum to the asymptote  j o i n i n g the  The seemingly h i g h s p e c t r a l e s t i m a t e s at low f r e q u e n c i e s , the p r e s s u r e s i g n a l w i t h s p i k e s  it  assuming the base  spectrum to have the slope of a s t r a i g h t  c r i b e d i n Appendix 1) i n d i c a t e  the spectrum  Whereas the Boundary Bay s p e c t r a show a  i s v e r y tempting to t r y to i n t e g r a t e under i t ,  t a m i n a t i o n of  effects  low-frequency "tail".  and the con-  at high frequencies  (des-  that any such attempt would not prove  fruitful. The p r i n c i p a l f i n d i n g p o i n t e d out by the p + f q^ s p e c t r a i s a  l a r g e p a r t of  t h i s p r e s s u r e which i s  coherent w i t h the waves.  to be c o n t r a s t e d w i t h the o b s e r v a t i o n s even a t r e l a t i v e l y s m a l l v a l u e s  show l i t t l e  might be expected  by Stewart  of kz (where k i s  waves and z the anemometer h e i g h t ) , ations  discussed  or no evidence to be the case;  of  s p e c t r a of  the  This  (1967),  the wave number of  turbulent v e l o c i t y  the e f f e c t of  the waves.  is that the  fluctu-  This  the rms v e l o c i t y f l u c t u a t i o n s  associ-  ated w i t h the wave motion observed by a probe even a t r e l a t i v e l y s m a l l values  of kz are of o r d e r ka-v/Uu, compared w i t h V u  fluctuations,  the  eu  f^^a  turbulent However  due to the waves a t the same v a l u e of kz should  (Uu) 2. whereas  those due to the t u r b u l e n c e are of order  2 , which i s  a t l e a s t an o r d e r of magnitude s m a l l e r .  the t u r b u l e n t v e l o c i t y waves,  f o r the  two b e i n g about equal i n t y p i c a l c o n d i t i o n s .  the p r e s s u r e f l u c t u a t i o n s be of o r d e r  2  fluctuations  effectively  the wave-induced p r e s s u r e f l u c t u a t i o n s  Thus even i f  mask those caused by the  could s t i l l exceed those  caused  141 by t u r b u l e n c e . One f u r t h e r f a c t should be mentioned.  As f a r as can be determined  from the s p e c t r a (which w i t h the important e x c e p t i o n of r u n 2a are contaminated by the e f f e c t s the e f f e c t s peak of  of  o f s p i k e s beyond 2 H z ) , they are m o d i f i e d by  the wave g e n e r a t i o n process  the l o c a l l y - g e n e r a t e d wave spectrum.  f o r which the i n f l u e n c e of s p i k e s wave g e n e r a t i o n process  w i t h dynamic p r e s s u r e s  (see  p. 91  above  the  In the case of run 2a,  can be i g n o r e d ,  appears to extend  beyond which p r e s s u r e measurements  7.4  at a l l f r e q u e n c i e s  the i n f l u e n c e  to 2 Hz, which i s  p r o b a b l y are g r o s s l y  of  the  the  limit  contaminated  ff).  The Phase S p e c t r a There are two phase s p e c t r a p r e s e n t e d :  wave e l e v a t i o n ,  that of p r e s s u r e  and t h a t of p r e s s u r e p l u s s t a t i c head  p 3' 2  F o r comparison w i t h theory the obvious  former,  i s much l e s s prone to e r r o r  it  a t the same time the r e s u l t s p + p^g i£  .  For this  reason both are d i s c u s s e d ,  the comparison, f o r 0.6  (1963),  and the  the  section);  b e a r i n g i n mind  phase r e s u l t s .  Table 7.2  the displays  i n v i s c i d laminar model w i t h a p p r o p r i a t e  f o r the wind p r o f i l e . which i s  the R e s u l t s  is  Hz waves f o r a l l r u n s , between measured phases  and those p r e d i c t e d from M i l e s '  Equation 4.1,  choice  of Shemdin and Hsu are presented i n terms of  l a r g e r e r r o r s i m p l i c i t i n the p + fjj'g  assumptions  (see  versus  )  a  wave e l e v a t i o n . since  versus  The p,>^  phases are computed from  i d e n t i c a l w i t h that used by L o n g u e t - H i g g i n s e_t _al  (p + ^ 3 ^ ) ) ^  phases from Q  are from M i l e s (1959a, F i g u r e s 4 and 5 ) . the runs d i v i d e themselves i n t o  It  three groups.  = tan  t  is  - 1  ft la.; <X and (5  immediately seen For the f i r s t  that  group  TABLE  7.2  Comparison of Observed Phase Angle of P r e s s u r e R e l a t i v e Waves a t 0.6 Miles'  Run No.  u  5  cm sec  _I  Hz, w i t h That C a l c u l a t e d from I n v i s c i d Laminar Model  et  c/u*  0  degrees tes:  to  (2)  (1)  (3) 10-2  8,  t  degrees  (4)  (3)  (4)  -180  mi  -179.7  -  1  220  34  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  10  -179.3  -174  -145  -152  6  570  13.3  -174.5  -131  -156  -111  4b  700  10.8  -173.5  -156  -163  -146  4a  800  9.4  -172.5  -152  -166  -142  Notes:  1. 2.  1 .5 x  10"  3  10-3 7 .5 x  c = 260 cm sec "1, n  - 3  10"  4  the ph ase  = §z / i = 0.565/U , 2  u  0  assumed,  v e l o c i t y of a 0.6 Hz wave.  where a l o g a r i t h m i c p r o f i l e i s  U-^ = 2.5 U ^ , and Z  Q  = 3.6  x 10-3  cm ( e q u i v a l e n t  assuming C^ = .0012). 3.  6  t  is  the phase lead of p r e l a t i v e  e.  tan"  -P(Ul/c) 1 - A(U]/c) _ 2  2  see L o n g u e t - H i g g i n s e t a l C*. and |3  is  (1963), E q u a t i o n 53.  are o b t a i n e d from M i l e s (1959a), F i g u r e s 4,  For the phase of 0  to ->£ computed from  (p -  ^^'t  ^ relative  to  the c o r r e c t e d measured phase a n g l e ,  ~t 2 ° f o r the pretation",  c o r r e l a t i o n (see p.  104).  ,  6  and i s  t  5.  = t a n - l P/ c accurate  "Data A n a l y s i s and I n t e r -  to  142 c  / , * > 30 ( i n u  2J 10.  c/u*  results  r  u  n  1);  f o r the second,  Run 1 i s  c/u*  of p a r t i c u l a r i n t e r e s t  for  various values  0.6  rad/sec, about 0.9  1 of  of CO .  The peak of  for  >v  t h e i r data is  d a t a at CO = 3.8  frequency the measured phase i s (Tl0°)  by M i l e s '  of  the p r e s e n t  results  1 a t the frequency +54°.  about - 1 7 2 ° ,  l e s s than 1 ° .  50.4.  Hz).  3 rad/sec,  A t a frequency  The phase s h i f t  can be c o n s i d e r e d r e l i a b l e i s  t h e i r phase s h i f t s  are not i n c o n s i s t e n t  10;  -126°,  are s t i l l  errors.  The presence  the p r e s s u r e  i n measurement  expected  predicted  c / u ^ at which  a shift  from - 1 8 0 ° of  essentially  data,  zero.  ( c / u ^ 20,  The d i f f e r e n c e s  technique or  of dynamic p r e s s u r e s  this  the phase i n run  a t low f r e q u e n c i e s  but d i f f e r markedly f o r the h i g h e r f r e q u e n c i e s . be put down to d i f f e r e n c e s  :  F o r run 1 a t  The h i g h e s t v a l u e of  is  ^  and l i e s w i t h i n the  182°.  (2 Hz) where c / u * 2 i l 0  at  Vc  the two r e s u l t s  the  differences  A t the c o r r e s p o n d i n g frequency a t which c/u .fiil0 i n t h e i r  which i s  in  phase  their  about equal to t h a t of r u n  (f a* 0.6  rad/sec  t h e i r r e p o r t e d v a l u e of  theory i s  present  t h e i r wave spectrum i s  rad/sec,  third,  They show i n  theoretical  f o r a 960 cm/sec wind of  the p r e s e n t  error  (1963).  g i v i n g a c/u^ c/u  and f o r the  i n comparing the  w i t h those of L o n g u e t - H i g g i n s e t a l  T a b l e 4 a comparison of measured v e r s u s  of  20 - 30,  Thus say)  must  unaccounted-for  i n a n t i p h a s e w i t h the waves  s i g n a l from t h e i r buoy may account f o r a major p a r t of  difference. For  inviscid angles of  20 < c/u^< 30 (runs 2 and 3 i n T a b l e 7.2) laminar model p r e d i c t s phase s h i f t s 5 to 2 0 ° .  Phillips  it  t h a t f o r c/u >20 the  t i o n process must be dominated by the i n t e r a c t i o n of since  that  the  i n p which are too s m a l l by  (1966) proposes  w i t h the t u r b u l e n t Reynolds s t r e s s e s ,  appears  Vi  the wavy a i r  at these v a l u e s  of  c/u  generaflow  i(  the  143  c r i t i c a l height:'is h i g h , and so i s the mean wind p r o f i l e i s  small.  i n a r e g i o n where the c u r v a t u r e of 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 t u r b u l e n t Reynolds s t r e s s e s or by some o t h e r mechanism such as transfer  t h a t proposed by Stewart  to the waves i s  horizontal  caused by the b a l a n c e over a wavelength between  t u r b u l e n t shear s t r e s s e s and g r a d i e n t s  The runs f o r which 10 Miles'  (1967), whereby the momentum  <.  c/u^ <  20 (4a,  i n v i s c i d mechanism should dominate.  phase s h i f t s  of normal p r e s s u r e .  4b,  6)  are those i n which  I n these runs , the  observed  are about 20 to 40 degrees l a r g e r than p r e d i c t e d .  i n t o account the phase a c c u r a c y of the measurements  it  is  Taking  clear  that  the i n v i s c i d laminar model i s not the o n l y mechanism g e n e r a t i n g waves for  this  range of c / u ^ e i t h e r .  observed r a t e s factors  of f i v e  Ps + 6 "J?2 a  (p. 159)  that  of wave growth exceed those p r e d i c t e d from the theory by to  eight.  I n c l u d e d i n Table 7.2 °f  I t w i l l be seen l a t e r  with respect  is  a comparison between the observed phases  to ">£  and the c o r r e s p o n d i n g phases p r e d i c t e d  from M i l e s '  i n v i s c i d laminar model.  the r e s u l t s  from the p r e s e n t experiment may be compared w i t h those of  Shemdin and Hsu (1966). Hsu i s used f o r t h i s  T h i s comparison i s  i n s e r t e d so  (The e a r l i e r m a n u s c r i p t r e p o r t of Shemdin and  comparison s i n c e  some r e s u l t s  presented  in it  not r e a d i l y a v a i l a b l e elsewhere have s i n c e been r e f e r r e d to i n literature.)  that  Shemdin and Hsu make a comparison of c/u-^ = 0.4  but  the c/u^  versus measured (p + PJ^^ ),•*£ phase and that computed from M i l e s ' (1957)  theory.  The v a l u e s  of c / u ^ f o r t h e i r study range from 3.0  13.5.  For c/u  Ca 10 t h e i r measured phase s h i f t s  exceed the  p r e d i c t e d ones by 10 - 2 0 ° i n t h e i r "best" run (the r e s u l t s  to  theoretically from which  144 are shown i n Shemdin and Hsu (1967) i n t h e i r T a b l e 2) o n l y other run p r e s e n t e d .  The phase d i s c r e p a n c i e s  and by 3 0 ° i n the  from runs 4 a , 4b, and  6 range from 1 7 ° to 4 5 ° , and are thus not i n c o n s i s t e n t observed by Shemdin and Hsu i n t h e i r wind-wave The wind-wave t u n n e l r e s u l t s  w i t h those  tunnel.  f o r lower v a l u e s  of  c / u ^ can be com-  pared w i t h the p r e s e n t  d a t a a t h i g h e r wave f r e q u e n c i e s .  ranges  i n runs 4 a , 4b, and 6.  from 3.8  to 5.3  observed and t h e o r e t i c a l phase s h i f t s runs  (see  F i g u r e s 36,  37,  and 39);  A t 1.5 Hz c / u ^  The d i s c r e p a n c i e s  between  are 8 ° , 2 3 ° , and 1 7 ° f o r the  Shemdin and Hsu f i n d  three  discrepancies  r a n g i n g from 7 ° i n t h e i r "best" r u n to 4 7 ° i n the other run a t c / u ^  =  4.5. Because of  the l a r g e s c a t t e r  even l a r g e r s c a t t e r  i n those of Shemdin and Hsu, i t  p u r s u i n g the comparison f u r t h e r . great differences  average  It  is  evident  i s not worthwhile  t h a t i n s p i t e of  i n the c o n d i t i o n s of measurement of  t u n n e l work and the p r e s e n t consistent,  i n the p r e s e n t phases and the a p p a r e n t l y  the wind-wave  f i e l d work, the measured phases are not i n -  both e x h i b i t i n g s i g n i f i c a n t l y  2 0 ° i n T a b l e 7.2)  the  larger shifts  than those p r e d i c t e d by M i l e s '  from - 1 8 0 ° (they (1957,  1959a)  theory.  7.5  The Energy and Momentum F l u x S p e c t r a These s p e c t r a ,  shown i n F i g u r e s 40 to 47,  major c o n t r i b u t i o n s of be d i s c u s s e d  this  thesis.  r e p r e s e n t one of  In the f o l l o w i n g s e c t i o n  the  they w i l l  and compared where p o s s i b l e w i t h other measurements.  to be d i s c u s s e d w i l l be the i n t e g r a l s under the s p e c t r a ; f o l l o w e d by a c o n s i d e r a t i o n of the s p e c t r a  themselves.  this w i l l  First be  145 7.5.1  Mean Energy F l u x The v a l u e s  of E and  the waves and i t s functions  respectively  the i n t e g r a t e d energy f l u x  s t a n d a r d e r r o r of the mean over the r u n , are shown as  of mean wind speed i n T a b l e 7 . 3 .  A l s o i n s e r t e d are  values  o b t a i n e d by K o l e s n i k o v and Efimov (1962) from t h e i r f r e e - f l o a t i n g i n c o n d i t i o n s of a c t i v e wave g e n e r a t i o n . the "Run Number" column. these r e s u l t s  to  however  the o n l y energy f l u x measurements  (p.  they are to the w r i t e r ' s  22)  their  that  knowledge  comparable w i t h those presented  and so they are i n c l u d e d because of From the t a b l e i t  These are marked w i t h a KE i n  I t has a l r e a d y been p o i n t e d out  are suspect;  buoy  here,  uniqueness.  can be seen t h a t both s e t s of r e s u l t s  a b l e s c a t t e r when compared w i t h wind speed a l o n e .  show c o n s i d e r -  Nonetheless  those of  K o l e s n i k o v and Efimov are w i t h i n an o r d e r of magnitude of those found i n this  experiment.  Lack of r e p o r t e d wave data i n the R u s s i a n p u b l i c a t i o n  p r e c l u d e s any more o b j e c t i v e The s t a n d a r d e r r o r s of  comparison. the mean ( (J*]| ) f o r each run are a l s o  given  i n T a b l e 7.3;  they r e p r e s e n t  of  of E o b t a i n e d f o r each data b l o c k i n the r u n , and are com-  the v a l u e s  puted from E q u a t i o n 5 . 5 . of  <T~^,  are l a r g e ;  the expected v a r i a b i l i t y over a g i v e n r u n  I t w i l l be immediately seen that the  i n run 4b <T"|.  i s more than double the v a l u e of E ,  w h i l e i n g e n e r a l i t ranges from o n e - h a l f It  to about one times E .  should be noted t h a t two runs (4 and 5,  (J-g/E are l a r g e s t )  values  showed p o s i t i v e  f o r which the  evidence of a l a c k of  ratios  stationarity  ( 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  as s p a t i a l l y inhomogeneous,  e x c e p t i o n of r u n 1 ) .  w i t h the p o s s i b l e  considered Run 4  146 was taken i n two s e c t i o n s and a l l s p e c t r a f o r the two runs are cantly different, w i t h time.  i n d i c a t i n g a development of  The l a r g e s c a t t e r  signifi-  the wave and wind  fields  from b l o c k to b l o c k i n the energy and  momentum f l u x s p e c t r a appears to h i d e any trends w i t h i n runs 4a and 4b. Run 5 d e f i n i t e l y increase  shows such a t r e n d ; the wind speed was observed  from s t a r t  the course of  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 r u n .  The d i s t r i b u t i o n s of  the E v a l u e s  f o r each r u n , and compared u s i n g the  about t h e i r means have been  deviations  i n any of  No  although a v i s u a l kurtosis  them.  although i t  waves,  are taken to mean t h a t the wave g e n e r a t i o n p r o -  causes on the average a p o s i t i v e  energy i n p u t to  i s b a s i c a l l y a random process w i t h l a r g e s c a t t e r .  appears from t h i s  Further,  d a t a to be no evidence of s t r o n g " i n t e r m i t t e n c y "  the sense t h a t a l l of  the energy i n p u t occurs a t r e l a t i v e l y  i n t e r v a l s d u r i n g which l a r g e t r a n s f e r s  take p l a c e ;  cess should r a t h e r be c o n s i d e r e d as one which i s which at a g i v e n time i s  there in  infrequent  the g e n e r a t i o n  active  the  pro-  a l l the time but  e i t h e r enhanced or d i m i n i s h e d 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 p r e s s u r e and the waves. 7.5.2  dis-  statistically  the E d i s t r i b u t i o n s showed no obvious skewness o f  The above f i n d i n g s cess,  no  from the normal d i s t r i b u t i o n s were found.  attempt was made to compute h i g h e r - o r d e r moments, i n s p e c t i o n of  plotted  C h i - s q u a r e d t e s t a g a i n s t normal  t r i b u t i o n s w i t h the same means and standard d e v i a t i o n s ; significant  to  The Wave-Supported Wind One of  Stress  the most important questions  which must be answered  about  TABLE  7.3  Comparison of Observed Values of E w i t h Those Obtained by K o l e s n i k o v and Efimov  5 cm sec "-  E  U  i No.  -  (1)  Note:  1.  (1962)  erg  cm-^sec (2)  -1  1  dyne cm (2)  _o  (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 f o r K o l e s n i k o v and Efimov  (KE i n "Run No.")  e x t r a p o l a t e d assuming a l o g a r i t h m i c p r o f i l e w i t h Cp =  are  .0012  from g i v e n speeds at 2.5 m. 2.  E and Tvw are the mean i n t e g r a l s under the E ( f ) f o r each r u n ; ^"~"E and 6% are standard e r r o r s of the number of data b l o c k s done (always  ^  20).  and TTv/f)  spectra  the mean over  147 the wave g e n e r a t i o n process  is,  "What f r a c t i o n of  - £> Uur  e x e r t e d by the wind on the water  waves?".  Stewart  ft  (1961) c o n s i d e r e d  this  the Reynolds  s u r f a c e goes d i r e c t l y  question,  t h a t a lower l i m i t to the f r a c t i o n '^-nj/t, where =  - p^U-xw , i s  7T  W  about 0.2.  is  was wave momentum p r e s e n t  d i s s i p a t i o n mechanisms flux  to the waves.  TTw/tr  is  at a given f e t c h ,  i n f a c t o n l y a p a r t of  H i s c o n c l u s i o n was  the mean wavethis  i n determining  which because of wave the  t o t a l momentum  t h a t i t would be p o s s i b l e  for  to be n e a r l y one.  Phillips  (1966) estimated  T w / ' r  by computing "cX* from h i s v e r s i o n of includes  conclusion  He c o n s i d e r e d  to be a lower l i m i t because the parameter he e s t i m a t e d Tw  into  and by examining wave  data measured at known f e t c h e s and wind speeds came to the  supported s t r e s s and  stress  the e f f e c t s  of  f or the waves f o r which c > 5u,. the M i l e s - P h i l l i p s theory  the i n t e r a c t i o n of  (which  the t u r b u l e n t Reynolds  s t r e s s e s i n the a i r w i t h the flow over the waves) and by comparing t h i s computed v a l u e w i t h 4z 0 . 1 .  ^u^.  He found t h a t f o r these longer waves  S i n c e the waves c o n s i d e r e d by Stewart i n h i s  such t h a t c >  5u*, P h i l l i p s '  theoretical  which was based on o b s e r v a t i o n s .  analysis  finding contradicts  In view of  tw/'t* were a l l  Stewarts',  the o b s e r v a t i o n s  by Snyder  and Cox (1966) and B a r n e t t and W i l k e r s o n (1967) which show t h a t  observed  wave growth r a t e s exceed those p r e d i c t e d by the M i l e s - P h i l l i p s theory by about one o r d e r of magnitude, from the p r e d i c t i o n s  of  this  Phillips' result,  which was  calculated  t h e o r y , must be regarded as d o u b t f u l .  The average momentum i n p u t to the waves has been computed f o r the r u n s .  These v a l u e s  p l a y e d i n Table 7.4  for  "cXw are shown i n T a b l e 6.5;  as f r a c t i o n s  of  2 P^QU^,  they are  where C^ = .0012.  all  redisAlso  148 shown a r e v a l u e s  of c p / u . ,  and a "wave-drag c o e f f i c i e n t "  y  The mean C y f o r the runs i s .0014, and a l l v a l u e s D  combined s t a n d a r d d e v i a t i o n s (1967):  of the mean v a l u e s  .0010 + .0003, and W e i l e r and B u r l i n g  which were o b t a i n e d from d i r e c t measurements a l t h o u g h the observed s c a t t e r  of (1967):  a t the same  1  sight,  i t i s n o t l a r g e when compared w i t h other o b s e r v a t i o n s .  appears l a r g e a t f i r s t  5 and 6 - ^ U.ur was , measured w i t h a s o n i c anemometer, -  It  indicates  f o r the s t r e s s g i v e n by the buoy measurements  means t h a t the most d i r e c t evidence  The l a r g e e x p e r i m e n t a l u n c e r t a i n t i e s  p a r t i c u l a r l y i n the p r e s s u r e sensor e r r o r of ~J"207 i n t h i s o  = .0012  that  g i v e as  ^ U '.  from which the v a l u e of  be i n f e r r e d i s that of r u n 6, which i n d i c a t e s  It  and the observed  i s p a r t i c u l a r l y heartening.  a c c u r a t e a measure of 1£* as does the formula  0.8.  In runs  of the s t r e s s measured by the buoy w i t h the  s o n i c anemometer measurements t h a t the v a l u e s  5-  .0015 ± .0004,  Thus,  (see T a b l e 6.5)  u a  quoted by Smith  site.  agreement  ^  shown f a l l w i t h i n the  of - ^ U l t /  in  = f-^/  2  T  w  This / f  can  the r a t i o i s about  i n the buoy measurements,  c a l i b r a t i o n , i n d i c a t e an expected  figure.  thus becomes f a i r l y  c e r t a i n that S t e w a r t ' s  e s t i m a t e that a  l a r g e f r a c t i o n of the wind s t r e s s 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 c o r r e c t ; is  0.8.  Since t h i s  the v a l u e i n d i c a t e d f o r the f r a c t i o n  f i n d i n g disagrees  with that of P h i l l i p s , i t s t r o n g l y  suggests t h a t h i s t h e o r e t i c a l e s t i m a t e f o r order of magnitude, measurements  i n agreement w i t h the f i n d i n g s o f the wave growth  of Snyder and Cox, and B a r n e t t and W i l k e r s o n .  The r e s u l t o b t a i n e d here f o r * ? ^ , / f , 0.8, (0.1 - 0.4)  i s too low by about one  i s l a r g e r than t h a t  observed i n a r e c e n t study by Wu (1968).  H i s measurements  TABLE F r a c t i o n of  Run No.  y  tes:  the Wind S t r e s s  Supported by the Waves  c„/u p *  5  u  7.4  DW  U  (2)  (i)  (3)  (4)  1  1.7  49  1.6  .0018  2b  0.9  26  1.1  .0013  2a  0.9  25  1.6  .0019  3  0.7  19  1.1  .0013  6  0.5  13.8  0.8  .00095  4b  0.4  12.6  0.6  .00075  4a  0.3  9.6  0.7  .00083  Mean Values  1.1  .0014  Standard E r r o r of Means  0.4  .0005  Notes:  1.  Cp i s of  the wave spectrum.  2.  — 5/29 u  of 3.  the wave phase v e l o c i t y a t the l o c a l l y - g e n e r a t e d peak  , s  r;  (C  = .0012); U  D  5  i s mean a i r speed at a h e i g h t  5 meters. w  is  the mean s t r e s s measured by the buoy system;  c a l c u l a t e d from *7J" = P^CnU^* except C  "f  i n r u n 6, where T*  c  C  is is  r e p l a c e d w i t h the s t r e s s "~£ measured by the s o n i c anemometer. s  % = '  V  e u a  2 5  .  149 were made i n a wind-water t u n n e l over n a t u r a l l y - g e n e r a t e d waves, and Cp/u,v i n his t u n n e l never exceeded 7 . 5 , lowest Cp/u,v observed i n the p r e s e n t his results of  "Cw  a p p l y to a d i f f e r e n t  study.  surfaces  w i t h the same measured roughness  increase  i n wave momentum w i t h f e t c h .  S i n c e the p r e s e n t  He f i r s t  length;  values  results  took the  differ-  solid  he a l s o observed  The l a t t e r o b s e r v a t i o n was  the then  are i n terms of momentum f l u x to the waves and s i n c e  the c / u ^ v a l u e s  the d i s c r e p a n c y between the two e x p e r i m e n t a l v a l u e s  are  so  of  T* / ""/*• W  not c o n s i d e r e d to be an i m p o r t a n t one. It  is  interesting  i n Table 7.4.  to note the apparent decrease  :<<>If t h i s were r e a l i t would i n d i c a t e  t i o n was becoming more e f f i c i e n t i n c r e a s i n g Cp/U^; with s u f f i c i e n t all  H i s measured  that  the former.  not observed momentum i n c r e a s e ,  different, is  results  can be assumed  measured over waves and over  used to c o n f i r m the v a l i d i t y of  than the  n e i t h e r of which can g i v e  comparable w i t h those p r e s e n t e d h e r e .  ence between drag c o e f f i c i e n t s  and  Thus i t  g e n e r a t i o n regime.  were o b t a i n e d i n two ways,  strictly  a v a l u e somewhat l e s s  of  7.5.3  Tj"^ , however,  of Z^/^  that  c  with p / U ^ C  the wave genera-  a t a b s o r b i n g the wind s t r e s s w i t h is  computed from C^, which i s not known  a c c u r a c y ; i t may indeed be t h a t v a r i a t i o n s i n  the observed v a r i a t i o n i n X /'X VI  C  cause  '.  The Energy and Momentum F l u x S p e c t r a Three f r e q u e n c i e s  associated  s i n g l e d out f o r comment.  These are the f r e q u e n c i e s  generated maxima i n wave s p e c t r a ; spectra;  and f^,  w i t h the energy f l u x s p e c t r a w i l l  fg,  fp,  of  the  a t the maxima of energy  be  locallyflux  the frequency a t which the wind speed a t a h e i g h t  1/k  150 =  A/2TT  equals  the wave phase speed.  The c h o i c e of  h e i g h t used to determine f^ i s p u r e l y a r b i t r a r y ; enough i n these measurements so t h a t i t  it  *X/2Tf  is  for  the  g e n e r a l l y small  i s w i t h i n the h e i g h t range over  which anemometer wind speed measurements are a v a i l a b l e but not so s m a l l t h a t the c u r v a t u r e of in  the p r o f i l e i s  the e x t r a p o l a t i o n of  are  large.  I n t h i s way i n a c c u r a c i e s  the wind p r o f i l e by assuming i t  to be l o g a r i t h m i  minimized. The v a l u e of f^ = ^ / 2 t T  file  is  assumed ( f  speed of  is  k  d e f i n e d as the frequency a t which the phase  the wave equals  U  a  " k = U  U  i s r e l a t e d to u * i f a l o g a r i t h m i c wind p r o -  a  the mean wind speed  " S/«k  =  2* K  l n  a t a h e i g h t 1/k =X/21T)  (^) = U /  l  K  k  n  ( E £ k ) > V § / 2  where K i s Von Karman's constant and the s u b s c r i p t a denotes "at anemometer h e i g h t " , cendental  case 5 m e t e r s .  (gK/2lT f  2  k  where  = u^/U^2 i  0.0012.  If u  s  k  u*)  =  (g/4fl  p  can be expected  r a t i o f ^ / f p can be expected k  The  z ) a  exp  (K/ V c p ) = constant  assumed to be  p  is  occurring,  then f^ w i l l  to i n c r e a s e w i t h f e t c h .  It  the  s h o u l d a l s o be  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 above.  Observed and P r e d i c t e d T r a n s i t i o n Fetches q u e s t i o n of the v a l u e of  the  be con-  to decrease w i t h i n c r e a s i n g f e t c h ,  r e l a t i o n f o r the waves and E q u a t i o n 7.3 7.5.3a  trans  can as a f i r s t a p p r o x i m a t i o n be taken as c o n s t a n t f o r  Since f  noted t h a t f / f  2  the d i m e n s i o n l e s s drag c o e f f i c i e n t ,  f e t c h over which wave g e n e r a t i o n i s  stant.  T h i s can be put i n the  form  f exp  full  in this  the f e t c h a t which the growth w i t h  151 time of  a g i v e n wave component s h i f t s  r a t e of i n c r e a s e  is  many measurements  from a l i n e a r to an e x p o n e n t i a l  s t i l l not s a t i s f a c t o r i l y  of  answered.  the frequency of the steep f r o n t face of  spectrum f o r g i v e n f e t c h e s and wind c o n d i t i o n s B a r n e t t and W i l k e r s o n , 1 9 6 7 ) ;  however,  p r e d i c t e d by M i l e s '  instance, Almost  the observed t r a n s i t i o n f e t c h e s are l e s s than those i n v i s c i d laminar model.  to t r a n s i t i o n , v e r s u s  c/u,;  if  through Frj, = c^N^/f^, where c^ i s the frequency of  for  II  g i v e s a p l o t of N^, the t h e o r e t i c a l l y  ff  (see,  the wave  these show a g r e a t d e a l of s c a t t e r . W  invariably,  There are now  is  Phillips  4.6)  p r e d i c t e d number of wave p e r i o d s related  to the  the group v e l o c i t y  the f r o n t f a c e of  (1966, F i g u r e  transition of  fetch  the waves and  the wave spectrum,  then the  fetches  a t which the observed s p e c t r a 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 7.5.  predicted fetches;  this  comparison i s  The observed t r a n s i t i o n f e t c h e s are l e s s  two cases:  shown i n T a b l e  than p r e d i c t e d except i n  r u n 1 and r u n 3.  Run 1 has a long f e t c h  (40 km), and was  taken i n l i g h t  winds.  S i n c e no i n f o r m a t i o n on c o n d i t i o n s upwind of  the sensors i s  available  it  is  not known how homogeneous the wind f i e l d r e a l l y was;  reason the r e s u l t with  for  i s not i n t e r p r e t e d as showing s i g n i f i c a n t  this  disagreement  theory. Run 3,  fetch  i n which the wind d i r e c t i o n was 8 0 ° ( T r u e ) , has a much l a r g e r  than runs 2 or 4 i n which the wind was from 1 1 5 ° .  i n f e t c h w i t h azimuth occurs a t about 1 0 5 ° ; see F i g u r e 2. alone makes the comparison w i t h theory d i f f i c u l t ; of ^10°  i n wind d i r e c t i o n or u n n o t i c e d s h i f t s  A large This  change fact  unaccounted-for  just before  could b r i n g the observed and p r e d i c t e d t r a n s i t i o n f e t c h e s  errors  the r u n began into  good  TABLE  7.5  Observed and P r e d i c t e d T r a n s i t i o n Fetches  f Hz  f cm/sec  f  Run No. Notes:  C  T (km)  o (km)  (3)  (4)  F  cm/sec  C$/u,  T  N v  (1)  (2)  F  1  .3  520  7.6  68.5  6 x 10  2  5.2  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 s u b s c r i p t f means " e v a l u a t e d a t the low-frequency of  2.  face  the l o c a l l y - g e n e r a t e d wave spectrum".  Nrp i s  the number o f wave p e r i o d s to t r a n s i t i o n .  o b t a i n e d from P h i l l i p s (1966, F i g u r e 3.  40.  F^ is  It  4.6).  the t h e o r e t i c a l l y p r e d i c t e d t r a n s i t i o n f e t c h ,  g i v e n by F ^ = (N^Cg), where Cg i s  is  and i s  the group v e l o c i t y of  the  ff waves at the low-frequency f a c e of 4.  F  0  is  Table  the wave spectrum.  the observed f e t c h f o r the g i v e n wind d i r e c t i o n ; 6.1.  see  152 agreement or throw them i n t o g r e a t e r disagreement. wind-drivenwave f i e l d s  Furthermore a l l  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  t h a t the wave s p e c t r a from runs 2 and 4 may be p a r t l y made up of 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 and hence over f e t c h e s as l a r g e as 7 km. l a r g e f r a c t i o n of  the e f f e c t i v e  one,  runs 2 and 4.  effect  and i n the p r e s e n t  N^'s i n t h i s  however,  case.  the observed t r a n s i t i o n  b r i n g i n g them i n t o c l o s e r agreement w i t h theory; a t l e a s t p a r t i a l l y o f f s e t by l a r g e r  the waves a t f r e q u e n c i e s  peak were t r a v e l l i n g a t an angle to the wind.  while results  run 4 show s i g n i f i c a n t  near the s p e c t r a l is  considered that  the  Energy T r a n s f e r  from r u n 2 are e q u i v o c a l , and those from  disagreement.  i n the Wave Spectrum  A glance a t the E ( f ) cases the frequency fp of  s p e c t r a ( F i g u r e s 40 to 47) the peak of  shows t h a t i n a l l  the wind-generated wave spectrum  o c c u r s a t or s u b s t a n t i a l l y below f g and f , x  of  It  for  from r u n 3 i n d i c a t e approximate agreement between p r e d i c t e d and  observed f e t c h e s ,  7.5.3b  theoretical  By the same token the p r e d i c t e d t r a n s i t i o n f e t c h  r u n 3 may be too s m a l l i f  results  locally-  circumstances may be too s m a l l i n  T h i s e r r o r would i n c r e a s e  is,  near the  T h i s means t h a t the downwind f e t c h may not be  f e t c h e s i n runs 2 and 4, its  the w i n d ,  These waves may produce a  the t o t a l energy a t low f r e q u e n c i e s ,  generated s p e c t r a l peak.  t h a t of  energy  the frequency of  the energy and momentum f l u x s p e c t r a r e s p e c t i v e l y .  t h a t of growth v i a normal p r e s s u r e s  swell  peaks  Two f a c t o r s  can c o n t r i b u t e to t h i s .  wind speeds had r e c e n t l y dropped or i f  the  besides  First,  if  from another source had  entered the r e g i o n , damping of s w e l l a t f r e q u e n c i e s  j u s t below the wave  153 peak c o u l d cause a s h i f t of f £ and f is  thought to be s m a l l , s i n c e  generally at frequencies The second f a c t o r i s  to h i g h e r f r e q u e n c i e s ;  T  first  l e s s than f p / 2 . the presence o f some mechanism whereby energy  nonlinear interactions which i s of  among wave components.  p r o b a b l y the most important wave-wave  (Benjamin and F e i r ,  being unstable,  generates  One such t r a n s f e r  is  sidebands o f  lower f r e q u e n c i e s ,  and thus move the f r o n t f a c e of  lower f r e q u e n c i e s ;  t h i s process  (1967),  it  t h e i r own a t  1969b).  (see  pp. 136,  7);  is  the wind.  also possible  both of  He  c o n s i d e r e d and e n l a r g e d upon r e c e n t l y by L o n g u e t - H i g g i n s (1969a, The f i r s t mechanism, which i s  has a l r e a d y been d i s c u s s e d on p. (1969b).  In essence i t  of momentum from s h o r t e r waves; similar  for  these have been  caused by a v a r i a t i o n a l o n g the  wave p r o p a g a t i o n d i r e c t i o n of the Reynolds s t r e s s on the water  Higgins  still  the spectrum to  the wind to generate waves w i t h o u t the a i d of normal p r e s s u r e s . suggests two such mechanisms  wave",  frequencies.  c o m p l e t e l y independent of  As has been p o i n t e d out by Stewart  process,  t h a t of sideband  sidebands a t lower and h i g h e r  is  via  i n t e r a c t i o n near the peak  1967), whereby a " c l a s s i c a l Stokes  The l o w - f r e q u e n c y sidebands generate  further  The  energy t r a n s f e r from h i g h e r f r e q u e n c i e s  the spectrum of a w i n d - g e n e r a t e d wave f i e l d ,  feeding  the spectrum other than  A number of such mechanisms are a v a i l a b l e .  and most obvious i s  effect  the s w e l l peaks i n the wave s p e c t r a are  can be t r a n s f e r r e d to the waves a t the peak of by normal p r e s s u r e s .  this  136.  The second i s  involves because  o u t l i n e d i n Longuet-  the "sweeping up" by long waves the a c t i o n of  to t h a t of a maser, L o n g u e t - H i g g i n s r e f e r s  mechanism".  surface,  the mechanism i s  to i t as a "maser  The s h o r t waves are presumed to grow v i a normal wind p r e s s u r e s ;  154 these become steep and may break a t of  the longer waves.  the convergences  on the f r o n t  Whether they break or not momentum i s  to the l o n g e r waves, but the mechanism i s most e f f e c t i v e waves a c t u a l l y do break; presumably most energy i s quencies  near the peak of  faces  transferred  if  the  short  t r a n s f e r r e d to  fre-  the wave spectrum.  An energy f l u x spectrum measured by c o r r e l a t i n g normal p r e s s u r e w i t h wave v e r t i c a l v e l o c i t y would not show t h i s of  energy  t r a n s f e r a t the peak  the wave spectrum, but would have a maximum a t a h i g h e r  T h e r e f o r e the i n t e g r a t e d  energy f l u x measured i n t h i s way would i n c l u d e  the energy t r a n s f e r a s s o c i a t e d that  w i t h the "maser" mechanism, p r o v i d e d  the maximum i n the energy f l u x spectrum was not a t such a h i g h  frequency t h a t i t was o u t s i d e  the frequency range of  the  In o r d e r f o r the maser mechanism to be e f f e c t i v e , of  waves. c /c of  sensors.  the energy  the s h o r t waves to the l o n g ones v i a b r e a k i n g must exceed  energy g a i n through the working of  s  frequency.  For this  to o c c u r , L o n g u e t - H i g g i n s shows that  must be much l e s s  £  than 1, where c  the s h o r t and long waves.  equals  c /c g  g  and so must be  In f a c t  above about 4-fp, the maser process by t h i s  and c  is  the  the v e l o c i t y  are the phase  e  long ratio  velocities ratio  large. is  always one or g r e a t e r ,  the p r e s e n t measurements  and so i t might be expected  experiment.  measurements  g  their  S i n c e c = g/co , t h e i r frequency  The observed r a t i o f g / f p exceeds two.  the Reynolds s t r e s s e s of  loss  but never  cease to be  that  accurate  the energy i n p u t i n  from waves w i t h f r e q u e n c i e s  h i g h e r than any measured  Two p o s s i b i l i t i e s  either  are a gross underestimate  are open; of  the  present  the energy f l u x becuase  they  cannot measure f l u x e s ' t o h i g h - f r e q u e n c y waves, or the maser mechanism  155 is relatively inefficient,  and accounts  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 ,  view of  the e f f i c i e n c y  l a r g e phase s h i f t s the f a c t values  7.6  that  a t 3-4  of  of  f o r o n l y a s m a l l f r a c t i o n of  the normal p r e s s u r e s  e x h i b i t e d by the  the in  observed  the p r e s s u r e from - 1 8 0 ° towards - 9 0 ° near f p , and  the energy and momentum f l u x s p e c t r a do f a l l  to  low  times f £ .  The i* S p e c t r a A d i s c u s s i o n of  the s p e c t r a o f  energy per r a d i a n , w i l l be put o f f  £ , the f r a c t i o n a l i n c r e a s e  i n wave  i n o r d e r to b e g i n the s e c t i o n w i t h a  comparison of  the mean v a l u e s  of £ w i t h those p r e d i c t e d by t h e o r y .  comparison i s  given i n Table  7.6.  7.6.1  Mean Values of £  present  t  , the measured mean v a l u e s case s i n c e  This  of  C, are g i v e n as E / o E ; i n  the measured E and E = f ^ g ig " are i n t e g r a t e d  the  values  5  f o r the energy f l u x and wave s p e c t r a f o r a g i v e n r u n , the r a d i a n f r e quency cjp of f o r cj .  the peak of  the l o c a l l y - g e n e r a t e d wave spectrum i s  The t h e o r e t i c a l p r e d i c t i o n s  £  t  c o r r e s p o n d i n g to  ^  m  chosen are com-  puted from M i l e s (1957, E q u a t i o n 2 . 1 0 ) :  the v a l u e s whereas  of |3  are o b t a i n e d from M i l e s (1959a, h i s F i g u r e 4 ) .  t  E and 1£ are i n t e g r a t i o n s under the c o r r e s p o n d i n g s p e c t r a , 1  t h e o r e t i c a l p r e d i c t i o n s are f o r a s i n g l e quency of the peak of  the wave spectrum.  s i n u s o i d at  Thus the  the observed f r e -  The comparison may t h e r e f o r e  TABLE  7.6  Comparison of Observed and P r e d i c t e d Values of  {  Run No.  = E/wE  ; y^2  <^p rad sec-1  E Erg -2 sec" c m  Cp/U!  CI  P  Notes:  cm  t  2  (1),(2)  1  2. 6  19 .6  1 X  10-2  2b  3. 6  10 .4  5 X  10"  2a  3. 6  10 .0  3  4. 4  6  <10" 2  13  3  0. 25 14  ::29  5 X 10-  3  0. 3  18  7 .6  4 X  10"  3  1. 9  3. 6  5 .5  1..5 X  10"  3  4b  3. 2  5 .0  1 X  4a  3. 7  3 .8  1.  2. 3  -  J t  (3)  18  Notes:  J m  1  (3)  2.8 X 10"  4  2. 5 X  10  5.9  x: i l O "  4  2. 8 X  10"  43  6.8  X  10"  4  3. 0 X  10-6  25  30  2.8  X  10"  4  4. 1 X  10-5  3. 2  6  60  2.8 X  lO"  10-3  3. 5  21  90  1.4  X  7..5 X 10"  3. 6  23  154  1.8  X  4  4  10-3  1. 7 X 10"  4  10  3. 1 X 10"  4  - 3  to v a l u e s  inviscid  laminar theory;  the s u b s c r i p t "m" r e f e r s  measured  values.  t  Cm  is =  c a l c u l a t e d from M i l e s '  o b t a i n e d from M i l e s (1959a, F i g u r e 4 ) .  E/^gcopT ; 2  r  t  ~ejt fnyp . v  6  1. 3 X 10"  3  The s u b s c r i p t "t" r e f e r s  ^  - 8  t  to  156 be u n f a i r ;  it  i s made p r i n c i p a l l y because  peaked q u i t e near <jp  (the  arbitrary;  the v a l u e of  i n s t e a d of  the o t h e r ) ;  c h o i c e of <Jp i n s t e a d of  i n Table 7.6,  of  they can i n some sense be thought of as  J * and  show t h a t  about 100)  fetch  Cp/U^,  represent-  g i v e n f o r a l l the runs  the measured i n t e g r a t e d v a l u e s ° sinusoids  except f o r r u n 1,  which must be c o n s i d e r e d an e x c e p t i o n . 1 and the f a c t  is  sinusoid. j'j. and of  m  t h e o r e t i c a l ones f o r s i n g l e r a t i o is  = 2 It f g  i s not changed s i g n i f i c a n t l y by u s i n g one  i n g the s p e c t r a from a s i n g l e The v a l u e s  the E s p e c t r a are s h a r p l y  by f a c t o r s  ( exceed •> m  of 8 - 200 (the  f o r which the f a c t o r i s The l a r g e v a l u e of  average  10^ and  Cp/U^ f o r r u n  t h a t wind c o n d i t i o n s were unknown over the long (40 km)  indicate f i r s t ,  caused by M i l e s '  t h a t any l o c a l g e n e r a t i o n was d e f i n i t e l y  i n v i s c i d mechanism and second,  that i t  is  not  probable that  the waves p r e s e n t were generated by s t r o n g e r winds upstream from measurement s i t e . Two f a c t s  T h i s second i n d i c a t i o n i s  shown by T a b l e 7.6  the  considered h i g h l y probable.  i n d i c a t e the presence  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  c r e p a n c i e s between measured and p r e d i c t e d v a l u e s is  the  of  of a mechanism  is f  the l a r g e  , and the  dis-  second  the much s m a l l e r range-.and percentage v a r i a t i o n between runs i n the  measured v a l u e s phenomenon i s  as: compared w i t h the t h e o r e t i c a l ones.  as s t r i k i n g as the former; whereas  runs by a f a c t o r of 100,  not enough v a l u e s  The T The  v a r i e s over  C v a r i e s by o n l y a f a c t o r of 5. m  to judge the form of  the v a r i a t i o n of  i t does not appear to be s i m i l a r to t h a t of 7.6.2  f  The l a t t e r  £  £  the  There are  w i t h c/U-^;  w i t h c/U-^.  Spectra  !i s p e c t r a themselves,  shown i n F i g u r e s 48-55, d i s p l a y a remark-  157 a b l e resemblance This i s fits  to curves drawn a c c o r d i n g to E q u a t i o n 6.9  (p.  130).  an e m p i r i c a l r e l a t i o n suggested by Snyder and Cox ( S C ) , which  t h e i r data and those of B a r n e t t and W i l k e r s o n (BW) q u i t e w e l l .  frequency where the SC curve c r o s s e s the the wave phase v e l o c i t y equals wavelength;  it  some evidence  of  c/u& always  is  greater  ^  differs  observed  As was mentioned i n " R e s u l t s " ,  i n the p r e s e n t d a t a , however,  occurs a t  values  l e s s than 10, w h i l e the lowest c / u * i n the SC and BW d a t a  than 10.  Since t h e i r measurements  would e x p l a i n the f a l l o f f p r e s e n t measurements,  of  a t s h o r t wavelengths) this  their results  ceased  is  to grow;  input w i l l  The  T h i s means resolution  energy i n p u t to the system due presumably come from d i f f e r e n t ranges of  c/u .);  i n d i f f e r e n t frequency bands  (different  to v a r y i n v e r s e l y as c / u * i t  should i n c r e a s e w i t h frequency up to  V(  if  it  the frequency a t which the waves r e a c h t h e i r minimum phase the h i g h - f r e q u e n c y f a l l o f f  i n t e r p r e t e d i n some o t h e r way.  in  Stewart  ^ is  to be b e l i e v e d  spectra  this  at high frequencies.  (by an i n s t r u m e n t w i t h s u f f i c i e n t  the t o t a l  it  in their  on the o t h e r hand, are of e n e r g y . f l u x .  t h a t what should be observed  normal p r e s s u r e s ;  are of wave growth,  t h a t the h i g h - f r e q u e n c y wave components  were approaching e q u i l i b r i u m and had t h e r e f o r e  If  is  of s i m i l a r h i g h frequency behaviour i n the SC  The f a l l o f f  c o u l d be expected  Hz,  "cross-  curves which  the h i g h - f r e q u e n c y r e g i o n , where the  below the SC p r e d i c t i o n s .  and BW d a t a .  one  ^ curve as i s UJy., a l t h o u g h n e i t h e r  The o n l y p a r t o f the observed  from the SC r e l a t i o n i s  there i s  t h a t a t which  appears not to be as good an a p p r o x i m a t i o n to the  p a r t i c u l a r l y good.  fall  is  the wind v e l o c i t y a t a h e i g h t of  over frequency" of the observed  values  ^ = 0 axis  The  is  to sources assumed 13.4  velocity.  i t must be  ( p e r s o n a l communication)  suggests  158 that of  the behaviour of  a "sheltering"  peak of  of  £ at high frequencies short-wavelength  can be e x p l a i n e d i n terms  waves by the  the wave spectrum, so t h a t over most of a long wave c y c l e  energy can be added to the s h o r t waves by the w i n d . i n g " immediately b r i n g s J e f f r e y s ' b e i n g c o n s i d e r e d here i s It  is  not,  theory  however,  The term  (1925) to mind;  of  the  the  little  "shelter"sheltering"  the type c o n s i d e r e d by  Jeffreys.  concerned i n s t e a d w i t h flow r e v e r s a l i n the c o o r d i n a t e system mov-  i n g w i t h the wave phase v e l o c i t y This reversal  c a t the  c r i t i c a l h e i g h t where U = c.  can be c o n s i d e r e d as a type of  separation,  " s e p a r a t i o n bubble" formed by the flow i n t h i s i n the trough of near the  c r e s t as suggested by P h i l l i p s  keeps pace w i t h the  energy  buoy i s  is  their  c o u l d be added to  the time, and thus  time i n r e g i o n s  i n f e r e n t i a l evidence  the measurement  a l a r g e f r a c t i o n of  S i n c e the momentum t r a n s f e r  system.  to the waves i s  flow  is  r e a l and  to the waves measured by the surface.  c a l c u l a t e d from the same i n £ were not  to the waves would be much l a r g e r  and the v a l u e s  of  would become much g r e a t e r  that  From the d i s c u s s i o n on p.  the observed f a l l o f f  the computed momentum t r a n s f e r s  momentum t r a n s f e r s  the a i r  the t o t a l wind s t r e s s on the water  , then i f  those g i v e n i n T a b l e 7 . 4 ,  of  could  e x i s t s which suggests  c l e a r t h a t the momentum t r a n s f e r  i n f o r m a t i o n as i s f  then  them.  i n the observed f s p e c t r a at h i g h f r e q u e n c i e s  not an a r t i f a c t of it  (1967) r a t h e r than  (1966; h i s F i g u r e 4 . 3 ) ,  long waves f o r most of  Another p i e c e of the f a l l o f f  the  waves i n the troughs would be i n an "eddy" which  indeed spend a l a r g e p r o p o r t i o n of where l i t t l e  and i f  c o o r d i n a t e system l i e s  the waves as suggested by Stewart  the s h o r t - w a v e l e n g t h  148,  longer ones near  real than  computed from these than those observed by  large  159 Smith (1967) and W e i l e r and B u r l i n g the t o t a l  (1967) .  t r a n s f e r from the a i r to the s e a ,  S i n c e these workers measured i t would be d i f f i c u l t  how the measured momentum i n p u t to the waves v i a normal p r e s s u r e  to  see  could  be l a r g e r . S i n c e the observed t s p e c t r a f a l l  below the e m p i r i c a l curve suggested  by Snyder and Cox at h i g h f r e q u e n c i e s , curve o v e r e s t i m a t e s quencies .  the energy  i t may be t h a t t h e i r e m p i r i c a l  (and momentum) t r a n s f e r a t h i g h f r e -  Snyder and Cox i n t e g r a t e d under t h e i r curve to compute  momentum t r a n s f e r  to the waves i n t h e i r experiment;  t h a t t h e i r anomalously h i g h wave drag c o e f f i c i e n t  the  i t may t h e r e f o r e  (C  T T  — .007)  be  is  DW accounted f o r by t h i s  o v e r e s t i m a t i o n of  The i * s p e c t r a p r e d i c t e d by M i l e s ' are a l s o  shown i n F i g u r e s 48-55.  the energy  flux.  (1957) i n v i s c i d l a m i n a r model  They c l e a r l y l i e f a r below  the  observed / * s p e c t r a and those c a l c u l a t e d from Snyder and Cox's e m p i r i c a l formula, Hz);  except a t the h i g h e r f r e q u e n c i e s  a t the lower f r e q u e n c i e s  d i e t e d from M i l e s '  the observed v a l u e s  of  i(  A Dimensionless At the s u g g e s t i o n  of  5 to 8.  The  the peak of  range covered by the data i s  the i n v i s c i d laminar model of M i l e s (1957) i s  to account f o r the observed f l u x e s  1.6  exceed those pre-:  t h a t a t a l l v a l u e s o f c / u ^ (at  the r e l e v a n t wave s p e c t r a the c^/u  7.6.3  — 10 r a d / s e c or f  i n v i s c i d laminar mechanism by f a c t o r s  c o n c l u s i o n to be reached i s  Cp/u^<50)  (  10 <  insufficient  of energy and momentum to the waves.  R e l a t i o n between of D r . A . E . G i l l  versus U ^ / c has been p r e p a r e d , and i s  (f)  and Wind Speed  a dimensionless  shown i n F i g u r e 55.  p l o t of It  immediately be seen that the s p e c t r a are f a i r l y w e l l - g r o u p e d i n  ^(f)  can the  160 r e g i o n 0.5 ^ U ^ / c ^ 5; e s t i m a t e of v a l u e of  the l i n e drawn through the p o i n t s  w  , the d e n s i t y r a t i o of a i r to water,  passes through zero a t U ^ / c at 1 . 1 , a drag c o e f f i c i e n t at a height  of  .0012  z a 2 meters. T(f)  is  is  assumed,  It  a logarithmic p r o f i l e with  it  is  found t h a t U^/Ug ~  1.0  7.5,  the mean wind speed a t a h e i g h t of 2 meters.  The e q u a t i o n  a  /  e  0.5<U /c<6*  at higher values  w  ( U / c - 1) 2  5  £ (f)  should p r o b a b l y be s e t  to  zero  of U ^ / c to a v o i d the p o s s i b i l i t y of computing e x c e s s i v e  energy and momentum t r a n s f e r s  of  =0.  T h i s suggests t h a t a r e a s o n a b l e f o r m u l a f o r  i s not a p p l i c a b l e above U^/c= 6;  It will  and i f  at U ^ / c  might be  f ( f ) - ?  where  a visual  the "best f i t " which has been f o r c e d to pass through the  fa/P  computing  is  be n o t i c e d t h a t  Snyder and Cox (1966).  at high this  frequencies.  empirical r e l a t i o n is  s i m i l a r to  I n o r d e r to compare the two r e l a t i o n s ,  that the  c a l c u l a t e d Snyder and Cox curves f o r the runs were put on the dimensionl e s s p l o t w i t h the measured v a l u e s  f(f);  of  i n g e n e r a l i t was found  that  the Snyder and Cox curves f i t t e d  the d a t a as w e l l as r e l a t i o n 7.5 d i d ;  differences  the o n l y advantage claimed f o r E q u a t i o n  were s m a l l .  In fact  7.5 over the Snyder and Cox f o r m u l a i s within its  range of v a l i d i t y  ^*(f)  that i t  is  slightly  simpler,  can be c a l c u l a t e d from i t w i t h o u t  need of matching a l o g a r i t h m i c p r o f i l e to a computed wavelength.  *The range 0.5 < U ^ / c < 6 t r a n s l a t e s e f f i c i e n t of .0012 i s assumed.  to 50 < c / u ,  v  <. 6 i f a drag co-  since the  SECTION 8 :  CONCLUSIONS  8.1  Introduction  161  8.2  The Power S p e c t r a  162  8.3  The C r o s s - S p e c t r a  162  8.4  The Energy and Momentum F l u x S p e c t r a  8.5  Mean F l u x of Energy to  8.6  Mean F l u x of Momentum to  8.7  £  .  163  the Waves  163  the Waves  , the F r a c t i o n a l Energy I n c r e a s e of  163 the Waves  per  Radian  164  8.8  The Boundary Bay Experiment  165  8.9  A i r Flow over Waves Moving i n t o the Wind  166  SECTION 8:  .8.1  CONCLUSIONS  Introduction A measurement system c o n s i s t i n g of a wave sensor and a buoy i n  which was mounted a p r e s s u r e sensor has been d e s i g n e d , b u i l t , tested,  thoroughly  and used to o b t a i n simultaneous r e c o r d i n g s of wave h e i g h t and  normal p r e s s u r e at the water s u r f a c e i n the presence of w i n d - d r i v e n sea waves.  Special analysis  large spurious spikes  techniques have been developed f o r d e a l i n g w i t h  i n the p r e s s u r e s i g n a l , and i t has been shown t h a t  these s p i k e s do not m a t e r i a l l y a f f e c t range of  interest,  the r e s u l t s  which i s between 0.1  i n the f r e q u e n c y  and 2 H z .  The r e c o r d i n g s o b t a i n e d w i t h the measurement system are the  first  simultaneous f i e l d measurements of p r e s s u r e and waves i n which the sensors have been c o n s t r a i n e d to move o n l y v e r t i c a l l y  (quasi-Eulerian  measurement), and they are the most comprehensive of a l l the  existing  measurements.  They s h o u l d thus p r o v i d e the n e c e s s a r y b a s i s f o r f u t u r e  p r e d i c t i o n s of  the magnitude and phase ( r e l a t i v e  normal p r e s s u r e s which appear to be the p r i n c i p a l  to the waves) of source of  the  the energy  t r a n s f e r from the wind to the waves. Power s p e c t r a o f  the p r e s s u r e and wave h e i g h t s i g n a l s  and the  c r o s s - s p e c t r a between them have been p r e s e n t e d f o r s i x s e p a r a t e which cover a wide range of wind and wave speeds. f l u x s p e c t r a and the i n t e g r a l s E and t  w  £ ,  the f r a c t i o n a l  Energy and momentum  under these s p e c t r a have been  computed and are p r e s e n t e d f o r a l l the r u n s . of  The f r e q u e n c y d i s t r i b u t i o n s  i n c r e a s e i n wave energy per r a d i a n , 161  runs,  and a d i m e n s i o n -  162  f(f)  l e s s p l o t of  versus  U 5 / c have been c a l c u l a t e d and are a l s o p r e -  sented .  8.2  The Power S p e c t r a The wave power s p e c t r a behave as expected  h i g h - f r e q u e n c y s l o p e s between - 4 . 5 amount of  long-wavelength  by marine t r a f f i c  and - 5 .  f o r the s i t e ,  In some runs a c o n s i d e r a b l e  s w e l l was p r e s e n t which was presumably generated  i n the a r e a .  The p r e s s u r e power s p e c t r a appear to be made up of sisting which i s  exhibiting  two p a r t s ,  con-  of a "basic" spectrum s i m i l a r to those observed over l a n d , superimposed a wave-induced "hump".  A d d i t i o n of Pj]^  to  on the  p r e s s u r e s i g n a l has the e f f e c t of removing the superimposed hump a t frequencies  whereas  i t has l i t t l e  t h a t a t h i g h f r e q u e n c i e s most of w i t h the waves i s  8.3  effect  at high frequencies,  the p r e s s u r e s i g n a l which i s  low  indicating coherent  i n quadrature w i t h them.  The C r o s s - S p e c t r a The phase between the p r e s s u r e and the waves has been compared w i t h  the phase p r e d i c t e d by M i l e s ' generation.  At frequencies  wave s p e c t r a ,  near the l o c a l l y - g e n e r a t e d peaks i n  observed phases are s h i f t e d  h i g h over wave troughs) 20°;  (1957) " i n v i s c i d laminar" model of wave  the e s t i m a t e d  e r r o r i n t h i s measurement i s  wave phase v e l o c i t y a t the peak of velocity.  (that  is,  pressure  by an amount which exceeds p r e d i c t e d v a l u e s  d i s c r e p a n c y occurs over a range 9.5  tion  from -180  the  £  l e s s than ~t 5 ° .  C p / u * 4 35, where Cp i s  the wave spectrum and u^ i s  by  This the the  fric-  163 8.4  The Energy and Momentum F l u x S p e c t r a S p e c t r a of  the f l u x e s of energy and momentum from the wind to  waves are p r e s e n t e d here f o r the f i r s t  time.  s h a r p l y a t a frequency a t or ( u s u a l l y )  s l i g h t l y h i g h e r than f p ,  frequency of spectrum i s  the peak of  the wave spectrum.  They both are peaked  c o n t r a s t e d w i t h the r a t h e r broad -  *^(f)  f^EW spectrum measured 1967;  W e i l e r and  1967).  The f a c t quencies  the  The s h a r p l y - p e a k e d  a t r e l a t i v e l y s h o r t d i s t a n c e s above the waves (Smith, Burling,  the  that  higher  the peaks of  the E ( f )  than fp suggests e i t h e r  and 7j* (f) w  s p e c t r a are a t  that wave-generation  fre-  mechanisms  are a c t i v e which t r a n s f e r energy by means other  than normal p r e s s u r e s ,  or t h a t energy i s  to lower  b e i n g t r a n s f e r r e d from h i g h e r  i n the wave f i e l d  8.5  frequencies  itself.  Mean F l u x of Energy to the Waves The i n t e g r a l s  second  intervals  means of  under the E ( f )  s p e c t r a have been determined a t  throughout each r u n ; the E v a l u e s  these averages.  The i n d i v i d u a l i n t e g r a l s  presented  are  20the  under the E ( f )  spectra  are found w i t h a C h i - s q u a r e d t e s t to be a p p r o x i m a t e l y n o r m a l l y d i s t r i buted about t h e i r means, these f a c t s  indicate  nature and v a r i e s  8.6  that  and are found to have l a r g e s t a n d a r d the wave g e n e r a t i o n process  c o n s i d e r a b l y i n time (or  is  deviations;  random i n  space).  The Mean F l u x of Momentum t o . t h e Waves The mean v a l u e s  of f  , the wave-supported momentum f l u x ,  have been  computed f o r each r u n 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 the mean f l u x e s  2 computed from ^ Op U 5 , where Cp, the drag  from  coefficient,  164 is  taken to be  .0012.  from the T"' i» w. and i s ment the  .0014 w i t h s t a n d a r d e r r o r  has been computed .0005.  In one measure-  t o t a l momentum i n p u t to the a i r - s e a boundary f  measured; results,  the r a t i o ^ w / ^ it  is  <, 0.1  .8.7  A "wave-drag c o e f f i c i e n t "  felt is  that  for  this  case was 0 . 8 .  = -  (1966)  and compared w i t h the p r e d i c t i o n s  of  that  the Waves per Radian  of M i l e s '  of !> have been prepared  i n v i s c i d laminar model and  w i t h an e m p i r i c a l curve suggested by Snyder and Cox (1966). of  £  are g r e a t e r  of between 5 and 8,  The  than those p r e d i c t e d by M i l e s '  and are about  the same s i z e as  interest.  theory  those  p r e d i c t e d by Snyder and Cox's e m p i r i c a l r e l a t i o n , over most of quency range of  these  probably i n c o r r e c t .  Frequency d i s t r i b u t i o n s of measured v a l u e s  by f a c t o r s  a  On the b a s i s of  the a s s e r t i o n made by P h i l l i p s  , the F r a c t i o n a l Energy I n c r e a s e  measured v a l u e s  £ u w was  the  fre-  Thus the p r e s e n t d a t a support the f i n d i n g of  Snyder and Cox t h a t M i l e s '  (1957) i n v i s c i d laminar mechanism i s  inadequate  to e x p l a i n observed r a t e s of wave growth. At high frequencies and Cox c u r v e . waves i n t h i s  the measured v a l u e s  This indicates  that  the slow-moving  frequency range are not generated  nearer the peak of  the wave spectrum;  t h a t s h o r t e r waves may be " s h e l t e r e d " by the  long waves a t the peak of  as suggested by Stewart such t h a t  of C f a l l  (1967),  below the Snyder  short-wavelength  as e f f i c i e n t l y  the p o s s i b i l i t y  therefore  f o r a l a r g e p o r t i o n of  the spectrum.  as  those  exists  the  This .could happen  time if,  the flow d i s t r i b u t i o n over the waves  the "cats-eye"- flow d i s t r i b u t i o n ( i n the c o o r d i n a t e  moving w i t h the wave phase speed) which was f i r s t  is  system  d e s c r i b e d by L i g h t h i l l  165 (1962) i s  situated  i n the wave t r o u g h .  A dimensionless observed v a l u e s  of  p l o t of ^ (f),  v e r s u s U5/Cp has been p r e s e n t e d ;  £ (f)  for U5/c >  a l t h o u g h somewhat s c a t t e r e d  the  4,  are approximated q u i t e w e l l by  f(f)  where U2 i s  =  ?  a  / ?  (U /c -  w  2  1),  the wind speed a t a h e i g h t of  relation fits  the d a t a as w e l l as  the range 0.5 < U^/c ^ 6;  This  simple  the Snyder and Cox curve does over  f o r U ^ / c ^ 6 i t does not a p p l y , s i n c e  values  of U ^ / c the measured v a l u e s  zero.  The f a l l o f f  in  two meters.  ,T(f)  of  ^(f)  may f a l l  at high frequencies,  be taken i n t o account i n any computation of  if  rapidly it  is  at  these  towards  real,  integrated fluxes  must  of  energy  or momentum to the waves.  8.8  The Boundary Bay Experiment T h i s experiment was f i r s t  sensor  and a sensor  a standard sensor. flat,  conceived as a f i e l d  developed by J . A . E l l i o t t of  t e s t of  the buoy  the I n s t i t u t e ,  The t e s t , which was made on a g e n t l y - s l o p i n g  against sand  i n d i c a t e d t h a t a l l three instruments produced r e s u l t s which were  e s s e n t i a 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 the pressure-waves Besides  this  experiment result  (0.1  atmospheric boundary l a y e r . 1.  the p r e s s u r e f i e l d  These r e s u l t s  interesting  i n the  turbulent  are l i s t e d below.  The p r e s s u r e power s p e c t r a o b t a i n e d show r e a s o n a b l y good  ment w i t h o t h e r measurements,  in  to 2 H z ) .  the experiment p r o v i d e d some  i n f o r m a t i o n on the s t r u c t u r e of  interest  both i n the f i e l d  (Priestley,  1965)  agreeand  166 i n a wind t u n n e l 2.  ( W i l l m a r t h and W o o l d r i d g e , 1962).  The p r e s s u r e power s p e c t r a have been found to group w e l l when 2 2 €a *) f o r r a d i a n wave-numbers g r e a t e r  non-dimensionalized with ( 10"" 2 cm"l; t h i s  indicates  the e x i s t e n c e of a r e l a t i o n s h i p between the  p r e s s u r e and the Reynolds s t r e s s 3. is  2 C = £ u * i n the boundary l a y e r .  The v e r t i c a l " s c a l e s i z e " of  about o n e - h a l f ,  5.  about 0.6 U,*,  }  where U,,, ±  At high frequencies  l a r g e phase l e a d (90  about o n e - t e n t h ,  of the  (1965).  The broadband a d v e c t i o n v e l o c i t y of is  eddies  The l a t t e r r e s u l t agrees r o u g h l y w i t h  more d e t a i l e d f i n d i n g s of P r i e s t l e y  eddies  the p r e s s u r e - g e n e r a t i n g  and the crosswind s c a l e s i z e i s  t h e i r downwind s c a l e s i z e .  4.  than  u  s  the  pressure-generating  taken to be 30 u , . v  (2-5 Hz) the p r e s s u r e i n the a i r shows a  to 1 8 0 ° ) over the downwind a i r speed a t the  same  elevation. 6. vertical  There i s  some evidence  accelerations  t h a t low-frequency  (0.05  - 2 Hz)  e x i s t i n the shear flow near the ground;  i s based on the apparent presence of a v e r t i c a l g r a d i e n t of the p r e s s u r e coherent w i t h the downwind a i r  8.9  A i r Flow over Waves Moving i n t o  the p a r t of  speed.  the Wind  A f a i r l y d e t a i l e d study has been made of between wave e l e v a t i o n ,  surface pressure,  the phase  relations  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 f o u r - s e c o n d p e r i o d which were l i n g against a l i g h t wind. to be a p p r o x i m a t e l y  this  travel-  The a i r flow over the wave group was found  g i v e n by p o t e n t i a l flow  theory;  the observed  differ-  167 ences,  a l t h o u g h s m a l l , are i m p o r t a n t .  +165° relative  to the waves i n s t e a d of  The phase of the p r e s s u r e was 1 8 0 ° , and the phase of  v e l o c i t y a l o n g the waves was + 1 2 ° i n s t e a d of 0 ° ; the phase of v e r t i c a l v e l o c i t y was 9 0 ° , as p r e d i c t e d .  the  The energy f l u x from the waves 9  to the wind was computed and found to be 20 e r g cm - sec -z  represents  the wind  a Q f o r the waves of about 4000.  1  , which  The momentum f l u x  from - 9  waves to wind has a l s o been computed and found to be 0.04 the t o t a l momentum f l u x from water a t a h e i g h t of  dyne cm  ,  to a i r was s i m u l t a n e o u s l y measured  1.75 meters w i t h a s o n i c anemometer,  and was 0.002 dyne  _o  cm  .  T h i s may i n d i c a t e  magnitude -2 x 10  the presence  dyne cm  /cm.  of a v e r t i c a l s t r e s s g r a d i e n t of  The observed p e r t u r b a t i o n s from  p o t e n t i a l flow are much s m a l l e r than those which would be from the p r e d i c t i o n s of P h i l l i p s (1966); however i t the flow regime over the waves i n t h i s the case c o n s i d e r e d by P h i l l i p s .  is  extrapolated  considered  case was q u i t e d i f f e r e n t  that  from  APPENDIX 1:  SPIKE REMOVAL  Al.l  Introduction  168  A1.2  The Spike Removal Process  169  A1.3  Definitions  171  A1.4  The E f f e c t s  A1.5  The E f f e c t of  the Spikes  on the Power S p e c t r a  177  A1.6  The E f f e c t of  the Spikes on the Phase S p e c t r a  178  A1.7  The E f f e c t of  the Spikes on the Energy F l u x S p e c t r a  A1.8  Conclusions  of a Regular Spike Window  175  .  .  180 182  APPENDIX 1:  Al.l  SPIKE REMOVAL  Introduction T h i s appendix deals w i t h the a n a l y s i s  presence spikes  i n the p r e s s u r e  s i g n a l of  large,  problems i n t r o d u c e d by  spurious "spikes".  occur i n the data whenever water washes over the  diaphragm at  the s u r f a c e of  the buoy.  to s a t u r a t e  p e r i o d s of 0.1  - 2 seconds.  be f u l l y random,  since  Their effects  a l s o on some of  the s m a l l ,  i n the a c t u a l a n a l y s i s  its  c r e s t s of  are r e p l a c e d w i t h zeros  p (t)  =  s  where p ( t ) g  is  spikes,  is  for to  the buoy on the  the l a r g e waves  set  for  to z e r o ,  one;  so wherever a s p i k e  removed,  the d u r a t i o n of  the  digitized  the s p i k e .  This  written  S(t)  Al.l,  the observed p r e s s u r e s i g n a l from which the spikes  been removed, p ( t ) of  p(t)  last  analysed must be a m o d i f i e d  mean v a l u e i s  is  enough  steep waves.  occurs i n the o r i g i n a l p r e s s u r e s i g n a l and i s  modified pressure signal  large  on the s i g n a l  they are caused by the motion of  Thus the p r e s s u r e s i g n a l which i s  data v a l u e s  often  the normal  T h e i r occurrence i n time cannot be s a i d  waves; they occur most o f t e n a t or near the and  pressure-sensing  t h a t of  t h e i r amplitude i s  the r e c o r d i n g system.  These  They are c h a r a c t e r i s e d by g e n e r a l l y  l a r g e amplitudes and a r i s e - t i m e much f a s t e r . t h a n turbulent pressure f l u c t u a t i o n s ;  the  the s i g n a l which would be observed i n the  and S ( t )  is  S(t)  =  0  during a spike  =  1  a t a l l other  have  absence  a "Spike f u n c t i o n " d e f i n e d by  168  times  A1.2.  169 E q u a t i o n A l . l cannot be i n v e r t e d to r e c o v e r p ( t ) ; pressure information l o s t during a spike i s  t h e r e f o r e w i l l be devoted  the observed p r e s s u r e The f i r s t  account of  this  the i n n e r workings of  appendix w i l l  the s p i k e removal p r o c e s s .  follow-;; the f i r s t  The next i s  is  the  This  spikes,  distortions  a comparison of  detailed  This  of  s p i k e s have been  the same s i g n a l s  which were measured a t the same time.  Three  analysis  the power s p e c t r a of wave  h o l e s have been i n s e r t e d a t times when s p i k e s  will  and d e r i v a t i o n s .  a cross-spectral  o b t a i n e d i n the f i e l d w i t h power s p e c t r a of  signals  of  deal with a f a i r l y  computer-generated s i n e waves i n which a r t i f i c i a l inserted.  represents  spectra.  s e c t i o n of  analyses  This  the s p i k e - i n d u c e d  be f o l l o w e d by a s e c t i o n g i v i n g some d e f i n i t i o n s different  the  the p r e s s u r e d a t a .  to e x p l o r i n g the e f f e c t s  i n the hope of g a i n i n g an u n d e r s t a n d i n g of of  to say  irretrievable.  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 section  that i s  occur i n the Lastly,  signals  i n which  pressure  phase and quad-  r a t u r e s p e c t r a between m o d i f i e d p r e s s u r e and m o d i f i e d wave s i g n a l s  from  complete runs made i n the f i e l d are compared w i t h phase and quadrature s p e c t r a between unmodified p r e s s u r e and unmodified wave s i g n a l s scattered  d a t a b l o c k s i n the same r u n s , i n which no p r e s s u r e  from  spikes  occurred.  A1.2  The Spike Removal Process A brief  o u t l i n e of  the f u n c t i o n s  of  the s p i k e removal s u b r o u t i n e  SPKSKP has been g i v e n i n "Data A n a l y s i s and I n t e r p r e t a t i o n " , p. 86.  This  o u t l i n e w i l l be e n l a r g e d upon i n the f o l l o w i n g p a r a g r a p h s . The d e t e c t i o n of s p u r i o u s d a t a by the r o u t i n e i s  on the b a s i s  of  170 slope;  if  the a b s o l u t e magnitude of  successive data points  is  l a r g e r than a t h r e s h o l d v a l u e  must be determined s e p a r a t e l y zero a p r e s e t  first  are checked a g a i n s t  the d a t a p o i n t s  detected.  the maximum p r e s s u r e v a l u e  is  succeeding  point is  exceeded, SPKSKP r e s t a r t s  NSKIP p o i n t s  The  c h e c k i n g continues  SPKSKP i s  p (t), Q  which a l l d a t a v a l u e s S(t)  0  "Spike  If  the observed p r e s s u r e d a t a , i n i t i a l l y to one.  (NSKIP + 1)  they  likely  NSKIP s u c c e s s i v e the  and S ( t ) , It  limit  points  (NSKIP +  l)  detected. reached.  and accepts  two  an a r r a y  s e t s both p ( t )  in and  Q  successive values  the p r e s s u r e w i t h spikes  either  a further  u n t i l the end of a g i v e n d a t a b l o c k i s  are set  at  during spikes.  removed, and S ( t ) ,  the  f u n c t i o n " of E q u a t i o n A 1 . 2 . The  computer program (FORTRAN IV) f o r SPKSKP i s  Appendix l a .  In the F o r t r a n program there i s  mentioned here because i t spikes, straight  is  line  reproduced i n  i n c l u d e d an o p t i o n not  not used i n the f i n a l a n a l y s i s whereby  i n s t e a d of b e i n g r e p l a c e d by z e r o e s ,  the  can be r e p l a c e d by a  i n t e r p o l a t e d between t h e i r b e g i n n i n g and end data  SPKSKP r e t u r n s ing  If  the z e r o i n g procedure and s k i p s  pass unmodified u n l e s s a new s p i k e i s  to zero f o r a t l e a s t  SPKSKP r e t u r n s p ( t ) ,  are zeroed  checked a g a i n s t DIFMAX.  c a l l e d once f o r every d a t a b l o c k a n a l y s e d ,  data a r r a y s :  the p o i n t  between each p o i n t to be  beyond a "spike" pass both c r i t e r i a they are a l l z e r o e d ; following points  automatically  (VALMAX) which i s  beyond the newly d e t e c t e d s p i k e .  and  (DIFMAX) which  succeeding  Before these p o i n t s  to o c c u r ; at the same time the d i f f e r e n c e zeroed and i t s  between  f o r each r u n , then SPKSKP w i l l  number (NSKIP) of  which the s p i k e i s  the p r e s s u r e d i f f e r e n c e  to the main program v i a one of  the z e r o i n g procedure the number of data p o i n t s  two r o u t e s .  points. If  dur-  zeroed i n one b l o c k  s  t  171 exceeds a preset"- number of p o i n t s number of  data p o i n t s  (JFRAC;  i n the b l o c k ) ,  u s u a l l y set  to o n e - h a l f  the main program i s  instructed  i g n o r e the b l o c k b e i n g a n a l y s e d and s t a r t on the next one. normal r e t u r n process  SPKSKP p r i n t s out the t o t a l  skipped and then d e l i v e r s program FIDDLE. each b l o c k ) Fourier  A1.3  p (t) s  and S ( t )  number of d a t a  and i s  to the  points  to the d i g i t a l p r e c o n d i t i o n i n g (over  r e t u r n e d to ther.main program f o r  analysis.  Definitions In o r d e r to make the f o l l o w i n g accounts  tions  are g i v e n f i r s t .  uncertainties  as  It  i s hoped t h a t  their  to the l o c a t i o n of f a c t o r s  We wish to perform a s p e c t r a l a n a l y s i s begins a t time t = 0 and extends to t = T . fixed intervals  tj where  A t is  frequency f quencies is  During  There the p r e s s u r e s i g n a l has any r e s i d u a l mean  removed from i t  the  g  of  inclusion w i l l  of Tt , 2,  defini-  eliminate  etc.  on a s i g n a l v ( t ) , This s i g n a l  is  which  sampled  at  time  = jAt,  j = 0,  1,  2,  the sampling i n t e r v a l , and i s ( v(t)  exceeding  c l e a r e r some b a s i c  is  I/At the r e c i p r o c a l of  arranged to c o n t a i n no s i g n i f i c a n t  f /2).  A1.3, the  sampling  energy a t  The r e s u l t i n g c o l l e c t i o n of d a t a p o i n t s  s  frev  (tj)  then d i v i d e d i n t o equal time i n t e r v a l s T , where  N A t  =  T  «  T  These s e t s of N A t data p o i n t s The  spectral analysis  A1.4.  are h e r e a f t e r  called  "blocks".  technique used i n the p r e s e n t work i s  one  developed by Cooley and Tukey (1965), and commonly r e f e r r e d to as  the  172 " F a s t F o u r i e r Transform" PKFORT). the s e t  T h i s program i s of  describes  coefficients  name of  of  are complex,  Af  and extending  =  the d i s c r e t e F o u r i e r s e r i e s which s i g n a l w i t h i n that b l o c k .  and occur a t f r e q u e n c i e s  cross-spectral  exactly These co-  separated by. i n t e r v a l s  A1.5 the f o l d i n g or N y q u i s t f r e q u e n c y .  s  the d a t a p o i n t s  points  !/-£•  from f = 0 to f / 2 ,  The r e a l p a r t of  the a c t u a l computer program i s  used to produce f o r each b l o c k of d a t a  the sampled ( d i s c r e t e )  efficients  is  (the  the F o u r i e r  coefficient  i n the b l o c k .  at f = 0 i s  the mean v a l u e of  From these F o u r i e r c o e f f i c i e n t s  e s t i m a t e s a t each frequency are determined.  repeated f o r each complete  power or c r o s s - s p e c t r a l  b l o c k of data p o i n t s  power or  This  analysis  i n the r u n ; the  estimates f o r a complete r u n are the  final  averages  over the number of b l o c k s i n the r u n . The F a s t F o u r i e r Transform program thus accepts an a r r a y v ( t j ) data points A(f ) k  and r e p l a c e s  where f  them w i t h an a r r a y of  = k A f is  k  the frequency of  complex F o u r i e r  the k ^ harmonic of  of  coefficients the  signal  w i t h i n the b l o c k . N-l v(tO  where N i s  exp (-2TT i j k / N )  the number of data p o i n t s  k/2TI  = k/NAt  -iw.tj  =  I  A(f )  =  B(f )  k  =V - 1 .  Putting  and u s i n g A 1 . 3 and A 1 . 4 ,  A(f ) k  i n the b l o c k and i  A1.6;  k  +  i  C(f ) k  At  A1.7.  A1.8  173 is  complex;  it  harmonic at In  contains  amplitude and phase i n f o r m a t i o n on the  the frequency  f . k  the s p e c t r a l a n a l y s i s  Interpretation",  program SCOR (see  p. 82) e s t i m a t e s of  (t>e(fk)  k<£kf  =  =  2Af  (power)  X . U f 2  k  "Data A n a l y s i s and  spectral  )  2  which the e s t i m a t e a t each frequency i s  The f i n a l set  of  ^) (fk)> e  some k i n d of "ensemble"  averages  ( J ) (f ) . k  the average over s e v e r a l r e a l i z a t i o n s of p h y s i c a l c o n d i t i o n s ;  that i s  begun on s e v e r a l o c c a s i o n s of  the  (identical)  itself)  the^^ (f ) e  n  its  estimated  should each be s t a t i s t i c a l l y  or of  their  The a n a l y s i s v(t)  would a r i s e  the v ( t )  measurement following  then c e r t a i n parameters  power spectrum ^ ( f e  k  )  were  the onsuch  (and  v(t)  d i s t r i b u t e d i n the same f a s h i o n f o r  We seek to f i n d "good estimates" of  these p a r a -  distributions. of  v  =  1  =  0  equivalent  from  which d e r i v e from the same  (tj)  i n a s e r i e s of b l o c k s i s  equivalent  to m u l t i p l y -  by the "data window" D(t)  This is  to say i f  are estimates  k  The ( J )  a t i d e n t i c a l time i n t e r v a l s  each r e a l i z a t i o n of v ( t ) . meters  of v ( t )  causal processes,  as the v a r i a n c e of v ( t ) ,  ing  ^  complete b l o c k s i n the r u n .  (fk)  set  estimate of  averaged over the t o t a l number  S i n c e the b l o c k and run d u r a t i o n s are f i n i t e  set  A1.9  k  J  the power spectrum f o r a r u n c o n s i s t s of a s i n g l e  of  C(f ) l  +  2  densities  1  are computed a t each frequency f o r each b l o c k .  of  Fourier  0  -  t  ^  T  otherwise  0 X ( f ) w i t h the " s p e c t r a l window" to c o n v oL l vviinnggU £  A L I O  174  (J) (f)  =  D  this  is  exactly  Al.ll;  2  the " r e c t a n g u l a r l a g window" d i s c u s s e d  Tukey (1959; pp.  i n Blackman and  68-70).  The m o d i f i e d is  Tsin (21Tf^/2) (2TTfT/2)2  (spike-contaminated)  pressure s i g n a l p ( t )  = p(t)S(t)  s  the o n l y one a v a i l a b l e w i t h which to form a c r o s s - s p e c t r u m w i t h  wave s i g n a l  7£ (t) .  =  which i s  ?£(t)S(t)  The F o u r i e r  The q u e s t i o n a r i s e s : used to o b t a i n the  transforms F T p ( f )  should i t be * £ ( t ) or  *£ (t) s  cross-spectrum?  and F T * £ ( f )  s  the  s  of p ( t )  and^ (t)  s  are  s  convolutions:  FT  (f)  =  FT (f)* FT (f)  FT, (f)  =  FT^(f)*FT (f)  p s  p  s  and A1.12,  s  where F T ( f ) g  quency, arc  is  s  the " F o u r i e r t r a n s f o r m window" of S ( f ) .  the phase spectrum  tangents of  FT^ (f).  the r a t i o s  Because F T p ( f )  s  Q p of  g  7£ (f) s  is  At a given  g i v e n by the d i f f e r e n c e  imaginary to r e a l p a r t s of F T p ( f ) i n p r a c t i c e not a v a i l a b l e )  have both been convolved w i t h the same t r a n s f o r m window, imaginary to r e a l p a r t s of F T ( f ) p  and F T ^ ( f )  same way by the c o n v o l u t i o n and t h e r e f o r e mation the same as ^ ^ ( f ) . with  ^(t),  Fourier the  ^(t)  the r a t i o of  If,  are both m o d i f i e d i n  ^Pg^gCf)  i-  s  the  and F T t j ( f )  the r a t i o s  of  the  tb^a f i r s t a p p r o x i -  on the other hand, p ( t )  is  s  the imaginary to  of and  g  (which i s  fre-  the r e a l p a r t of  correlated  the  p (t) s  t r a n s f o r m i s m o d i f i e d by c o n v o l u t i o n w i t h the s p i k e window w h i l e transform r a t i o is  s p e c t r a are used e x c l u s i v e l y  not.  For this  reason the p ( t ) ^ ( t )  f o r the p r e s e n t a t i o n of  s  the  s  results.  cross-  175 A1.4  Effects  of a Regular Spike Window  In o r d e r to assess the i n f l u e n c e r e s u l t i n g from the removal of s p i k e s , signals  have been generated  of "holes"  i n the p r e s s u r e  signal  h y p o t h e t i c a l p r e s s u r e and wave  on the computer and analysed w i t h the same  procedures used on the f i e l d  data.  Both s i g n a l s  c o n s i s t of a  single  s i n u s o i d w i t h zero mean;  they have the same amplitude and frequency but  differ  T h e i r frequency i s  i n phase by 3 0 ° .  does not c o n t a i n an i n t e g r a l number of s i g n a l may be zeroed f o r a f i x e d b e i n g any p o s i t i v e  integer;  arranged so a d a t a b l o c k  cycles.  l e n g t h of  exactly  both L and the d u r a t i o n of  are a n a l y s e d ;  the p r e s s u r e and wave " s i g n a l s "  presented,  c o n s i s t of a s i n g l e c y c l e of  have been  this  sinusoid.  and v a l u e s  different  of L have  been  from o n l y one com-  and the s p i k e f u n c t i o n repeats  i n t r o d u c e d by the r e p e t i t i v e  both  once per  T h i s combination was chosen because i t  shows the s p e c t r a l d i s t o r t i o n s  pro-  s p e c t r a of  f o r which the "pressure" and "wave" s i g n a l s  sinusoid,  L  analysed  A number of  F o r the sake of b r e v i t y the c r o s s - s p e c t r a  bination is  the "holes"  power and c r o s s  are o b t a i n e d .  combinations of s i n e waves, h o l e d u r a t i o n s ,  either  for a given run.  p r e s s u r e and wave s i g n a l s  as the f i e l d s i g n a l s  analysed.  desired,  time once every L c y c l e s ,  duced by the z e r o i n g procedure can be p r e s e t These h y p o t h e t i c a l  If  spike  clearly function  used and because the c o n t a m i n a t i o n i n t r o d u c e d by removing d a t a from every c y c l e of  the s i g n a l  o c c u r r e d i n any of  i s more severe than the contamination which  the runs  analysed.  The p a r t i c u l a r " s i g n a l s " p'(t)  =  5.0  used were  sin (21TNf /f ) 0  s  _2 dyne cm ,  176  In  •^'(t)  =  5.0  S  =  0  =  1  1  (t)  E q u a t i o n A1.13 N i s  sin (21TNf /f o  (Mf / f s o  the c y c l e number.  + 30) 4  of  o  is  g  the sampling frequency  chosen to be 0.4 H z , and M = 0,  1,  or (2TT x 20/125) r a d i a n s ,  and p ' ( t ) S ' ( t ) v e r s u s  (t)S'(t).  are d i s p l a y e d i n F i g u r e s 56 to  and * £ ' ( t ) ;  phase later.  The r e s u l t s  the power s p e c t r a of  p (t)S'(t) 1  the b l o c k  The v e r t i c a l b a r s on the  power s p e c t r a l e s t i m a t e s are s t a n d a r d e r r o r s about the mean over 10 b l o c k s f o r each s p e c t r a l e s t i m a t e ;  of  ...  63.  F i g u r e 59 c l e a r l y shows the envelope of  s p e c t r a l window g i v e n by E q u a t i o n A l . l l .  width.  2,  1024 samples each were a n a l y s e d f o r two c r o s s - s p e c t r a :  F i g u r e s 56 and 59 show r e s p e c t i v e l y = P (t)  s  and a "hole" o c c u r r e d once per c y c l e a t the  versus ^ ' ( t ) ,  these analyses  N L. (Mf It + 50) — s o  the sample number and f  2 If (M + 30/125) and ended 20 samples,  p'(t)S'(t)  A1.13  Thus from E q u a t i o n A1.13 the p e r i o d 6"fthe s i n u -  soids? was 125 samples,  Ten b l o c k s of  + TT/6)cm, and  otherwise.  (chosen here to be 50 H z ) , f is  s  the  the h o r i z o n t a l bars g i v e the band-,  The p r e s s u r e power spectrum i n F i g u r e 56 shows t h a t the  effect  m u l t i p l y i n g the p r e s s u r e s i g n a l w i t h the "Spike f u n c t i o n " (which  r e p e a t s once per c y c l e s t a r t i n g a t 2 IT (1.24) and ending a t 2TT (1.4) radians)  is  the g e n e r a t i o n of harmonics of the fundamental; r e p l i c a s of  the b l o c k s p e c t r a l window appear a t 0 . 8 , spectrum i s  1.2,  and 1.6  Hz.  The coherence  shown i n F i g u r e 57 and the phase spectrum i n F i g u r e 58.  w i l l be remembered from E q u a t i o n s A1.13 t h a t the phase of the signals  lags  the wave s i g n a l by 3 0 ° .  It  pressure  A l t h o u g h the phase spectrum i s  177 almost 0.4  constant  w i t h frequency near 0.4  Hz the measured phase l a g  at  Hz i s 3 8 ° . T u r n i n g now to F i g u r e s 60 to 63,which d i s p l a y c r o s s - s p e c t r a  p (t)  and  s  0.1  s  (t),  the phase at 0.4 Hz i n F i g u r e 62 i s  of 3 0 ° , l e n d i n g credence  Ps  s  seen to be w i t h i n  to the arguments made on p.174  that  the  c o r r e l a t i o n should be used f o r f i n d i n g the phase spectrum r a t h e r  than the 0.4  between  fo ^  correlation.  /  s  Note the l a r g e coherences a t harmonics of  Hz i n F i g u r e 61 as compared w i t h the r a t h e r s m a l l coherences  the harmonics i n F i g u r e 57;  the h i g h - f r e q u e n c y  correlations  d a t a are a l s o found to be l a r g e r than those  of  the p  the  p ^  correlations.  s  ^ s  s  entirely  A1.5  from the f i e l d  s  P  e c t  ra  determined by the i n f l u e n c e  Because  is  in  the  spikes.  of  of  the p r e s s u r e only a  signal  in  qualitative  the s p i k e s on the p r e s s u r e  power  T h i s estimate has been made by m u l t i p l y i n g an observed wave ^(t)  by the s p i k e f u n c t i o n S ( t ) signal,  on the wave power  spectrum.  The  s p e c t r a of  ^(t)  and 6)  d e r i v e d from the  and o b s e r v i n g the e f f e c t of  and of  *£(t)S(t)  runs i n which a p p r e c i a b l e s p i k e  tion is  of  irretrievably lost,  the e f f e c t  observed p r e s s u r e  3,  °7 s  s  t h a t at h i g h f r e q u e n c i e s  the i n f o r m a t i o n from the r e g i o n s  e s t i m a t e can be made of  field  P  the Spikes on the Power S p e c t r a  which s p i k e s are present  spectra.  the  from the f i e l d d a t a the observed phases are p r o b a b l y  The E f f e c t of  signal  This indicates  coherences of  at  i n F i g u r e s 64 to 67.  ^ s ^  a  the s p i k e window  r  e  s  n  o  w  n  f  o  r  t  c o n t a m i n a t i o n o c c u r r e d (runs  The g e n e r a l  seen to be a smearing out of  =  simultaneously  effect  of  the s p i k e  the uncontaminated  spectra,  n  e  1,  2b,  contaminawith  178  energy b e i n g removed from the peak and added at the h i g h energy i s  a l s o added a t f r e q u e n c i e s  peak i n runs 1 and 6, w h i l e i t 2a and 3.  assuming t h a t  The a c t i o n of  added to  T h i s apparent  on the s p e c t r a and the  the presence of s w e l l  the s p i k e s  to decrease t h e i r f a l l o f f  rate.  l a r g e r at higher frequencies. since neither  spikes.  frequencies  The contaminated s p e c t r a c r o s s  the  Hz, the contaminated s p e c t r a b e i n g Above 2.5 Hz the s p e c t r a d i v e r g e  the p r e s s u r e nor the waves are well-measured a t these l a r g e d i s c r e p a n c i e s  The E f f e c t of  spike  i n the wave f i e l d would  on the wave s p e c t r a a t h i g h  uncontaminated s p e c t r a near 1.5  A1.6  the s p i k e s  the l i k e l i h o o d of l a r g e numbers of  frequencies,  i n runs  the s w e l l peak may be r e l a t e d to the t o t a l number of  occurrences,  is  removed from these r e g i o n s  the s w e l l peak i s p r e s e n t .  c o r r e l a t i o n between the a c t i o n of presence of  spectral  below the wind-generated peak i f no s w e l l peak appears  the spectrum and removed i f  increase  below the wind-generated  I t appears t h a t i n the f o u r runs shown energy i s  the f r e q u e n c i e s in  is  frequencies;  are not d i s c u s s e d  rapidly;  these  further.  the Spikes on the Phase S p e c t r a  The r e c o r d e d p r e s s u r e s i g n a l c o n s i s t e d interspersed with regions  of s p i k e a c t i v i t y .  the s p i k e s have on the phase s p e c t r a ,  g e n e r a l l y of c l e a r  signal  As a check on the  effect  the s p i k e d e t e c t i o n and removal  program SPKSKP was a l t e r e d s l i g h t l y so t h a t the d e t e c t i o n of a s p i k e anywhere i n a d a t a b l o c k caused the e n t i r e b l o c k to be i g n o r e d .  T h i s a l t e r a t i o n i n SPKSKP allowed the a n a l y s i s  of b l o c k s of  clear  179  data;  these were,however,  out the r u n .  In p r a c t i c e ,  good s t a t i s t i c s for  scattered to get  a t random time i n t e r v a l s through-  a sufficient  number of b l o c k s  the b l o c k l e n g t h had to be shortened from 20 seconds  the "spikey" d a t a to 5 seconds f o r the c l e a r d a t a .  T h e r e f o r e the  lowest frequency which could be r e s o l v e d was r a i s e d from 0.05 the case of  the "spikey" d a t a to 0.2  " R e s u l t s " f o r d e t a i l e d d e s c r i p t i o n s of  F i g u r e s 68 to >71.  I n each f i g u r e  &  2b, 3, and  these runs) are shown i n  two curves are drawn c o r r e s p o n d i n g to  two ways of p r e c o n d i t i o n i n g the d a t a ; between p  Hz i n  Hz f o r the c l e a r d a t a .  Phase s p e c t r a between p r e s s u r e and waves f o r runs 1, 6 (see  the f u l l  curve shows the phases  and ^ f r o m a l l b l o c k s i n each r u n f o r data s u b j e c t e d  s p i k e removal p r o c e s s ,  to  v e r t i c a l lines  spikes  occurred.  at h i g h and low f r e q u e n c i e s  The  are the  quency l i m i t s beyond which the r e l e v a n t phases become u n r e l i a b l e ; is,  the f r e q u e n c i e s  d i s c u s s i o n of  this  the  the d o t t e d curve shows the phases between p and  from the b l o c k s d u r i n g which no d e t e c t a b l e d o t t e d and f u l l  for  where the coherence f a l l s coherence  below about 0.3  l i m i t i n "Results",  p.  frethat  (see  the  115).  The phase s p e c t r a computed i n the two d i f f e r e n t ways show subs t a n t i a l agreement.  The l a r g e s t d i s c r e p a n c i e s  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 g i v e n  fre-  quency.  T h i s comparison of  c l e a n and s p i k e - c o n t a m i n a t e d d a t a p r o v i d e s  180  d i r e c t evidence  t h a t the e f f e c t of  the s p i k e s on the phase s p e c t r a  is  quite small.  A1.7 The E f f e c t of  the Spikes on the Energy F l u x S p e c t r a  The p o s s i b i l i t y s t i l l tortions  exists  that s i g n i f i c a n t  can occur i n the quadrature spectrum Qu  the g e n e r a l l y s m a l l d i s t o r t i o n s the energy f l u x s p e c t r a E ( f ) through r e l a t i o n 5.9)  spike-induced d i s (f) , i n s p i t e of  i n the phase s p e c t r a ;  for this  (which are d e r i v e d from the  reason  Qxif^(f)  computed from the " c l e a r " and "spikey" p r e s s u r e  d a t a are a l s o compared, i n F i g u r e s 72 to 75". I t w i l l be seen t h a t a l t h o u g h the two E ( f ) closely  i n runs 2b and 6,  pancies between the  large--so  large i n fact  e r r o r of each o t h e r . scatter p.  146.  of  the s c a t t e r  between the two E ( f )  i n the E ( f )  agreesquite  there occur what appear to be l a r g e d i s c r e -  two s p e c t r a i n the other two runs (1 and 3 ) .  The s i g n i f i c a n c e c a l l y since  spectra  these d i s c r e p a n c i e s  cannot be assessed  statisti-  over the d a t a b l o c k s analysed i s i n v a r i a b l y that a l l of  the " l a r g e " d i s c r e p a n c i e s  noted  s p e c t r a i n runs 1 and 3 l i e w i t h i n one standard The p o s s i b l e spectra is  physical significance  of  this  large  taken up i n " D i s c u s s i o n of R e s u l t s " on  181 The observed d i s c r e p a n c i e s  i n runs 1 and 3 between the E ( f )  a r e hence w i t h i n the observed s c a t t e r  i n the r e s u l t s .  i n run 1 occurs at one frequency o n l y , of  the peak of  of  the " c l e a n " d a t a i n d i c a t e s  negative  E(f)  analysis.  the wave spectrum.  spectrum)  at  0.37  probably closer discrepancies  —9  erg cm  Hz there i s  1  T h i s agreement  is  little  It  is  interesting  the r e s u l t of  the w e i g h t i n g o f  of E makes i t unnecessary  f o r the d i f f e r e n c e s  the h i g h e r f r e q u e n c i e s ( F i g u r e 70)  indicates  are d i f f e r e n t  is  6.1)  generation is  that i n s p i t e  of  integrals  erg cm  9  sec  1  -  .  the h i g h e r - f r e q u e n c y i n the i n t e g r a t i o n p r o The good  agreement  to pursue f u r t h e r  the  spectra. The d i s c r e p a n c y noted  at  to 2 Hz) i n the phase s p e c t r a f o r the r u n  t h a t the average phase angles of  from those of  the s p i k e y d a t a i s  i n the two E ( f )  somewhat d i f f e r e n t .  (1.1  — E from the c l e a r d a t a i s  Table  the "spikey" data g i v e s E =  cess, which minimizes low-frequency d i s c r e p a n c i e s .  Run 3 ( F i g u r e 74)  a larger  s p e c t r a from the two s e t s of d a t a the  s p e c t r a l e s t i m a t e s (they have l a r g e r bandwidths)  reasons  analysis  l i k e l i h o o d that  and the "clean" data g i v e s E = 16.3  between the two v a l u e s  frequency  the r e s u l t from the "spikey" d a t a  to b e i n g r e a l i s t i c .  sec"  (has  r u n was lower (see  under these s p e c t r a are almost i d e n t i c a l ; 16.6  the  frequency than does the "spikey" d a t a  therefore  i n the E ( f )  is  I t w i l l be noted t h a t the  S i n c e the wind speed f o r t h i s  was a c t u a l l y o c c u r r i n g ;  The d i s c r e p a n c y  s t r o n g e r wave g e n e r a t i o n  this  than the wave speed at 0.37  Hz; t h i s  spectra  the s p i k e y d a t a .  62 e r g cm - 2 - 1  30 erg cm  sec  the " c l e a r " data  The i n t e g r a t e d energy  flux  - 2 - 1 sec , w h i l e the v a l u e determined  , 507, lower.  Both of  the E ( f )  spectra  from r u n 3 i n d i c a t e wave damping a t 0.2 H z , a t which frequency a l a r g e amount of s w e l l there i s  in this  energy was p r e s e n t case no o b j e c t i v e  i n the wave power spectrum.  Therefore  c r i t e r i o n by which to judge which of  18>2 the two a n a l y s e s expected via  i s more n e a r l y c o r r e c t .  to underestimate E ( f )  the s p i k e removal p r o c e s s ;  energy i n p u t was l a r g e s t  The "spikey" d a t a might be  s i n c e i t has had energy removed from on the o t h e r hand i f  A1.8  assumed  that  then s i n c e  the c l e a r data would miss  energy t r a n s f e r and hence might be expected  better  is  to the l a r g e s t - a m p l i t u d e waves,  l a r g e s t waves caused s p i k e s more o f t e n ,  only choice remaining i s  it  it  the  this  to underestimate E ( f ) . The  to assume t h a t E f o r r u n 3 i s not known to  than -50%.  Conclusions The o b j e c t of  removal of s p i k e s  this  appendix has been to study the e f f e c t s  of  the  from the p r e s s u r e s i g n a l on the power and cross  spectra  of p r e s s u r e and wave s i g n a l s which form the most important r e s u l t s  of  this  below.  thesis. 1.  The c o n c l u s i o n s r e s u l t i n g from t h i s  By computing the power and cross  s i n e waves of  study are l i s t e d  s p e c t r a of  computer-generated  the same frequency but d i f f e r e n t phase,  one or both of which  was m o d i f i e d to be r e p l a c e d by zeros once every c y c l e f o r a d u r a t i o n of o n e - s i x t h of a c y c l e first is  is  (a " h o l e ' ) , 1  two c o n c l u s i o n s have been r e a c h e d .  t h a t the power spectrum of  d i s t o r t e d so t h a t energy i s  The  the "hole-contaminated" s i n e wave  removed from the s p e c t r a l peak a t  the  fundamental frequency and i n s e r t e d a t the harmonics of  the s i g n a l .  second c o n c l u s i o n i s  the two s i n e waves  compared i s if  t h a t the phase d i f f e r e n c e between  a c c u r a t e l y reproduced by the c r o s s - s p e c t r a l a n a l y s i s  the cross  spectrum between  by the presence of h o l e s 2. field  The e f f e c t of  data i s  two s i g n a l s  only  contaminated i n the same way  i s used to determine the  phase.  the s p i k e removal on the power s p e c t r a of  a r e d i s t r i b u t i o n of energy,  The  as i n c o n c l u s i o n ( 1 ) .  the There  183 i s however no evidence of  the frequency of  3.  t h a t the excess energy i s  the peak of  shifted  i n t o harmonics  the wind-generated wave spectrum.  A comparison has been made between phase s p e c t r a between  p r e s s u r e and waves f o r two d i f f e r e n t methods of pressure data.  treatment of  the  In one case the d a t a were a n a l y s e d n o r m a l l y i n  sense t h a t when a s p i k e o c c u r r e d i t was r e p l a c e d w i t h z e r o e s ; other case o n l y data b l o c k s i n which no s p i k e s The comparison shows f a i r l y good agreement, s p e c t r a are a f f e c t e d conclusion 4.  appeared were  suggesting  that  the in  the  analysed. the  l i t t l e by the energy r e d i s t r i b u t i o n noted  phase in  (2).  A comparison has a l s o been made of energy f l u x s p e c t r a between  p r e s s u r e and waves f o r the aforementioned pressure data.  two methods of  treating  the  Of the f o u r runs w i t h a p p r e c i a b l e numbers of s p i k e s ,  the  agreement between the energy f l u x s p e c t r a computed from the d a t a a n a l y s e d i n the  two ways was v e r y good i n two cases (runs 2b and 6 ) ;  i n the other two cases fluxes  (run 1 and 3) was poor.  agreement  The i n t e g r a t e d  E were computed f o r the two runs where agreement was poor;  E values  computed from the c l e a n and s p i k e y d a t a agreed c l o s e l y  1,  the E computed from s p i k e y data underestimated  while  from c l e a n d a t a by 507o i n r u n 3.  As a r e s u l t  of  spike contamination i s  the computed r e s u l t s  present  f o r the run are  in this  the  f o r run  t h a t computed  the run 1 d i s c r e p a n c i e s  are c o n s i d e r e d to be unimportant w h i l e those i n r u n 3 i n d i c a t e serious  energy  that  r u n and t h a t the a c c u r a c y  suspect.  IC  IF  )C  "'-('  S P I K E O R ,  !  •  IF(  T H E  ROUTINE  W I L L  NEND DO C H E C K  C O R R E C T  F O R  T E S T  =  I  +  l+K  =  200  IF •C  GO T O  ALSO  A H E A D .  A L L . P O I N T S  C O R R E C T I O N - '  PROGRAM  W I T H  R O U T I N E  I E R S P  =••  2-  .).  T O  IF  A N O T H E R  N S K I P  S P I K E IT.  B E Y O N D  K  O C C U R S  DURIMG-NS-K-IF  T O T A L I Z E S  C O R R E C T E D  J =  K  + N S K I P  E N D  F O R  -  M S T A R T , O F  1 MEND  A R R ( N P T S )  J.GE.NPTS)  (  C H E C K  G O T O3 0 0  . S A T U R A T I O N -  A  O F  A H E A D  - ••  F O R  +  =  IF  D I F J . G T . D I F M A X  (  A R R C J + l )  T O  D  . GO  G O I N G  .  T O  25  .  .  .  .  ..  0  S P I K E  (,J)  - A R R )  GO  T O  25  0  =0.  SARR(J-) K= K+1 M = M +  20 0  N E X T  DIFJ  A R R ( J )  1  •  S T A R T  ING  310  C H E C K S  I F ( A B S ( A R R ( J ) ) . G E . V A L M A X ) C  G A L L  - P O I N T S . M S T A R T  C  C O U N T I N G ,  )  •C •  L  M.GT.JFRAC  C O R R E C T I O N  lC  S T A R T  1; R E T U R N T O ) GOT O 410  C  DO  ,  IFSKP I S IFSKP.GE.l  IF( 1=1  |l*9 [50  D E T E C T E D  IF  = - 0 . •  -  -  -.  .  .  1  C O N T I N U E GO  2 5 0  T O  =0.  A R R ( J ) SARR  \  v.  3 00  )  ( , ! ) - = •  0.  -  -  ••  ?  K=K+1 M  =  GO 3 0 0  ;  M  +  T O  1  5 0  C O N T I N U E GO  •310--  T O  320  NSTOP=  I  . ' W R I T E (  +  FORMAT ( 1 H SPIKE  3 1 5  K )  I E R S P  — 4 1 0  IS  I E R S P  = -330  T O  W R I T E C  415  R E M O V A L  R E T U R N E D  GO  I  FR  AS  0NABLB , 14, FRAC*IBL IS  Q U E S T  S T O P P E D 2  . ...  N S T O P  IF  I  MORE  A T  9 H T H  hl\\  T H A N V A L U E .  V A L U E S  S K I P  )  ,  E X C E E D E D  - -  -  R , 4 1 5  )  —  •  I  , 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  .  BLOCK.  IGN  )  S P  =  ?  R E T U R N 32  0--  I F L I N E  I F L I N • +-  =  W R I T E ( 6 , 3 2 5 )  FORMAT(  325 1  WITH  C  IERSP  3 30  - •  -  1H  =  0  U  R  E  END  1H . ,  0 MEANS  I E R S P T  I F L I N E )  , 2 X , 5 4 H S P ! K E  , A 8,  R  1-  A I F L I N C  N  1 X , I 5,  N O R M A L  -  2 1H  M  ,  R E M O V A L D A T A  C O M P L E T E D .  V A L U E S  SKIPPED  S K I P P E D .  V A L U E S  R E P L A C E D . •  )  -'  '  R E T U R N  - •  -  •  .  *  2  FORMAT(2H l O R E H  ...  J F R A C ,  0 , 2 X , 2 9 H D A T A  1 P E D . C  •-  6 , 3 1 5  -  .  .  ..  .  , SUBROUTINE S P K S K P C A R R , 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 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 PTS  =  N P T S  •  D I M E N S I O N  C  R E A D  C  AFTER  C  P O I N T S .  C  FRAC  IN  S E T S  C  IF  S P K S K P  D A T A  . 16  F(  O P E R A T I O N  S K I P S  (5,16  0  )  (  U X ,  B L O C K S  A IF L I N  C2 ) /  )  00  -TO.  ......  .-,  D I F M A X ,  F 6 . 5 ,  2(  '  CORRECTED  _  V A L M A X , ),  A  SKIPPED.  ON  E N C O U N T E R  S P I K E .  L I N E S /  .  ....  N S K I P ,  7X,  S P I K E S  S U C C E S S I V E  13,2  I F L I M ,  I F S K P  ID)  ( 9 X ,  PTS*FRAC  =  FORMAT  ( 1 H 0 , 2 X , I ^ H M A X I M U M  | i f , - l H .  /  )  J F R A C ,  -3X,  REMOVAL =  6H  Z E R O S ,  i * X , F 6 . 1  S K I P S  E N C O U N T E R I N G  . . .  A L S O  '  S P K S K P  ON  6H  F R A C ,  6,17  K  0,  2 0  W R I T E (  S P I K E  IF  WHOLE  A I F L I N ( l ) ,  READ  •1  20  OF  - I E R S P . • H E .  J F R A C  C  IS  MODE  F O R M A T  17.  •  IS B A S I C N U M B E R OF D A T A P O I N T S 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 MAXIMUM ALLOWED V A L U E OF D A T A P O I N T S . TO S E T M A X . ALLOWED NO. OF P O I N T S TO B E  M U L T I P L I E R  I F S K P 1,  ~  A I F L I N ( 2 )  AMD  S P I K E . . D I F M A X V A L M A X  IS  S A R R ( l ) ,  DATA.,  DATA.NSKIP  R E Q U I R E D  A-  C  -  A R R ( l ) ,  U R E  D I F M A X ,  I FMAX-  8HD  =,  V A L M A X ,  ALLOWED  F7  . 3 , 2X,  ROUT I M E . I N I T I A L I Z E  N S K I P  OF  NUMBER  8 H V A L M A X  =,  SKIPPED F 8 . 3 , 2 X,  V A L U E S  IS  ,  7 H N S . K I P . . = , .1.3  )  COUNTERS.  0  L = 0 M  =  0  MEND |C }C  =  |'C  BY  |C  CORRECTS  R E P L A C I N G  C  RANGE  C  T O T A L  •G  0  S T A R T -CH-ECK I HG GREATER THAN M A  I  IS  T H E M  P O I N T S  300  C  1=2,  I F L I N . E O . 0  I N I T  •=-  NEND  =  NEND  =  K =  + (  I  TO  •C  K =0 T E S T FOR-  2h  P O I N T S SHORT  OF  THE  TO A  B E Y O N D  OUT  T H E  S E R I E S  END  OF  S P I K E  THE  NEW  S P I K E  OF  S P I K E .  NEW  IT  OCCURS- IN  S P I K E . K  T E S T  C O R R E C T E D  T H E  C O U N T S  R O U T I N E  W H I L E  P O I N T S .  2 2  K  -—  -  •  .  .  .  + 1  1  )  =  TO  -  A R R ( I N I T )  ARR(  S E E  +  1  I  IF  ))  -  2  L A S T  )  +  )  /  RK  ADD  C O R R E C T E D  P O I N T  G O T O -2 5 - -  •• -  R E A C H E D  -- •  S P I K E S  -ARR(1-1)  A R R ( I )  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  .. - . ^ ? v ;  P R E S E N C E .GT.  I F C A B S C D I  F)  D A T A  R E S E T  GO  TO  TO  END  S T A R T  W I L L , I F  300  =  R E S E T  L=  N S K I P  U S E D  T H E  FROM  AND  BY  1  -  FOR  D l F  GO TO  A R R ( J E N D )  C O U N T E R  C H E C K  IF  L I M E  S P I K E ,  S U C C E S S I V E DATA. P Q.I . N T ! S E T T I N G T H E M TO ZERO O R  B E T W E E N  POINTS  2 5  )  GO -  IF-CL.-GT.CK  jC  C  )  J E N D  ARR(  2 5  TO  L . N E . 2  GO C  IS  TO  FF-ER-ENCES  DATA  N P T S GO  IF(  T E S T  A  C O R R E C T  IF(  RK  -2h  S T R A I G H T  C O R R E C T E D , L  I  L+1  ADD 2 2  A  D  CORRECTS  A F T E R  I N C R E M E N T E D  I F ( L . E O . O ) L =  W I T H  N S K I P ,  B E I N G  DO  B Y - L O O K I N O - F O R  POI NTS  N S K I P  OF  D A T A  X D I F . P R O G R A M  O K ,  0  OF  - I N I T I A L  D I F M A X L,  )  R E T U R N  -  SPIKE-  GO TO  TO  (+QR-  -GO l-NG)  -  -  '  -  If 9  D A T A  C H E C K I N G  LOOP •  TO  300  ' '  .-•  x  '  APPENDIX  la:  FORTRAN IV PROGRAM FOR SPKSKP  APPENDIX 2:  THE BOUNDARY BAY EXPERIMENT  A2.1  Introduction  184  A2.2  D e s c r i p t i o n of  A2.3  Equipment  186  A2.3.1  The Buoy Sensor  186  A2.3.2  The Reference  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  Site  185  Barocel  A2.4  Calibration  190  A2.5  Data A n a l y s i s  191  A2.6  Results  192  A2.6.1  Summary of Relevant I n f o r m a t i o n  192  A2.6.2  The Hot-Wire Data  192  A2.6.3  The P r e s s u r e Power S p e c t r a  195  A2.6.4  Non-Dimensional P r e s s u r e S p e c t r a  198  A2.6.5  Comparison of Power S p e c t r a w i t h Other Measurements  A2.6.6  The P r e s s u r e  A2.6.7  Pressure-Velocity A2v6.7a  A2.7  ,  199  Cross-Spectra  202  Correlations  Comparison w i t h F a v r e et  208 al  Conclusions  (1957)  .  .  .  210 215  A2.7.1  Sensor  Tests  A2.7.2  The S t r u c t u r e of  215 the P r e s s u r e F i e l d  216  APPENDIX 2:  A2.1  THE BOUNDARY BAY EXPERIMENT  Introduction In  the summer of 1968  the o p p o r t u n i t y arose of  p r e s s u r e sensor used i n the buoy w i t h others by Mr. J . A . E l l i o t t ,  a fellow  student  at  comparing the  b e i n g developed  and used  the I n s t i t u t e of Oceanography.  Two sensors were used i n the comparison w i t h the buoy s e n s o r . was an unmodified " B a r o c e l " t r a n s d u c e r , Datametrics Inc.  a commercial d e v i c e made by  of Waltham, M a s s . , U . S . A . ,  pressure f lu c tu ati on s  on a f a i r l y l e v e l  which was used to measure  sand s u r f a c e .  probe developed by E l l i o t t which used a B a r o c e l as i t s w i t h which p r e s s u r e measurements air  f l o w above  the sand  spectra,  w i t h each of  and thus  of  its  to e s t a b l i s h whether or not  to compare the  pressure  resulting  the buoy sensor and the The p o s s i b i l i t y  existed  two might be contaminated by dynamic  The unmodified B a r o c e l was taken as the standard f o r the  tests,  p o r t b e i n g arranged i n a s t a n d a r d manner f o r  measurements.  A secondary o b j e c t i v e was of  turbulent  o r i g i n a t i n g from the d i s t u r b a n c e to the flow caused by t h e i r  pressure-sensing  surface  the  the experiment were to o b t a i n  the t h r e e s e n s o r s ,  t h a t p r e s s u r e s measured by the  shapes.  t r a n s d u c e r and  were made i n the body of  E l l i o t t probe were measuring true p r e s s u r e s .  pressures  The o t h e r was a  surface.  The p r i n c i p a l o b j e c t i v e s measurements  One  the p r e s s u r e f i e l d  to make some p r e l i m i n a r y  investigations  i n the a i r , at v a r i o u s h e i g h t s and at v a r i o u s down-  184  185 stream and c r o s s - s t e a m  A2.2  D e s c r i p t i o n of A suitable,  the e f f o r t s  separations.  Site  easily  accessible  of another I n s t i t u t e  c o n d u c t i n g measurements  of  land s i t e was a t hand, thanks  student,  small-scale  M r . N. E . J . B o s t o n , who was  temperature s t r u c t u r e i n  atmospheric boundary l a y e r over a sand f l a t m i l e s by road from the  km seaward of level  c o m p l e t e l y covered a t h i g h t i d e , the d i k e a t low t i d e .  Its  are some s h a l l o w pools of r e s i d u a l water  plant;  the  experiment.  i s bare f o r about  s u r f a c e of f i n e less  sand i s  2.5 nearly  than 1 : 2,500.  There  ( a v e r a g i n g 1 - 10 m i n diameter)  Growing on the sand are numerous clumps of a l o w - l y i n g  these clumps are about 60 cm i n diameter and about 10 cm h i g h .  Thus the sand, a l t h o u g h smooth on the average, p r e s e n t which have h o r i z o n t a l s c a l e s of The h i g h t i d e s h o r e l i n e is  ten  Institute.  on the average w i t h a seaward s l o p e of  a t low t i d e .  the  a t Boundary Bay, about  F i g u r e 76 shows a map of the immediate a r e a of The sand f l a t ,  to  strewn w i t h l o g s ,  The d i k e i s  (about  has roughness  10 - 100 cm.  10 meters  from the base of  the  dike)  some as much as 20 m long and 1.5 m i n d i a m e t e r .  about 3 m h i g h and 10 m i n w i d t h .  on the shoreward s i d e ;  beyond i t  flat  There i s  to a l l o w us the use of  it.  the experiment to be s e t  the d e l t a of  km.  The three h u n t i n g shacks on the s h o r e l i n e 60 H z ) ; M r . H . R. H i p w e l l ,  a drainage d i t c h  farmlands ( p a r t of  the F r a s e r R i v e r ) s t r e t c h i n l a n d f o r 5 - 1 0  (115 VAC,  elements  a l l have e l e c t r i c  the owner of one,  Cables were c o n s t r u c t e d  up about 100 m from the  kindly  (Boston)  shoreline.  power  consented which allowed  186 A2.3  Equipment The t h r e e p r e s s u r e sensors and the ways they were deployed i n  experiment w i l l be d e s c r i b e d  A2.3.1  first.  The Buoy Sensor The t r a n s d u c e r , e l e c t r o n i c s ,  (see  the  "Experiment").  During the Boundary Bay experiment the apparatus  was arranged i n two ways. The microphone and i t s insulated  with a glass  A t f i r s t "case 1" the buoy was not u s e d .  o s c i l l a t o r were f i r m l y clamped i n t o a t h e r m a l l y  (5 cm t h i c k n e s s  ence s e n s o r .  and buoy have been d e s c r i b e d e a r l i e r  of styrofoam) wooden box a l o n g w i t h the  refer-  The aluminum backup volume used i n the buoy was r e p l a c e d j a r of  twice  w i t h 3 cm of s t y r o f o a m .  the u s u a l volume which was i t s e l f  insulated  The microphone was connected to a 0.8 mm  diameter p r e s s u r e p o r t which was f l u s h w i t h 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 p o l i s h e d h o l e i n a b r a s s p l u g (3 cm i n d i a meter) .  The p r e s s u r e c o n n e c t i o n to the sensor was made w i t h a 16 cm  l e n g t h of  2 mm I . D . s t e e l  tubing.  Long-term ( p e r i o d s of  100 seconds  o r more) p r e s s u r e v a r i a t i o n s were f i l t e r e d out by a pneumatic h i g h - p a s s filter  c o n s i s t i n g of  the b u i l t - i n leak around the diaphragm of  phone and the backup volume.  To make measurements w i t h t h i s  the e n t i r e box was b u r i e d f l u s h w i t h the sand. i n s u r e t h a t roughness  F o r the second s e t  the box.  Some care was taken to tap  than 1 cm).  of measurements  ("case 2")  and o s c i l l a t o r were p l a c e d i n a s e p a r a t e box, top of  arrangement  elements w i t h i n 2 m upstream o f the p r e s s u r e  were kept s m a l l ( s c a l e s i z e s l e s s  the  the m i c r o -  the buoy microphone  and the buoy was f i x e d  The buoy s u r f a c e was made as n e a r l y as  possible  to  187. i d e n t i c a l w i t h t h a t used d u r i n g t e s t s on waves, being that  the o n l y  the rubber diaphragm used i n the sea experiments was r e p l a c e d  by a c a r e f u l l y d r i l l e d 0.8 mm diameter p r e s s u r e which n o r m a l l y covered the e l e c t r o n i c s 5).  The e l e c t r o n i c s ,  2 mm I . D .  steel  package  tap i n the perspex (see  sheet  Experiment, F i g u r e  microphone, and backup volume were i n the t h e r m a l l y  i n s u l a t e d box below. of  difference  The tap was connected  tubing.  to the microphone by 20 cm  The experiments were conducted w i t h the buoy  and the box below i t b u r i e d i n sand to the depth a t which i t would normally f l o a t  i n calm water.  of v a r y i n g i t s  angle of  A2.3.2  No attempts  t i l t with respect  were made to see  The Reference B a r o c e l  I t has a r e s o l u t i o n of  less  than 0.1  dyne cm  of  used that was f l a t see F i g u r e 78).  - 2  i n the arrangement  Stated a c c u r a c y and l i n e a r i t y are 0.25% of f u l l  - 10  - 3  covering pressures  from 0 - 1 3  p l a c e d i n the same box as  m i l l i b a r s (1 mb  the r e f e r e n c e B a r o c e l was  the buoy sensor;  i t measured p r e s s u r e a t a  p r e s s u r e tap i d e n t i c a l w i t h t h a t used f o r the buoy sensor 5 cm crosswind from i t .  buoy sensor;  cm of  scale.  atmos.).  For the f i r s t s e r i e s of experiments  tap w i t h 13.5  type.  from 0 to 10 Hz (with some phase e r r o r above 7 Hz;  I t has 9 p r e s s u r e ranges dyne c m  the c a p a c i t a n c e  d i f f e r e n t i a l p r e s s u r e on  i t s most s e n s i t i v e range and a frequency response  3  effect  to the sand s u r f a c e .  The Datametries " B a r o c e l " t r a n s d u c e r i s  = 10  the  One s i d e of  the same  the t r a n s d u c e r was connected to  (2 mm I . D . )  the o t h e r s i d e was connected  s e r v e d as a backup volume.  and s i t u a t e d  t u b i n g as t h a t used f o r  the  the  to a s m a l l vacuum f l a s k which  Long-term p r e s s u r e changes were  equalized  188 w i t h a 4 cm l e n g t h of 0.15 between the high-pass  mm I . D . hypodermic n e e d l e ,  two p r e s s u r e p o r t s of  filter.  which was  connected  the t r a n s d u c e r and a c t e d as a pneumatic  T h i s arrangement and the  thermal box was designed and  b u i l t by E l l i o t t . F o r the second set  of measurements  the r e f e r e n c e  tap was i n the box, which was b u r i e d as f o r case 1; were b u r i e d a t d i f f e r e n t  times,  Barocel pressure  the buoy and i t s  crosswind and downwind from the  box  refer-  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  cation.  It  of  this  probe w i l l be d i s c u s s e d  i n a forthcoming p u b l i -  i s mounted on the end of a 50 cm " s t i n g " along which  pressures  sensed by the probe are t r a n s m i t t e d by tubes to a B a r o c e l t r a n s d u c e r w i t h a backup volume arrangement s i m i l a r to t h a t used f o r the sensor. O.D.,  reference  The t r a n s d u c e r and backup volume are c o n t a i n e d i n a 12.5 cm  35 cm l o n g aluminum c y l i n d e r w i t h a c o n i c a l nose;  t r a n s d u c e r output to the s i g n a l  conditioning electronics  l o c a t e d as f a r as 80 m from the probe (the  same i s  cables  carry  the  which can be  true f o r the  reference  Barocel).  A2.3.4  A u x i l i a r y Equipment Wind speed was measured w i t h two " C a s e l l a " cup anemometers;  returned e l e c t r i c a l  p u l s e s to counters  once per cup r e v o l u t i o n ;  total  a t the r e c o r d i n g a r e a which counted  counts were r e c o r d e d every f i v e  w i t h a one-minute break to w r i t e  these  the numbers down.  minutes  Wind speed was  o b t a i n e d from c a l i b r a t i o n curves p r o v i d e d by the m a n u f a c t u r e r .  Wind  189 speed was o b t a i n e d from c a l i b r a t i o n curves p r o v i d e d by the m a n u f a c t u r e r . Wind d i r e c t i o n was measured w i t h a s m a l l vane which turned the wiper of  a potentiometer,and  readout on a meter  c a l i b r a t e d at  five-degree  intervals. Downwind f l u c t u a t i o n s all  of  of  t u r b u l e n t wind speed were measured  the runs w i t h a "Disa" c o n s t a n t  temperature h o t - w i r e  for  anemometer;  the w i r e was p l a c e d v e r t i c a l l y i n the a i r flow at v a r i o u s h e i g h t s the  above  sand. The h o t - w i r e s i g n a l ,  were r e c o r d e d a t 1\  ips  the  three p r e s s u r e s i g n a l s ,  and a v o i c e  log  tape speed on a 14-channel "Ampex FR-1300"  p o r t a b l e i n s t r u m e n t a t i o n tape r e c o r d e r i n the FM mode ( v o l t a g e s are conv e r t e d to frequency-modulated audio s i g n a l s demodulated on p l a y b a c k .  and r e c o r d e d ; they  Frequency r e s p o n s e s i s f l a t  from 0 - 2  For some runs a mercury thermometer was used to measure a i r and sand  temperature  A p o r t a b l e mast 4.5 meters  e r e c t e d on the sand at low t i d e about 150 meters  tide  kHz.)  temperature.  F i g u r e 77 shows a t y p i c a l s e t u p . is  are  line.  S i g n a l cables  r e c o r d e r and a l l s i g n a l two c a b l e s  d i s t a n t which c o n t a i n s  c o n d i t i o n i n g equipment.  Power i s  l a i d from the h u n t i n g shack on shore^to  On the mast are three cup anemometers 0.5 meters  down from the h i g h  are l a i d from the sensors on and near  mast back to the p a n e l t r u c k 40 meters  A l s o on the mast,  the p a r t i c u l a r setup show, i s  the probe of  the  the  tape  o b t a i n e d from  the t r u c k .  at h e i g h t s of 4 . 5 ,  2.0,  (the m i d d l e anemometer d i d not f u n c t i o n r e l i a b l y and  from i t were not u s e d ) .  high  at a height the h o t - w i r e  and  results  of 4 meters anemometer.  for The  190 three p r e s s u r e instruments  are b u r i e d w i t h i n 2'meters of  the mast and o r i e n t e d i n t o  the wind.  sand i s  mast  The r o c k e t - l i k e o b j e c t on the  the c y l i n d e r c o n t a i n i n g the e l e c t r o n i c s  The wind d i r e c t i o n vane i s (the  experiments  the base of  f o r the E l l i o t t probe.  l o c a t e d halfway between the t r u c k and the  d i d not depend f o r t h e i r v a l i d i t y on an a c c u r a t e  measurement of wind d i r e c t i o n ) .  A2.4  Calibration All  three p r e s s u r e sensors were c a l i b r a t e d i n the l a b o r a t o r y w i t h  the equipment d e s c r i b e d i n the "Experiment" s e c t i o n  (p.  were i n every case c a l i b r a t e d w i t h the same p r e s s u r e used i n the f i e l d .  58 f f ) .  tubing  They  configurations  The amplitude and phase c a l i b r a t i o n f o r the buoy  sensor a t f r e q u e n c i e s  from 0.05  to 10 Hz are d i s p l a y e d i n F i g u r e 78.  These were o b t a i n e d by s i n u s o i d a l l y v a r y i n g the p r e s s u r e i n the t i o n drum at v a r i o u s f r e q u e n c i e s .  calibra-  The buoy sensor s i g n a l was compared  w i t h t h a t from a B a r o c e l which had one p o r t a t drum p r e s s u r e and the o t h e r a t atmospheric  pressure.  The amplitude and phase of  the B a r o c e l s i g n a l began to d i f f e r a  measurable amount from the buoy sensor a t f r e q u e n c i e s is  assumed  ance i s  to be caused by the B a r o c e l and not the buoy,  since  this  a reson-  known to o c c u r i n the drum at 45 Hz and there was no reason  suspect a r o l l o f f quency of 7 Hz.  above 7 Hz;  its  i n the buoy sensor below 100 Hz (the  forevolume  is  300 H z ) .  resonant  fre-  No c a l i b r a t i o n s were made above  Assuming the buoy sensor to be c o r r e c t at h i g h f r e q u e n c i e s  the B a r o c e l to be c o r r e c t at low f r e q u e n c i e s , the d r u m - d r i v e r system i s  to  shown i n F i g u r e 79.  and  the frequency response  of  191 A2.5  Data A n a l y s i s The d a t a was a n a l y s e d u s i n g the same g e n e r a l methods d e s c r i b e d i n  "Data A n a l y s i s and I n t e r p r e t a t i o n " .  The analog tapes were f i r s t  back i n t o a s i x - c h a n n e l c h a r t r e c o r d e r ; s e c t i o n s s u i t a b l e f o r were chosen u s i n g as c r i t e r i a the l e n g t h of  the s i g n a l s  (as  analysis  time d u r i n g which a l l  f u n c t i o n e d s i m u l t a n e o u s l y and the g e n e r a l s t e a d i n e s s of v a r i a n c e of  played  the  signals  "eyeballed"  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  s e c t i o n s were marked. These s e c t i o n s were then p l a y e d back a t e i g h t  times r e a l  f i l t e r e d w i t h c a r e f u l l y phase-matched low-pass  filters  aliasing),  500 samples per second  and d i g i t i z e d a t a sampling r a t e of  on the I n s t i t u t e  A/D converter  (62.5  (to  speed,  prevent  Hz " r e a l - t i m e " d a t a 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 f o r p a r i t y e r r o r s and other l i k e l y d i g i t a l f a i l u r e s .  If  they proved s u i t a b l e  they were  a n a l y s e d w i t h the F a s t F o u r i e r Transform package developed a t Institute. and cross  The computer output from t h i s package gave power  the spectra  s p e c t r a i n c l u d i n g Co, Qu, Phase and Coherence f o r a l l  a n a l y s e d (see  "Data A n a l y s i s and I n t e r p r e t a t i o n " ) .  Fourier  transformed i n b l o c k s of 2048 p o i n t s  time).  The f i n a l  of d a t a b l o c k s i n a g i v e n r u n .  signals  The data was  each (32.7  s p e c t r a l e s t i m a t e s were averages  then  seconds of  real  over the t o t a l number  A l s o p r i n t e d out were s t a n d a r d e r r o r of  the mean and average t r e n d f o r each s p e c t r a l e s t i m a t e and c r o s s - s p e c t r a l e s t i m a t e a t each f r e q u e n c y . i n d i v i d u a l frequencies mately \ octave)  For this  analysis  spectral densities  at  were averaged over r o u g h l y l o g a r i t h m i c ( a p p r o x i -  bandwidths.  192 A2.6  Results  A2.6.1  Summary o f R e l e v a n t I n f o r m a t i o n  A summary of i n f o r m a t i o n r e l e v a n t to the f i e l d runs i s i n Table A 2 . 1 .  The r e s u l t s  i n three s e c t i o n s .  from the d a t a of  I n the f i r s t  these runs w i l l  the d e t e r m i n a t i o n of  v e l o c i t y " u^ from the h o t - w i r e d a t a i s  described;  t i o n of p r e s s u r e s p e c t r a i n t o a form s u i t a b l e observations.  In the second s e c t i o n  the " f r i c t i o n  the s i g n a l s  from  These power s p e c t r a are  also  I n the t h i r d  the buoy sensor d a t a w i t h the d a t a from the  p r e s s u r e sensors and from the h o t - w i r e anemometer are  A2.6.2  section  two  presented.  The Hot-Wire Data  For a l l of  the runs simultaneous h o t - w i r e anemometer data were  available.  Power s p e c t r a were computed f o r the d a t a taken,  cases cross  s p e c t r a w i t h the p r e s s u r e d a t a were a l s o  power s p e c t r a are d i s c u s s e d In F i g u r e 80  the l o g of u,  is  p l o t t e d versus  f o r one of  the runs a n a l y s e d .  be t y p i c a l of a l l  the v e l o c i t y  spectra obtained.  is  of s l o p e  -5/3;  computed.  the spectrum of  a hot-wire height,  the p o i n t s  and i n some The  first.  log  downwind speed f l u c t u a t i o n s  fits  presented  f o r comparison w i t h o t h e r  compared w i t h those o b t a i n e d by other i n v e s t i g a t o r s . the c r o s s - s p e c t r a o f  be  t h i s permits n o r m a l i z a -  power s p e c t r a o f  the three p r e s s u r e sensors are compared.  presented  log (kz), It  where z  is  considered  to  The l i n e f i t t e d  as f o r a l l the v e l o c i t y  the p o i n t s v e r y w e l l over one decade of  is  turbulent  spectra this  to line  frequency.  These s p e c t r a have been used to o b t a i n a v a l u e f o r u ,  the " f r i c t i o n  193 TABLE A 2 . 1 I n f o r m a t i o n Summary f o r Boundary Bay Runs:September,  Wind Speed a t 4m (cm sec 1)  Date Time (Sept. of tun 1968) day  -  totes:  la  Wind Direction  Sensor  1530  390  NW  30  Pa  27?  1430  310  NW  5 cw  400  200 cw  30  60 cw  2  60 cw  u'  30  50 cw  Pa  30  60 cw  2  60 cw  30  50 cw  30  60 dw  2  60 dw  1  Pa Pb (buoy)  5b  1500  27  340  NW  Pb (buoy) u 5c  1600  27  400  NW  1  Pa Pb (buoy) u'  5d  1700  27  420  NW  Pa Pb (buoy)  1730  27  450  NW  Pa Pb (buoy)  1.  60 dw, 10 cw  30  60 dw  2  60 dw  0.2  0.2  0.2  0.2  60 dw, 10 cw  100  60 dw  2  60 dw  100  u'  Notes:  30  30  u' 5e  0.0  0  0  u  Cloud Cover (tenths)  (3)  Pb (ground)  5a  Horizontal Separation (cm)  (2)  (1)  4  Height (cm)  1968  0.2  60 dw, 10 cw  Mean wind speed measured w i t h cup anemometers  at heights  of  50  and 400 cm. 2.  p  a  =  E l l i o t t a i r p r e s s u r e sensor;  ground or i n buoy on ground); u' 3.  cw =  crosswind from r e f e r e n c e  =  pb = buoy p r e s s u r e sensor hot-wire  (on  anemometer.  p r e s s u r e sensor; dw = downwind from r e f e r e n c e p r e s s u r e s e n s o r .  194 velocity"  (see,  for instance,  v a l u e of  CJ)uu.(k) , i n the k ^ ^ -  by T a y l o r i n 1960 u  (see,  where K 2i 0.4  r e g i o n of  3  the spectrum.  f o r example, W e i l e r and B u r l i n g  (mz) v u ; 1.13  ^ )  d)  . k^/3  2/3  -  2  -  and U i s  Lumley and Panofsky (1964) ) from the  2  /  3  f  5  ~i  /  i s Von Karman's c o n s t a n t ,  I t was shown (1967) )  that  ua(k)>  0ua<f).  3  z is  the mean wind speed a t t h a t h e i g h t ;  the h e i g h t of the anemometer = 0.48  i s Kolmogorov's  u n i v e r s a l c o n s t a n t f o r a i r i n the U n i v e r s a l I n e r t i a l Subrange of lence scales  (the v a l u e of  B u r l i n g (1963)  k  t h a t g i v e n i n Pond, Stewart, and  );  =  2  T  T  X is  is  turbu-  =  21T f U  the wavenumber and X the s c a l e s i z e of the t u r b u l e n t e d d i e s .  l a s t statement i s  The  the "Frozen Turbulence" assumption, or T a y l o r ' s hypo-  thesis. o In the arguments l e a d i n g to the e x p r e s s i o n f o r u, i s v  i n c l u d e d the  assumption t h a t i n the I n e r t i a l Subrange under d i s c u s s i o n the r a t e of p r o d u c t i o n of m e c h a n i c a l energy i s  equal to the r a t e of energy d i s s i p a t i o n .  F o r the Boundary Bay runs no r e c o r d of the mean temperature p r o f i l e or of  the mean h u m i d i t y p r o f i l e was k e p t .  Both of these q u a n t i t i e s