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Field observations of frequency domain statistics and nonlinear effects in wind-generated ocean waves Garrett, John Frederick 1970

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FIELD OBSERVATIONS OF FREQUENCY DOMAIN STATISTICS AND NONLINEAR EFFECTS IN WIND-GENERATED OCEAN WAVES by JOHN FREDERICK GARRETT B.A., Harvard University, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April, 1970 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree tha permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physi cs  The University of British Columbia Vancouver 8, Canada Date April 13. 1970 ABSTRACT The objectives of this study were to show how far the statistical assumptions usually made when dealing with the theory of continuum nonlinear interactions were relevant to the case of fetch limited wind generated wave fields, and to observe some consequence of the nonlinearity as a check on the other results. The usual assumption, that the f i r s t order wave f i e l d is composed of Gaussian random wave components stationary in space and time, is shown to require that the real and imaginary parts of the complex Fourier coefficients of the observed surface elevation at a fixed point are normally distributed with zero mean, and that the real and imaginary parts of the complex coefficient for a given frequency band are independent. Coefficients from different frequency bands must also be independent. Four wave fields were observed and described in terms of their power spectra and frequency-wavenumber spectra. Although their directional spectra were found to be strongly influenced by the particular arrangement of fetch at the observing site, their power spectra agreed well with both theory and other observations, at least after correction for the observed currents. This agreement is important from the point of view of establishing the relevance of the results of this work to other f i e l d situations. Observations of the frequency domain statistics for these four wave fields indicated that the assumptions of stationarity and of normal distributions for the Fourier coefficients were correct to good accuracy. Two methods were tried for observing direct evidence of nonlinear effects in the wave f i e l d . One of these, the attempt to detect products of second i i i order interactions i n the frequency-wavenumber spectra, f a i l e d because of the poor r e s o l u t i o n of the array of sensors. The other, the bispectrum, succeeded, with reasonable agreement being achieved between the observed bispectra and bispectra predicted from the power spectra using the r e s u l t s of perturbation of theory. i v TABLE OF CONTENTS page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS x CHAPTER I: INTRODUCTION 1 CHAPTER I I : THEORY AND THEORY OF MEASUREMENTS 6 Steady waves 6 Stationary random wave f i e l d 8 Surface elevation s t a t i s t i c s 9 Relations between t h e o r e t i c a l and observed variables 11 S t a t i o n a r i t y and the observed Fourier c o e f f i c i e n t s 13 Power spectrum to second order 16 Cross spectrum to second order 16 Frequency-wavenumber spectrum to second order 18 Power spectrum to fourth order 19 Third moment to fourth order 21 Bispectrum 22 The 'window' and 'hanning 1 23 Purpose of observations 26 CHAPTER I I I : THE OBSERVED WAVE FIELD 27 Observing conditions 30 Power spectra 32 page Frequency-wavenumber spectra 41 Summary of observations 66 CHAPTER IV: FREQUENCY DOMAIN STATISTICS 67 The populations of c o e f f i c i e n t s 67 Tests on c o e f f i c i e n t s at s i n g l e frequencies 68 Tests of the independence of c o e f f i c i e n t s at d i f f e r e n t frequencies 81 Conclusions 84 CHAPTER V: OBSERVATIONS OF NONLINEAR EFFECTS 89 Second order n o n l i n e a r i t i e s i n the frequency-wavenumber spectrum 89 The bispectrum and second order n o n l i n e a r i t i e s 95 CHAPTER VI: SUMMARY OF CONCLUSIONS 113 LIST OF REFERENCES 115 APPENDIX A: EXPERIMENTAL TECHNIQUE 117 P h y s i c a l arrangement at s i t e 117 Surface height sensors 126 S i g n a l handling 131 Data processing 133 C a l i b r a t i o n 145 Measurement technique f o r wavenumber spectra 160 APPENDIX B: POTENTIAL FLOW PERTURBATION ANALYSIS 170 vi LIST OF TABLES Table page I Field conditions during observations 31 II Least squares analysis of mean spectra 39 III Variability of wavenumber spectral estimates 52 IV Ratio of reflected and incident amplitudes for various reflectors 123 V Effect of reflections on spectra 125 VI Sensitivity of sensors 150 VII Nonlinearity of sensors 153 v i i LIST OF FIGURES Figure page 1 Map of observing site 28 2 Arrangement of sensors 29 3 Envelope and mean of power spectra from April 11, 1967 run 33 4 Spectrum and 95% confidence limits from April 11, 1967 run 34 5 Mean spectra from a l l runs 36 6 Mean spectra x f from a l l runs 38 7 Mean spectra x "f5" corrected for current 40 8 Broadband wavenumber spectral window 42 9 Frequency-wavenumber spectrum for 0.88 Hz band, April 11 44 10 Frequency-wavenumber spectrum for 1.26 Hz band, April 13 45 11 Frequency-wavenumber spectra for April 11 data 46 12 Frequency-wavenumber spectra for April 13 data 47 13 Frequency-wavenumber spectra for April 25 data 48 14 Frequency-wavenumber spectra for April 28 data 49 15 Summary wavenumbers for April 11 observations 60 16 Summary wavenumbers for April 13 observations 61 17 Summary wavenumbers for April 25 observations 62 18 Summary wavenumbers for April 28 observations 63 19 Observed wavenumbers vs. frequency 64 20 Fourier coefficient statistics vs. harmonic number, April 11 73 21 Fourier coefficient statistics vs. harmonic number, April 13 74 22 Fourier coefficient statistics vs. harmonic number, April 25 75 v i i i Figure page 23 Fourier coefficient statistics vs. harmonic number, April 28 76 24 Coherence among sensors for various harmonics 77 25 Statistics for the f i r s t 10 Fourier coefficients 78 26 Correlation coefficients for (\\ and X, T- unhanned, April 11 85 27 Correlation coefficients for R., (£• and I, I : hanned, April 11 86 28 Correlation coefficients for (C; T; unhanned, April 11 887 29 Correlation coefficients for (l; £• hanned, April 11 88 30 Ratio test for second harmonics, April 11 90 31 Ratio test for second harmonics, April 13 91 32 Ratio test for second harmonics, April 25 92 33 Ratio test for second harmonics, April 28 93 34 Simulated bispectrum for sinusoidal surface wave 98 35 Simulated bispectrum for sinusoidal surface wave, hanned 99 36 Simulated bispectrum for four random sinusoids 100 37 Directional distributions of energy used in modeling 102 38 Observed and simulated bispectra for April 11 data 104 39 Observed and simulated bispectra for April 13 data 105 40 Input spectrum and angular deviations for April 11 model 107 41 Input spectrum and angular deviations for April 13 model 108 42 Effect of angular deviation on simulated bispectrum 110 43 Effect of directional distribution on simulated bispectrum 111 44 Photographs of observing site 118 45 Coherence vs. separation for various frequencies, April 11 124 i x F i g u r e page 46 Observed s p e c t r a and p r e d i c t e d e f f e c t s o f r e f l e c t i o n s 127 47 S e n s i n g and r e c o r d i n g system 128 48 Sensor system c i r c u i t s 129 49 C h a r a c t e r i s t i c s o f d i g i t i z i n g f i l t e r s 134 50 B l o c k d i a g r a m o f a n a l y s i s system 135 51 B l o c k d i a g r a m o f FTOR 136 52 B l o c k d i a g r a m o f SCOR 142 53 C a l i b r a t o r 147 54 C a l i b r a t i o n s p e c t r u m 148 55 J o i n t d i s t r i b u t i o n o f skewness and s t a n d a r d d e v i a t i o n from c a l i b r a t i o n s 158 56 J o i n t d i s t r i b u t i o n o f skewness and s t a n d a r d d e v i a t i o n from f i e l d d a t a 159 57 Comparison o f c a l i b r a t i o n s p e c t r u m w i t h mean s p e c t r a o f f i e l d d a t a l 6 l 58 C a l i b r a t i o n s p e c t r u m and mean s p e c t r a o f f i e l d d a t a n o r m a l i z e d t o b r i n g peaks t o same f r e q u e n c y 162 59 C a l i b r a t i o n s p e c t r u m and mean s p e c t r a o f f i e l d d a t a n o r m a l i z e d i n f r e q u e n c y and energy 163 60 Arrangement o f s e n s o r s and r e s u l t i n g s e p a r a t i o n s 167 61 Wavenumber s p e c t r a l window f o r harmonic s i n e 168 X ACKNOWLEDGMENTS This work was done in conjunction with the Air-Sea Interaction program at the Institute of Oceanography of the University of British Columbia. The f i e l d work was supported by the U.S. Office of Naval Research under Grant No. NT 083-207, while the elaborate d i g i t a l computations were possible because of the support of the U.B.C. Computing Centre by the National Research Council of Canada. I have been supported personally by the Institute of Oceanography and by a National Research Council Postgraduate Scholarship. I am indebted to Dr. R.W. Stewart and Dr. R.W. Burling for their help and advice, and particularly to Dr. Burling for his criticisms of the theoretical sections of this thesis. Although most of the students and staff of the Institute have at one time or another helped in the planning and execution of my grandiose projects, Mr. J.R. Wilson and Dr. F.W. Dobson are especially deserving of thanks, the f i r s t for his patience in sharing his programming expertise with a neophyte and the second for his willingness to set out for the platform in the most abominable weather. Finally I would like to thank my wife Toni for the steady good cheer with which she has faced the formidable task of managing a household on the meagre income of a graduate student. 1 CHAPTER I INTRODUCTION The study of the s t a t i s t i c a l mechanics of ocean waves has made much progress i n recent years, so that some problems, such as that of wave forecasting, are for present p r a c t i c a l purposes solved. However most of t h i s progress has been made on a semi-empirical b a s i s . No s a t i s f a c t o r y explanation i s a v a i l a b l e for e i t h e r the process by which waves are generated by the wind or for the process leading to the d i s t i n c t i v e way i n which energy i s d i s t r i b u t e d among d i f f e r e n t wave components. The subject of t h i s i n v e s t i g a t i o n i s r e l a t e d to the l a t t e r problem. It i s commonly observed that when waves are produced by wind blowing o f f a nearby shore, i . e . under conditions of limited fetch, the energy i n the wave f i e l d i s mostly contained i n a narrow band of frequencies. There i s l i t t l e energy at high frequencies with the energy density gradually increasing towards low frequencies. In the frequency range where g r a v i t y produces the main r e s t o r i n g force,., i . e . below about f i v e Hertz, the energy density r i s e s r a p i d l y toward lower frequencies u n t i l i t reaches a peak. Below t h i s peak i t abruptly drops to a comparatively low l e v e l . P h i l l i p s (1958) deduced from dimensional arguments that for frequencies somewhat higher than the low frequency cutoff, but below the c a p i l l a r y region, the energy density should depend only on the frequency, and i n t h i s 'equilibrium region' should be inversely proportional to the f i f t h power of the frequency. This agrees quite well with what i s observed, except that i n most observations the energy density seems to decrease s l i g h t l y less r a p i d l y towards high frequencies. P h i l l i p s r e l i e d upon the observations to 2 provide the constant of p r o p o r t i o n a l i t y , but Longuet-Higgins (1962a), using a d i f f e r e n t approach, succeeded i n deducing t h e o r e t i c a l l y a p r o p o r t i o n a l i t y factor i n f a i r agreement with the observations but with a weak dependence on both wind speed and fetch. One i n t e r e s t i n g feature of the wave spectrum i s the manner i n which i t changes with increasing wind speed or fetch. The low frequency cutoff simply s h i f t s toward lower frequencies with the power law behavior at frequencies higher than the cutoff being extended to the new c u t o f f . Thus the energy density at the peak of the spectrum, just above the cutoff, depends more or less on the inverse f i f t h power of the frequency of the cutoff, which i n turn depends on the wind speed and fetch. For t h i s to take place the wave component with a frequency i n i t i a l l y j u st below that of the cutoff must experience very rapid growth ( ' t r a n s i t i o n ' ) . The energy for t h i s must come from the wind e i t h e r d i r e c t l y or v i a nonlinear transfers from other wave components. No e x i s t i n g theory for wave generation by the wind i s adequate to explain the observed growth rate of the waves (cf. Snyder and Cox, 1966, and Barnett and Wilkerson, 1967), or the observed aerodynamic fluxes of energy and momentum to the waves (Dobson, 1969), but observations suggest that the maximum inputs of energy and momentum are at frequencies s l i g h t l y higher than the frequency of the peak of the spectrum, as shown i n Dobson (1969). Thus by default i t can be i n f e r r e d that the nonlinear transfers play a s i g n i f i c a n t r o l e . There are two types of nonlinear mechanisms operating within the wave f i e l d . In one of these the n o n l i n e a r i t y arises i n the surface boundary condition applied i n the usual p o t e n t i a l flow development of the wave equation. This type, hereafter referred to as 'continuum n o n l i n e a r i t i e s ' , i s amenable to treatment by perturbation analysis, examples of which may be 3 found i n P h i l l i p s (I960), Hasselman (1962), and Benjamin and F e i r (1967). The observations to be described are pr i m a r i l y r e l a t e d to th i s type of non l i n e a r i t y . The other type of no n l i n e a r i t y present i s that associated with wave breaking. Although Longuet-Higgins (1969b) has made a reasonably successful attempt to explain the t r a n s i t i o n of the low frequency components as being due to energy transfers through wave breaking at the expense of shorter waves, th i s type of no n l i n e a r i t y has generally proved both t h e o r e t i c a l l y and observationally quite i n t r a c t a b l e . There seems l i t t l e doubt that the p o t e n t i a l flow perturbation analysis of the continuum nonlinear e f f e c t s i s correct as far as i t goes. However i n order to use the r e s u l t s to make predictions about the importance of the possible transfers w i t h i n the wave f i e l d which r e s u l t i t i s necessary to make some assumptions about the s t a t i s t i c a l nature of the wave f i e l d and about the d i s t r i b u t i o n of the energy i n a given frequency band among various vector wave numbers. The assumption that i s usually made i s that the surface elevation i s normally d i s t r i b u t e d about some mean value and that the l i n e a r or f i r s t order wave f i e l d i s stationary i n both space and time. The s t a t i s t i c s of the surface elevations have been the subject of several i n v e s t i g a t i o n s , e.g. Kinsman (I960), which have generally shown the assumption of normality to be correct to the degree that might be expected by neglecting the known nonlinear e f f e c t s . On the other hand only a few f i e l d observations of the d i r e c t i o n a l spectrum of fetch l i m i t e d waves have ever been made ( G i l c h r i s t , 1966), and no observations have been made which were suitable for examining the s t a t i o n a r i t y assumption or attempting to detect nonlinear e f f e c t s . The aim of t h i s study then was to 4 provide s u f f i c i e n t information about a few observed wave f i e l d s to enable the c a l c u l a t i o n s of the importance of continuum nonlinear e f f e c t s to be made with some confidence. In addition i t was hoped to observe d i r e c t l y some consequence of the n o n l i n e a r i t y as a check on the other r e s u l t s . The method selected for achieving these objectives was to record the sea surface elevations as a function of time at a number of c l o s e l y spaced points using a two-dimensional array of surface height sensors. The records obtained could then be analysed to y i e l d the spectra of energy with frequency and with wavenumber. Although a l i n e a r , or one dimensional, array would have given equivalent or better d i r e c t i o n a l r e s o l u t i o n for waves obeying the assumed dispersion r e l a t i o n , the two dimensional array was used i n the hope that i t would be possible to resolve the products of second order i n t e r a c t i o n s , which have a d i f f e r e n t r e l a t i o n s h i p between frequency and wavenumber than free g r a v i t y waves. In addition the frequency domain s t a t i s t i c s of the records could be analysed to test the s t a t i o n a r i t y assumption. F i n a l l y , b i s p e c t r a could be computed to give the desired d i r e c t estimate of nonlinear e f f e c t s . Chapter II deals with the theory of the measurements, and i s followed by Chapter I I I describing the frequency and wavenumber spectra of the observed wave f i e l d s . The tests of s t a t i o n a r i t y using frequency domain s t a t i s t i c s are contained i n Chapter IV, which i s followed by a chapter dealing with the observations of nonlinear e f f e c t s and the comparison of these observations with t h e o r e t i c a l r e s u l t s . Only the most e s s e n t i a l d e t a i l s of the techniques of observation and analysis are presented i n these chapters, but a more complete discussion i s contained i n Appendix A. Appendix B gives a resume of Hasselmann's (1962) treatment of the perturbation 5 a n a l y s i s theory which has been used to p r e d i c t the nonlinear e f f e c t s f o r comparison w i t h the observed bispectrum. 6 CHAPTER II THEORY AND THEORY OF MEASUREMENTS Steady Waves Interactions among surface gravity waves due to nonlinearities in the surface boundary condition applied in the usual potential flow theory have received much attention in the literature. Of the various theoretical treatments the one best suited for dealing with wave fields with continuous spectra is that of Hasselman (1962, 1963). He represents the velocity potential and surface elevation as perturbation series 4> = d «- x$ + ••• - (2.i) and 3 = ^ " iS +_,) * • • • (2.2) Each order 'n' of the potential is taken to be the sum of a Fourier series, which after considering the bottom boundary condition for constant depth h takes the form n<P " 2_ 2- rtyk C O S U JeV, 4 C O S K A C , * * ^ (2.3) where Jjt is the vector wavenumber of magnitude M. , and 5 is a sign parameter with the conditions + and — . The f i r s t order equations (see Appendix B for a more complete development) give the dispersion relation for frequency w ^ - % k ^ ^ (2.4) If the surface elevation is similarly represented: then the f i r s t order equations show that The requirement that "5 be real means that x - (,z; r and As is shown in Appendix B, the second order solution for surface elevation is X H 3 e (2.8) where Jk0 =• and to„ = ^ J50+aw^ it 0W 4= C S . 1 0 ' * ^ " z f " " # T h e expression between the brackets ^•••^ w i l l hereafter be written as , so that (2.8) becomes ^ = I (- L L l 4 i ^ A t e -fe. *' 5- ~ ~ ~ ^ (2.9) Special notice should be taken of the fact that ^ has the character of a forced rather than a free oscillation and that i t s components do not obey the dispersion relation (2.4). An attempt which was made to use this latter fact as a means of identifying the second order components in the wavenumber spectrum of a given frequency band is discussed in Chapter V. Stationary Random Wave Field To carry the theory over to random wave fields such as are found on the ocean the significance of the notation must be extended. The Fourier sums are now to be viewed as Fourier-Stieltjes sums with the ^ cB^ and 5^-in. becoming random interval functions of intervals A i i centered on Jt (This and the matters discussed in the following paragraphs are explained very clearly, although not in connection with this application, in Chapter 2, of Yaglom, 1962). The assumption which is usually made is that (<j> and ^ are random functions of X and "t" and that they are stationary in both space and time. It is obvious that this can be true only approximately as ocean waves are usually either growing or decaying in space and time because of spatial and temporal variations in the wind f i e l d . One of the aims of this experiment is to investigate whether the assumption is accurate enough to be useful for predicting nonlinearities in wind wave fields. For a random function to be stationary, statistics of the form )^(&>"0 J aX A* A"0 must depend only on A x and A \ , where the overbar denotes the limit of an ensemble average. For this to be true, i t must also be true that z 5 ' is": = o (2.10) or the equivalent ^ k > K ~ ® for a l l Sj. + s i and for disjoint (nonoverlapping) intervals centered on A ( and M.L . If we introduce the notation (2.11) 9 where , R^. and are real random interval functions the requirement that ( 7 ? - 7 in order for \ to be real then becomes TS - - vl Now from the stationarity condition (2.10) Thus stationarity implies that ,R1 ,L* - 0 and ^ = I*7- (2.12), Finally, since " Z s ' 7 ~ 5 ' = D s ' T 5> J -The origin of the z-axis has been defined to make thus insuring that (2 .13 ) Surface Elevation Statistics The variance of the sea surface height to second order w i l l be 10 ft ' ' 1-1 1 On the basis of the preceeding considerations, for a stationary random sea this reduces to = f _ i , z . ' ; , z ~ ; <2.i5>. In a similar way, the mean elevation of a stationary random sea surface is given to second order by ^ ~~ %r% (2.16), where jj i - s defined in (2.8) and (2.9). In order to carry these statistics to a higher order i t is useful to note another requirement for stationarity, namely that 0 (2.17). This is true because there are no combinations of (j, j kz-i. > -%3 J S I J 3 J - J S 3 for which i t , +- )(l, and s , " , * SJ.<JJ_+ Sa'* 1? can simultaneously be made to vanish, as would be required i f were to depend only on L\, X A.,.* 4 & . The last is of course a requirement for stationarity. With this in mind, the variance of a stationary random sea surface about 1-0 , correct to fourth order, is 11 + C 5 ' "S< r S i _ S l ? Z s ' " Z " 5 ' - 2 ^ = X T - J J * l * - * ' ^ ~ (2.18) , Likewise, the third moment of the stationary random sea surface correct to fourth order w i l l be T - 3 - 3 L E L XL (ic 4v; t * c >: ?;,) * 3 I E t t I. C t , K J-\ Xl JZ" (2 .19) , The skewness coefficient to this order is T 3,1,1,5 Relations between Theoretical and Observed Variables In practice we observe "% at some point * 0 , over some limited period H - 6. we might thus write V*o^ = H TL X* ®k * _—. ~— 0-\ S — i & t-o"V uj 5 (7 (2.20) where we define * r s t o (2 .21) , 12 Here the summation is over a l l vector wavenumbers which contribute to the frequency o-> , and n^ .u> i-s a random interval function of to . Since O is real, and since w (2.22), Actually, in the analysis of the observations, the continuous elevation is sampled dig i t a l l y , so that the f i n i t e length of time corresponds to a finite number of samples KJ taken at intervals of A."V* . In the analysis of the data we transform from the time domain to the frequency domain by means of a discrete Fourier transform of the form: N-1 _ i ITT jpAV ^ "IP ° (2.23) where co. - UL-i * N M" For simplification we suppose we have chosen y\l At" such that the io„ - -^ILP for p =• 0,1.1, - • - • are equal to the used previously, in (2.20) etc. (In practice of course the OJ are defined in terms of the p rather than vice versa.) In this case F^c-o ^ £ - 4 - 0 ( 2 - 2 4 ) In fact, in the particular analysis system used the Fourier coefficients were defined so that the data could be reconstituted from <D*^ = H R e * \ p * r + ( / t - ^ £ ' W p * J (2.25) 0< N-1 <«->f 13 where u p - ?• ^  P ^  p - o,i i ... and C^p) *-s t n e complex Fourier coefficient •X' as defined in the analysis programs. To find the relationship between y4(w\ and we substitute from (2.20) and (2.25): f - I UA Ol- not"-n 'o Thus /4M - (2<26) Stationarity and the Observed Fourier Coefficients We wish to relate the probability structure of the y4 C^) to that of the <n^~ K • Noting from (2.26) that " JR.: <o ~ to f i r s t order (2.27) 14 In most cases we w i l l be dealing with observations taken at a single point. Because of the assumed spatial stationarity of ^(Xo^'H in such cases we may as well assume that point is X „ •=• O so that - i and and further ReaJ part t/\ = X YL ,R>L * (2.28), Thus, on the basis of (2.10), (2.12) and (2.14) we would expect Real p.c-r • = l E Jj = 0 (2.29) I w * * r « u ^ pa tt •/(.*>) - Z E = O (2.30) and (Rati pa.ft ./cu* X 1 ***a' m a L r3 P * f t 'M"S) = 4 <L_ H. i fyk tXt - O (2.31). Also from (2.10) we see that >A^\>AC^) ~ 1JS >_ , Z l = O (2.32) Similarly from (2.17) we get for the f i r s t order coefficients y4lu,1j/(->Oy4(wO = O (2.33), To second order since ^ (o 1 O by (2.29) and (2.30) 2- Z_ 1— ^— E. ^Z-k Cjk A 15 with S, co, +• ^ j . - <^> • From (2.10) i t can be seen that this must average zero, as the only mean product of fzE.^ 's with a non-zero average is , i/L ^-XL > which contributes only to to = S i o - <,UJ — o . To fourth order (2.35), We have already shown that the f i r s t three terms on.the right hand side vanish, which leaves t^ <o, which is proportional to H E £ £ IL I- £ Zl ^ ^ ZVr (2.36), where to, - 6Ato^ + ioe and cot - + % 6 0 £ (2.37). The only combinations with non-zero averages are pairs of complex conjugates where and V - -h . These together with the two conditions (2.37) require that either co, = o = tOj. ; or co, = - to t . The latter could only occur i f the product being formed were J L ^ W , L^ui t rather than a^io, i^iOj. > s o t n a t w e must conclude that to fourth order, for co^o^O (2.38) 16 Power Spectrum to Second Order The power spectral density at f ~ is computed from - NA* A, . k? (2.39). To second order this is The integral under the second order spectrum is (2.40), •f > O A 5 (2.41), from (2.15). Cross Spectrum to Second Order The covariance between the surface elevations at two points x, and x t i s (2.42). To second order in a stationary random sea this is k s (2.43), 17 where (2.10) has been used. Recognizing the symmetry between M. and -/TL , this might also be written as a sum over positive wavenumbers only: *>° s *" ~ (2.44). The covariance may be represented as the integral of a cross spectrum in a manner similar to the representation of the variance as the integral over a spectrum. Suppose we define the complex cross spectrum as (2.45) To second order A.; u> c s i s . ^ o = IN6V n , z ; , z ^ e (2.46), where the sum is again over a l l wavenumbers, (2.10) has been used, and 2-TT-The real part of this is the cospectrum: The sum over this, using (2.44) is L i t ^ s U - ^ = L 2 . 1 , z ; , Z - t e co*c*.i4.-»^i = 1 1 ^ z ; ; ^ t ^ . t i . - ^ i --ILL , z s A l z ; s A c u . u . - w l k>o 5 - . V * . ^ , V * » * ) ( 2 - 4 8 ) 18 Frequency-Wavenumber Spectrum to Second Order Suppose we were to measure the cross spectra between each pair of points on a grid, such that would take a l l the values t f ^ (3A-j } with * = -r\r**\, ••• ' , , * • and where i and j are the unit vectors in the x and y directions (2.49) (2.50), respectively. We can write the complex cross spectrum (2.45) for each pair of points as (2.51). If we perform a two-dimensional discrete complex Fourier transform on the spatial array of cross spectra we w i l l produce a frequency-wavenumber spectrum: (2.52). In practice of course cross spectra are available from relatively few separations which are distributed in a way intended to give the greatest 19 range of separations for the fewest sensors and therefore provide a poor approximation to a uniform rectangular grid. In these practical cases the window in fact controls the resolution. This is discussed further in Chapter V and in the section of Appendix A describing the wavenumber spectrum measurements. Power Spectrum to Fourth Order Next let us consider the effect of the next higher terms in the spectrum: right hand side is just the second order spectrum the only term remaining to be considered is the last. Substituting from (2.9) we find that (2.53). tri p l e products of the ,~Z>i a n c* thus vanishes. As the f i r s t term on the 5 ^ = 5L YL *L E_ C A ^  ~^LK e (2.54) where and the sums are over a l l M.- contributing to ut • and a l l contributing to u/: at f i r s t order. Noting from (2.8) that (2.55) 20 and recalling (2.10), (2.11), (2.12) and (2.13), after a considerable amount of manipulation, (2.54) may be reduced to (2.56). In a similar way we can work out the contribution to the cross spectrum by fourth order terms: loll «r« i . •> / . u4n.tr*. (2.57) It may be seen from this that a spatial Fourier transform as in (2.52) w i l l correctly associate the fourth order energy of the second order forced waves with their wavenumber. Since the relationship between frequency and wavenumber is not the same as for free waves, the energy of the forced, waves should be distinguishable from that of the free waves in the wavenumber spectra of observed surface heights. Third Moment to Fourth Order Suppose we consider next the third moment of the observed surface elevations for a stationary random sea, at Xo - O ; The only terms which contribute to the average are those in which the time dependence vanishes, i.e. Z^i-° • Thus, we are lef t with 22 Substituting from (2.26) this becomes Further substitution from (2.21) gives I ,ZAI 2^.Z^ S + JL^.Z^ Z-A^ + Ui^ker orcier -^eri^S where A, X, A, contribute respectively to <*J, to to Noting from (2.17) that the third order term vanishes, and introducing (2.10), we are left with •frvW (2.59) Bispectrum The bispectrum as computed for the surface elevation observations is given by (2.60) where fn = •LTT-23 From (2.58) i t may be seen that (2.61) . The bispectrum so defined may be i d e n t i f i e d with that discussed by Hasselmann, Munk and MacDonald (1963), with (2.59) and (2.6l) being comparable to t h e i r equation (16). (Their (16) i s i n terms of frequency t h e i r bispectrum would have to be summed over both p o s i t i v e and negative frequencies to give the t h i r d moment, whereas the development presented here has used only p o s i t i v e frequencies.) The 'Window' and 'Hanning' The convolution theorem for complex Fourier i n t e g r a l transforms states that i f %(^) and are r e a l functions with the transforms rather than wavenumber components l i k e "J. and respectively, then the transforms of w i l l be the Suppose i n t h i s case that = o so that _ , 1 T T f T Thus i f ^-(+) i s some function which i s sampled from Y=0 to T - T mu l t i p l i e d by JkOfl , the apparent transform of , based on the i . e . w i l l be 24 (2.62). At any frequency (J~ contains contributions from £r for a l l other frequencies. The function is usually referred to as the 'window' associated with the sampling function M. (+^ . Suppose now we imagine that is the result of d i g i t a l l y sampling some function \n[?) at intervals with a spacing A t" . If we approximate the sampling function with a series of Dirac St functions at intervals of ^ t " , we find that its transform w i l l be another series of & functions in frequency space, with a spacing interval of • Applying the convolution theorem leads to the familiar result that i f rUc") (the trans-form of h 1+) ) vanishes outside some interval - r - < ,° j °o - j^p -then (2.63), Recognizing that our sample from *t"= o to V - f corresponds to a limited number of di g i t a l samples j NJ » X_ We then see that ^ ^ (2.64). The discrete Fourier transform, which is not to be confused with the infi n i t e Fourier series approximation, depends on the realization that N 25 numbers, corresponding to the N samples in the period O — V } can be exactly related to —- complex Fourier coefficients by means of a system of N) independent linear equations. The resulting coefficients correspond to a di g i t a l sampling of G tf") with an interval . If they are used to attempt to reconstruct W (,+^  outside the interval O^: t ^ T , the resulting function w i l l be periodic in time with a period of hi A't- , in a manner completely analogous to the periodicity in frequency space of the transform of a discrete time series. The complex Fourier coefficients w i l l be related to VAC*^  by flW ( - i XX ( p -<rubt) \ I W t ) I L T-TT (-£- - «7-\ W (2.65). The effect of the window is to allow leakage from the Fourier coefficient at one frequency into a l l other frequencies. For the coefficients the - t envelope of the window f a l l s as ^ — ^ . in the power spectrum this becomes | ' w n ;"- c n ^ s n o t r a P i d enough for many purposes. There are several methods of changing the shape of the window. The method in the analysis system used here is to convolve the coefficients with the weights -±, +£, - i , as suggested in Godfrey et a l . (1967), i.e. (2.66). This operation, which is known as 'hanning', has the effect of multiplying the window by a factor 1 1 - ( p - er N M^** (2.67), so that the envelope f a l l s as ( — v~ ^  and in the power spectrum 26 A discussion of these effects is also included in Appendix A, with more specific reference to the actual analysis system. Purpose of Observations The objective of the observations described in the following chapters is to investigate the degree of validity of the stationarity assumptions under f i e l d conditions, by means of the relations (2.29), (2.30) and (2.31). Also observed are the power spectrum (2.39), and wavenumber spectrum (2.52). A f i r s t order wave f i e l d deduced from these is used to compute a model bispectrum (2.60) which is compared with that observed. 27 CHAPTER III THE OBSERVED WAVE FIELD Four sets of wave observations were made. This chapter describes these in terms of their power spectra and frequency-wavenumber spectra. The observations were taken at the Spanish Banks site maintained by the Institute of Oceanography of the University of British Columbia, using an array of nine capacitance wire surface height sensors. Figure 1 shows the location of the site and Figure 2 the arrangement of the sensors in the array. The site and the construction and calibration of the sensors are discussed in detail in Appendix A. It should immediately be obvious from the asymmetrical fetch distribution for the easterly winds observed that the wave f i e l d is probably not going to be quite like that which would be expected for the theoretically attractive case of the wind blowing directly off a straight shoreline. Although this is true, as w i l l be seen from the observed wavenumber spectra, the observed frequency (power) spectra do closely resemble those observed elsewhere. The aim of the experiments is to provide a basis for an understanding of the observed wave f i e l d , with the idea that such an understanding could easily be transferred to cases with different fetch distributions. The observations were recorded in analog form on magnetic tape and were sampled and digitized during playback. A l l of the results shown were obtained on the di g i t a l computer, transforming to the frequency domain by means of a Fast Fourier Transform technique. Each data run was broken up into a sequence of short sections or blocks usually about ten seconds long. 28 FIGURE 1 : MAP OF OBSERVING SITE 29 & array support Y<>> - @ — i i n m LV ar ray support ARRAY: nn -© sensor locations indicated by roman numerals 1 meter rx-o 1 f o o t ! H O R I Z O N T A L SECTION at 9 foot water level 85° N uzzzza P L A T O R M SUPPORTS I I I I I I I I FIGURE 2 ARRANGEMENT OF SENSORS 30 Each of these was transformed and its spectra and cross spectra computed. The spectra and cross spectra for the entire run are then means of these quantities from the individual blocks. Their va r i a b i l i t y about these means provides a way of estimating confidence limits for the means. The analysis system is discussed in more detail in Appendix A. Observing Conditions One of the challenges of f i e l d work is that for any given observing site conditions w i l l almost certainly never be ideal but w i l l at best approach the ideal only to some degree. At the Spanish Banks site the ideal conditions for the experiments contemplated here would have been for the water depth during a period of slack water, i.e. zero t i d a l current, to be greater than three meters but low enough so that the wave crests did not reach the horizontal cross brace on the platform, with a steady wind of from four to ten meters per second blowing from a direction between 80° and 110° (true) for long enough to guarantee that the waves observed were locally generated. Suitable t i d a l conditions are approached at the site during about twenty minutes on each day for periods of four or five days twice each month. An acceptable wind blows about 25% of the time during winter months but this drops to about 10% by midsummer. Thus suitable observing conditions should occur about twice a month in winter and something less than once a month in summer. Although some data were collected in December, 1965, these proved unsuitable for analysis. The four sets of observations under discussion were a l l taken in April, 1967, and as might be expected from the above, they were not a l l taken under ideal conditions. Table I summarizes the conditions actually 31 TABLE I FIELD CONDITIONS DURING OBSERVATIONS Run number 1 2 3 4 Date (1967) April 11 April 13 April 25 April 28 Number of sensors 9 8 9 9 Length of data analysed (min.) 10 6 10 6 Wind speed (m/sec) 4.6 4.7 6.4 5.6 Wind direction (true) 110° 125° 110° 110° Surface current speed (cm/sec) 3 7 15 10 Surface current direction N N NW NW Water depth (meters) 2.1 3.0 3.3 2.9 32 present. The wind speeds were measured with a Thornthwaite cup anemometer mounted three meters above the water surface on a mast and are averages of readings taken periodically during the run. The wind direction is from a vane mounted on the platform and is also an average of periodic readings. The surface currents were measured by determining the time required for a piece of wet paper towel floating in the water to move a known distance. This technique is more useful for determining when the current is nearly zero than for measuring the actual speed. On April 13 one of the wave sensor oscillators (at position VIII) would not oscillate because of the low ambient temperature (3°C.) so that run was made with only eight sensors. Power Spectra Spectra were computed for each of the sensors for each of the runs. Figure 3 shows the envelope and mean of the spectra from the individual sensors for one of the runs, while Figure 4 shows the 95% confidence limits on the mean for that run as computed in two ways. From the var i a b i l i t y of the individual spectra about their mean, the 95% confidence limit on the mean at the frequency "f is where cr(f) is the standard deviation of the individual spectra about their number of blocks and N the number of sensors. The uncertainty of the spectrum from each individual sensor was estimated from the observed variability among the sections (blocks) into (3.1) 33 J a I I » » I I I I I Lg FREQUENCY, Hz FIGURE 3 ENVELOPE AND MEAN OF POWER SPECTRA FROM APRIL 11,1967 RUN 20 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i N I E CM E u 10 10 X 10 95%confidence f rom observed variability 95^con f idence f rom X 4 assumption O 1 i I I i j i t I I I i i i i I i i I i | i i i I I i I l I M i 0 1 2 3  2 FREQUENCY (f) Hz FIGURE 4 SPECTRUM AND 95% CONFIDENCE LIMITS FROM APRIL 11,1967 RUN •35 which each sensor's record was divided and at a l l frequencies was very close to that expected from the "X. distribution. Thus the increase in observed uncertainty in the mean spectra towards high frequency is due to increased variability among the sensors rather than increased temporal variability in the wave f i e l d . Although i t is not certain, the cause of this is probably reflection from the platform and array supports. (See Appendix A.) Although the general agreement between theoretical v a r i a b i l i t y and observed variability is comforting, the confidence limits used hereafter w i l l be based on observed variability unless otherwise stated. Figure 5 shows the mean spectra from a l l four of the runs. The most striking feature of Figure 5 is the power law (linear on the log-log plot) drop off in energy density towards higher frequencies. This is a, common characteristic of wind wave spectra obtained under fetch limited conditions. (Cf. Phillips, 1966, Chapter 4) As mentioned in Chapter I, using an argument based on an equilibrium between energy input and wave breaking for the waves at frequencies higher than that of maximum energy density, Phillips (1958) found that dimensional considerations would require that the spectrum in this region be given by a universal law of the form S(0« K f * (3.2) with oL *=• -5" and - |3 <^x where @ is a universal constant and ^ is the acceleration due to gravity. The observational evidence so far available has generally been considered to support such a law and to give a value of around Scm^ sec-'--'*' for K ( a s calculated from data compiled in Chapter 4 of Phillips, 1966). F R E Q U E N C Y , Hz. FIGURE 5 MEAN SPECTRA FROM ALL RUNS 37 Observed values of <A range from -5.5 to -4.5. The observations on which these values are based have usually consisted of several series of measurements at a single fixed point, with scatter between runs being considered a reflection of s t a t i s t i c a l uncertainties due to sampling. In the case of the observations made with the array however, we are able to directly estimate the s t a t i s t i c a l uncertainties associated with a given run. Figure 6 shows the mean spectra for a l l four runs with their 95% confidence limits indicated by shading, while Table II gives the results of least squares f i t s for K and d for two frequency ranges within the spectra'. One of the frequency ranges includes the entire spectrum above the peak, while the shorter was chosen to minimize effects of reflections. An attempt was made to determine the inherent nonlinearity of the sensors by using a dynamic calibration method in which the effects of a sinusoidal wave train were simulated by moving the capacitance wire orbitally (i.e., the wire remains vertical while every point on i t describes a cir c l e about a horizontal axis) through a s t i l l water surface. The broken line in Figure 6 is an example of a spectrum produced in this way, using a simulated sinusoid of 20 cm peak to peak amplitude. This technique is described in some detail in Appendix A. Taken by i t s e l f , this data could only lead to the conclusion that the value of K varied significantly from day to day, as was shown in Garrett (1969). At the time that paper was written the importance of the effects of the currents on the observed spectra was not properly calculated. Since then however i t has been discovered that ;the correction of the observed spectra for the observed tidal currents removes most of the variation in K , with the uncertainties in the current measurements leading to the 38 0 1 2 3 FREQUENCY (f) Uz FIGURE 6 MEAN SPECTRA x f^FROM ALL RUNS 39 TABLE II LEAST SQUARES ANALYSIS OF MEAN SPECTRA Run Number Frequency Interval ^ e< Hz cm2Hz4 1 0.77 - 3.08 6.36 -4.45 2 0.68 - 3.11 5.32 -4.68 3 0.68 - 3.10 8.54 -4.68 4 0.87 - 3.10 7.91 -4.57 1 0.77 - 1.64 6.51 -4.50 2 0.68 - 1.75 5.32 -4.73 3 0.68 - 1.74 8.45 -4.68 4 0.87 - 2.23 9.22 -4.27 40 0 I i i i i i i i i i i i i i i i i i 1,1 i i i i i i i i i i i i i . 0 , 1 2 3 . F R E O U E N C Y ( f ) H z FIGURE 7 r 5" MEAN SPECTRA x T CORRECTED FOR CURRENT 41 feeling that the remaining differences could probably be eliminated with better current measurement. Figure 7 shows the mean spectra of Figure 6 after those for the April 25 and April 28 runs have been corrected for the observed currents. The manner in which the corrections were made w i l l be discussed later in this chapter. Another feature of these spectra which should be noted is that a l l except that for April 28 show a change in mean slope at about 0.9 Hz., as may be seen in Figure 7. Below that frequency there is the suggestion that the energy density decreases more rapidly than- t , while above that frequency i t seems evident that the decrease is slightly less rapid. A similar effect has been noted by Barnett and Wilkerson (1967) under the name of "overshoot". Frequency-Wavenumber Spectra The frequency-wavenumber spectra were computed from a two-dimensional Fourier transform of the observed cross spectra from the array as in Equation (2.52). However since only a f i n i t e number (usually 71) of separations were available from the array of nine sensors the resolution is somewhat restricted. The details of this are discussed in Appendix A. Figure 8 shows the wavenumber spectral windowy or transfer function, for this array. As may be seen, the resolution is much better in the east-west direction than in the north-south, i.e. the wavenumber aligned with the long axis of array may be estimated more closely than that aligned with the short axis. This is a product of the asymmetry of the array, and is deliberate since the array was designed with the intention of looking for the second order components associated with the peak of the power spectrum, which should appear at different wavenumbers than the free waves at the same frequency 42 o N - S W A V E N U M B E R L ( C Y C L E S / M E T E R ) FIGURE 8 BROADBAND WAVENUMBER SPECTRAL WINDOW POSITIVE VALUES SHADED 43 (see Equation (2.9)). Since i t was expected that both the waves at the peak of the spectrum and those at twice the frequency of the peak would be t r a v e l l i n g more or less to the west, the array was constructed to have maximum re s o l u t i o n i n t h i s d i r e c t i o n . The observed wavenumber spectrum for a given frequency band i s the convolution of the true wavenumber spectrum with t h i s pattern, i . e . the observed wavenumber spectrum of a pure sinusoid of wavenumber h would consist of the pattern of Figure 8 centered on h . In t h i s figure the contour i n t e r v a l s are i n steps of 10% of the maximum value and areas with p o s i t i v e values have been shaded, with a heavier tone being used for values greater than 30% of the maximum. Wavenumber spectra were computed for each of the four runs from hanned c o e f f i c i e n t s for each of twelve frequency bands of width 0.1 Hz. between 0.39 and 1.46 Hz. The r e s u l t s appeared as printed output and as computer contoured p l o t s . Figures 11, 12, 13 and 14 have been prepared from the l a t t e r . In doing so the plots were s i m p l i f i e d by deleting a l l contours except those at -30%, 0, 30%, 60% and 90% of the maximum value of energy density on the plo t i n question. A l l areas of p o s i t i v e energy density have been shaded, with a heavier tone being used for areas with energy density greater than 30% of the maximum. As an i n d i c a t i o n of the s i m p l i f i c a t i o n involved Figures 9 and 10 show f u l l sized computer plots contoured at in t e r v a l s of 10% of the maximum which have been modified only by the introduction of shading. Each of Figures 11 through 14 shows a l l the wavenumber spectra for a given run. The center frequency appropriate to the frequency band i s given i n the lower r i g h t corner of each of the i n d i v i d u a l wavenumber spectra. In 44 O N - S W A V E N U M B E R L ( C Y C L E S / M E T E R ) FIGURE 9 FREQUENCY-WAVENUMBER SPECTRUM FOR 0.88 Hz PROPAGATION DIRECTION RADIALLY BAND, APRIL 11 INWARD 45 N~S WAVENUMBER L (CYCLES/METER) FIGURE 10 FREQUENCY-WAVENUMBER SPECTRUM FOR 1.26 Hz BAND, APRIL 13 Ml T O C <0<XQ,39 46 1.17 TRANSFER FUNCTION APRIL 11,1967 Hz .5 1 2 3 FIGURE 11 FREQUENCY-WAVENUMBER SPECTRA FOR APRIL 11 DATA 47 • • • I • . • • . . ' I • I •'! TRANSFER FUNCTION Hz .5 1 2 3 FIGURE 12 FREQUENCY-WAVENUMBER SPECTRA FOR APRIL 13 DATA TRANSFER FUNCTION Hz .5 1 2 3 FIGURE 13 FREQUENCY-WAVENUMBER SPECTRA FOR APRIL 25 DATA < < 1 0.48 )°/ iT>0.58 c ^0.68 TRANSFER FUNCTION 1 L - 5 / = > < — > >0.78 e I 2 J , 1.07 h 10-1-APRIL 28,1967 Hz .5 1 2 3 FIGURE 14 FREQUENCY-WAVENUMBER SPECTRA FOR APRIL 28 DATA 50 addition the position of the band is shown on the graph of the frequency spectrum in the lower center of the page. Small letters have been used to relate the various wavenumber spectra to the appropriate place on the spectrum. As the contours are in terms of relative rather than absolute energy density, the frequency spectrum also gives some idea of the amounts of energy in each of the bands. The transfer function T"6^ is also shown on each figure as a reminder of the limited resolving power of the array. Each frequency band for which a wavenumber spectrum was computed has associated with i t a group of wavenumbers within which the wave energy should be concentrated i f the dispersion relation is valid. This range of wave-numbers is indicated on each of the wavenumber spectra as the area between two concentric circles, one of radius and the other of radius where is the center frequency of the band and 0W ±s its bandwidth. Wavenumber spectra were computed for each block of data; the data in the figures is the average over a l l blocks. This technique is similar to that for computing the frequency spectra as described earlier. In addition to the mean wavenumber spectrum the standard deviation about the mean of the wavenumber spectra for the various blocks was calculated for use in establishing confidence limits. Unfortunately the mass of numbers so obtained is almost impossible to comprehend, let alone display. However to give some idea of typical values the standard error for the mean energy density has been computed at one vector wavenumber in each of several frequency bands. The vector wavenumber chosen has in every case been that of the maximum energy density in that frequency band. The frequency bands for which this computation has been done were those containing the peak of the power spectrum, and the bands centered on 0.29 Hz., 0.39 Hz., 0.49 Hz., and 1.46 Hz. The results for each of the four runs are shown in Table III. Generally speaking in the bands near the peak of the frequency spectrum the standard deviations about the mean energy density for wavenumbers near the peak energy density had roughly the same value as the mean. This is what would be expected i f the wavenumber spectral estimates were to behave like "XJ" variates with two degrees of freedom. On the other hand in these same frequency bands, but at wavenumbers somewhat removed from that of the maximum energy density, the standard deviations were considerably larger than the means. In the high frequency bands, represented in the table by that centered on 1.46 Hz., the standard deviations are larger than the means at a l l wave-numbers, including that of maximum energy density. This means that the wavenumber spectra of the 10 second sections of data differ among themselves more than would be expected on a s t r i c t l y s t a t i s t i c a l basis. This suggests that the angular width of the directional spectrum at these frequencies is probably not due to the continual and more or less simultaneous arrival of waves from a wide range of directions but is rather the result of waves arriving from a narrow range of directions which varies in time. To account for the observed s t a t i s t i c a l effect the mean direction would have to shift slowly compared to the 10 second length of the blocks. It can easily be shown that for waves which are spatially stationary over 52 TABLE III VARIABILITY OF WAVENUMBER SPECTRAL ESTIMATES April 11 April 13 April 25 April 28 Standard deviation Mean 1.06 0.90 0.84 0.95 Peak of Spectrum Number of blocks 57 . 35 57 33 Standard error, % 14 14 11 17 1.46 Hz. Standard deviation Mean 2.23 1.76 2.37 2.00 Band Standard error, % 29 30 32 36 0.49 Hz. Standard deviation Mean 1.08 1.00 1.06 1.39 Band Standard error, % 14 17 12 24 0.39 Hz. Standard deviation Mean 1.23 0.96 1.09 1.30 Band Standard error, % 16 17 15 23 0.29 Hz. Standard deviation Mean 0.98 0.53 Band Standard error, % 13 9 53 the size of the array, i.e. do not grow or decay significantly between the f i r s t sensor and the last one, the imaginary part of the complex spatial Fourier transform must vanish. The imaginary part was computed in the same way as the real part (the wavenumber spectrum) as a check on computational errors. In a l l cases the result was gratifying: the values obtained were at least six orders of magnitude smaller than the corresponding values of the real part. It was suggested that this result should cast some light on the growth rates of different components. For example in the case of an exponentially growing sinusoid propagating along the x-axis, the complex cross spectrum -of Equation (2.46) becomes CScW)= i w * t L A L A * «• e where Z_^lN - I'Vit £• . For simplicity let us take the wave to be unidirectional: 2_ 2- = e * e . I f then the separation is allowed to vary from -1_ to \_ , and taking X t = O , the spatial Fourier transform of the term in the cross spectrum with positive kB is then — I x x kt* - > Jk* = M ,A;. A A . v " t r m , - n The cross spectrum for -A.0 w i l l have a similar transform. The part of Equation (3.3) in brackets may be written as (3.3) (3.4) 54 The expression given in (3.4) has no imaginary part, while the imaginary part for (3.3) is + ^ ' - ^ (3.5) The vanishing of the imaginary part for A„ = M. means that any increase in the size of this part of the wave number spectrum to to exponential growth at some wavenumber w i l l not appear at that wavenumber. Despite this discouraging result, an attempt was made to detect growth by calculating as a function of frequency the ratio of the largest value of energy density in the imaginary part of the observed discrete wavenumber spectrum for a given band to the largest value in the real part for that band. The absolute values for these ratios lay between 2.5x10"^ and 4.5x10"^ f° r a l l frequency bands for a l l four runs. There was no evidence of any systematic increase in any frequency band, nor was there any evidence of any particular relationship between the positions of the largest imaginary value and the largest real value. The combined effects of the various terms in and functions of (Ji e-;k^ in the expressions for the real and imaginary parts of the wavenumber spectrum make i t d i f f i c u l t to use their ratior.to estimate Jf without certain knowledge of it„ . Bearing in mind the fact that the expressions given so far have been for a one-dimensional spatial Fourier transform while the wavenumber spectra actually computed involve a two-dimensional transform, i t is possible to estimate the order of magnitude of the minimum y which would 55 be consistent with a r a t i o of 3x10"^ which turns out to be on the order of 10" D per meter. As t h i s i s unreasonably low the most we can conclude i s that t h i s i s not a s a t i s f a c t o r y means of estimating Y, Although the wavenumber spectra for the various runs are c l e a r l y d i f f e r e n t , they do have a number of features i n common. For example, general speaking the energy i n the low frequency bands occupies a narrow range of wavenumbers so that the r e s o l u t i o n i s l i m i t e d by the array r e s o l u t i o n . This i s p a r t i c u l a r l y evident i n the band centered on 0.68 Hz. i n a l l of the runs from the strong resemblance of the observed pattern to the t r a n s f e r function. Also generally speaking the waves near the peak of the frequency spectrum come from the d i r e c t i o n of the greatest fetch rather than the wind d i r e c t i o n . This i s i n agreement with the observations at the same s i t e reported G i l c h r i s t (1966). (It should be emphasized that because of the p e c u l i a r fetch d i s t r i b u t i o n the d i r e c t i o n a l spectra are grossly unlike those expected i n the open sea.) Another general observation i s that the energy i n a given band i s concentrated at wavenumbers compatible with the deep water dispersion r e l a t i o n . Towards higher frequencies the r e s u l t s for the l a s t two runs might seem to b e l i e t h i s , but these w i l l be explained below i n terms of observed currents. At low frequencies the coherences between the records from the various sensors were high, allowing the r e l a t i v e phases to be measured with some pr e c i s i o n . In most cases the plot of these phase differences against vector separation for a given frequency band was simple enough to permit drawing s t r a i g h t l i n e s of constant phase, and hence to determine the d i r e c t i o n of propagation of the waves i n t h i s band. The d i r e c t i o n a l r e s o l u t i o n achieved i n t h i s way was at low frequencies much better than that given by the wave-56 number spectra themselves. Using t h i s technique i t was found that on three out of the four days the waves i n the lowest frequency bands ( w e l l below the peak of the spectrum) were coming from behind the p l a t f o r m , i . e . , from the west. On the A p r i l 11 and A p r i l 28 r e s u l t s the feathered arrows showing the d i r e c t i o n from the phase p l o t s i n d i c a t e t h i s very c l e a r l y . On A p r i l 13 the phase p l o t technique d i d not work because the band at 0.39 Hz. contains energy coming from both east and west as i s apparent from the bimodal wavenumber spectrum. This u n f o r t u n a t e l y does not show up as c l e a r l y on the reduced p l o t as on the f u l l s i z e one, but i t suggests t h a t at frequencies below the 0.39 Hz. band the energy l a r g e l y comes from the west. The source of these waves, which apparently move against the wind, i s unknown. One p o s s i b i l i t y i s that they are s o u t h e a s t e r l y s w e l l from the S t r a i t of Georgia r e f r a c t e d around Point Grey. Another i s that they are the waves of undetected shipping (ships were watched f o r and noted i n the data l o g , but some could have been missed.) The v a r i a b i l i t y i n these bands i s not markedly d i f f e r e n t than that of the peak of the spectrum, as may be seen from Table I I I , which argues against the ship wave hypothesis, since ship waves would be i n t e r m i t t e n t depending on the number of ships passing. I f these waves are not due to sh i p p i n g , t h e i r presence ca s t s serious doubt on the r e s u l t s of G i l c h r i s t (1966) w i t h regard to the d i r e c t i o n of the long waves supposedly generated by the P h i l l i p s resonance mechanism. The north-south l i n e a r a r r a y used i n h i s experiment would have been unable to d i s t i n g u i s h between waves coming from east or west. The f i r s t run, that of A p r i l 11 shown i n Figure 11 poses the fewest problems of i n t e r p r e t a t i o n . In the lowest frequency band the waves are 57 coming from t h e n o r t h w e s t . I n the n e x t h i g h e r band t h e y have s h i f t e d o v e r t o t h e n o r t h e a s t , r o u g h l y from t h e p o i n t o f maximum f e t c h . The band c e n t e r e d on 0.58 Hz., w h i c h c o n t a i n s t h e peak o f t h e f r e q u e n c y (power) s p e c t r u m , comes from the same d i r e c t i o n . As the c e n t e r f r e q u e n c i e s i n c r e a s e t h e d i r e c t i o n a l d i s t r i b u t i o n s g r a d u a l l y broaden, as i n d i c a t e d by t h e i n c r e a s i n g a r e a e n c l o s e d by t h e 60% c o n t o u r s . At t h e same time t h e d i r e c t i o n o f the peak g r a d u a l l y s h i f t s around u n t i l by 1.46 Hz. t h e wave d i r e c t i o n i s a l i g n e d w i t h the o b s e r v e d w i n d . The wavenumber s p e c t r a o b t a i n e d on A p r i l 13 shown i n F i g u r e 12 a r e a l s o f a i r l y easy t o i n t e r p r e t . The o b s e r v e d w i n d d i r e c t i o n on t h i s day d i f f e r e d f rom t h a t f o r a l l t h e o t h e r r u n s by about 15°. T h i s r u n i s a l s o d i f f e r e n t i n t h a t one o f t h e s e n s o r s r e f u s e d t o work due t o t h e c o l d so t h a t the t r a n s f e r f u n c t i o n i s b r o a d e r . The g r o s s b e h a v i o r o f t h e wavenumber s p e c t r a a t low and m i d d l e f r e q u e n c i e s i s much l i k e t h a t o f the A p r i l 11 d a t a , e x c e p t t h a t the e n e r g y i n each band h e r e comes from a d i r e c t i o n s l i g h t l y t o the r i g h t ( c l o c k w i s e , s o u t h ) o f t h a t f o r the e q u i v a l e n t band on t h e o t h e r day. The a p p a r e n t b i m o d a l i t y i n t h e 1.26 Hz. band i s an a r t i f a c t p r o d uced by a c o m b i n a t i o n o f a s i d e l o b e o f t h e t r a n s f e r f u n c t i o n t o g e t h e r w i t h l e a k a g e from th e f r e q u e n c y bands i m m e d i a t e l y above and below as a r e s u l t o f t h e han n i n g o f th e c o e f f i c i e n t s . The v e r t i c a l l y e l o n g a t e d peak i s a s w o l l e n s i d e l o b e o f t h e a c t u a l peak w h i c h i s t h e h o r i z o n t a l l y e l o n g a t e d one i m m e d i a t e l y below i t . T h i s s p e c t r u m has been shown w i t h o u t s i m p l i f i c a t i o n i n F i g u r e 10 so t h a t t h e b a s i s f o r t h i s i n t e r p r e t a t i o n c a n be i n s p e c t e d . The s i g n i f i c a n c e o f the f a c t t h a t the t r u e peak d i f f e r s m a r k e d l y i n d i r e c t i o n from t h o s e i n t h e bands a d j a c e n t i n f r e q u e n c y and does n o t seem t o f a l l n e a r the range o f wavenumbers e x p e c t e d from t h e d i s p e r s i o n r e l a t i o n i s n o t c l e a r . 58 The observations taken on A p r i l 25 are i n t e r e s t i n g f o r s e v e r a l reasons. This run was made w i t h the highest mean wind speed (6.4 meters/second) and l a r g e s t observed current (15 cm/sec) of any of the four, and i t i s the only one i n which there i s no evidence of waves coming from the west i n the lowest frequency bands. The peak of the spectrum seems to come from a d i r e c t i o n c l o s e r to the wind d i r e c t i o n than on the preceeding days. This i s d i f f i c u l t to r e c o n c i l e w i t h the M i l e s type wave generation mechanism which would have these waves coming from the maximum f e t c h regardless of the wind speed as long as the wind speed (6.4 m/sec) was greater than the phase v e l o c i t y (2.8 m/sec) at the peak. Towards the high frequency end of the spectrum the wavenumbers of the energy d e n s i t y peaks d i f f e r s i g n i f i c a n t l y from those p r e d i c t e d by the d i s p e r s i o n r e l a t i o n . This i s thought to be due to the e f f e c t of the current i n the d i r e c t i o n of wave propagation tending to s h i f t a given wavenumber to a higher apparent frequency. This aspect w i l l be discussed i n more d e t a i l l a t e r . The f i n a l set of wavenumber spec t r a , taken on A p r i l 28 and shown i n Figure 14, are e a s i l y understood only when viewed without reference to the observed wind d i r e c t i o n . Both the d i r e c t i o n a l spectrum and frequency (power) spectrum are very d i f f e r e n t from the others. Although the wind speed (5.6 m/sec) was comparatively high the peak of the frequency spectrum occurs i n the 0.68 Hz. band ra t h e r than that centered on 0.58 Hz. as i n a l l the others. I t i s worth noting that t h i s spectrum i s the l a r g e s t of the four at high frequencies as shown c l e a r l y on Figure 7. • A l s o the wave d i r e c t i o n s p e r s i s t e n t l y f a i l to swing over to the observed wind d i r e c t i o n . I t i s d i f f i c u l t to b e l i e v e that the measurement of the wind d i r e c t i o n at the s i t e could have been i n e r r o r as much as would be r e q u i r e d to account 59 for the observed wave d i r e c t i o n s . The other conceivable p o s s i b i l i t i e s are a s h i f t i n the wind d i r e c t i o n immediately before the recording was made, current shears s u f f i c i e n t to r e f r a c t the waves, or some s p a t i a l v a r i a t i o n i n the wind f i e l d such that most of the waves seen would be generated by a wind with a d i f f e r e n t d i r e c t i o n from the l o c a l wind. The f i r s t f a i l s to agree with e i t h e r notes taken at the time of the recording or memory, and there i s no information a v a i l a b l e for t e s t i n g the others, so that the matter must unfortunately remain unresolved. It i s desirable to summarize the information contained i n the various observed wavenumber spectra to the point where i t can be used for predictions and c a l c u l a t i o n s . An attempt has therefore been made to choose one vector wavenumber to represent the mode of propagation of energy i n each frequency band. This wavenumber was generally chosen to coincide with the peak energy density i n the wavenumber spectrum. Since t h i s i s a gross o v e r s i m p l i f i c a t i o n , p a r t i c u l a r l y at high frequencies, a range of v a r i a b i l i t y has been associated with these vectors. This range i s based loosely on the 60% contour with subjective compensation introduced for the e f f e c t s of the transfer function and leakage through hanning. The r e s u l t i n g vectors and ranges are shown to scale i n Figures 15 through 18. Two frequencies are associated with some of the vectors for the A p r i l 25 and 28 data. The frequency i n parenthesis i s an i n f e r r e d frequency i n which an attempt has been made to compensate for the e f f e c t s of the observed currents. This was done from the graph i n Figure 19 i n which the magnitudes of the wavenumbers for the various bands have been plotted against observed frequencies. Also on the figure i s a family of curves representing the e f f e c t s on the observed r e l a t i o n between frequency and wavenumber of various 60 1.46 1.27 1.17 1.07 center f requency, Hz 0.97 0.88 0 7 8 0.68 0 . 5 8 0,1 e a s t -w e s t ' 0.49 north-south 0.1 0.05 I „ I ,i, j SUMMARY WAVENUMBERS APRIL 11,1967 rad ian/cm FIGURE 15 SUMMARY WAVENUMBERS FOR APRIL 11 OBSERVATIONS "1.46 136 1.26 1.17 1.07 0.1 e a s t -w e s t center f r equency , Hz 0.97 0.88 0,78 0 .68 0 5 8 - 0 4 9 0.1 north -south 0£>5. J L J I L radians/ cm SUMMARY W A V E N U M B E R S APRIL 13,1967 FIGURE 16 SUMMARY WAVENUMBERS FOR APRIL 13 OBSERVATIONS 62 146 ' 1.36 1/26 116.' 1.07 (133) ; (122) (112) (1,03) (0.96). center f requency, Hz (inferred frequency) e a s t -west 0.97 0.87 0.78 0.68 0 .58 0.49-(0.88) (0.80) (0.72) (0:63) (0.54) (0.46) 0.1 L J L north-south 0.05 1__J L l I I L radian/cm SUMMARY WAVENUMBERS APRIL25/ I967 FIGURE 17 SUMMARY WAVENUMBERS FOR APRIL 25 OBSERVATIONS 63 '1.45 (1.27) 1.36 (1.20) 1.26 (1.12) 1,16-(1.04) cen te r f requency , Hz ( in fe r red f requency) 0.09 e a s t -west 1.07 0.97 0.87 0.78 0 .68 0.58 0.48 ; (0-98) (0.89) (0.80) (071) (0.63) (0.54) (0.46) 0.1 north-south 0 .05 J I I I I i i i rad ian /cm SUMMARY W A V E N U M B E R S APRIL 2 8 , 1 9 6 7 FIGURE 18 SUMMARY WAVENUMBERS FOR APRIL 28 OBSERVATIONS 64 2 3 4 5 6 7 8 W A V E N U M B E R IXI (x1Cf 2radian/cm) FIGURE 19 OBSERVED WAVENUMBERS vs. FREQUENCY 65 components of current in the direction of the wave propagation. The wave-numbers for the April 11 and 13 runs follow the zero current curve within the experimental error (except for the point at 1.27 Hz. on April 13). Those for the April 25 and 28 observations however generally f a l l between measured speeds of 15 cm/sec on the 25th and 10 cm/sec on the 28th. The inferred frequency for a given wavenumber is simply the frequency at which that wavenumber would have been observed were i t not being convected by the current. The inset graphs show the inferred frequency plotted against the observed frequency for the two days. The correction applied to the spectra already shown in Figure 7 was based on a current of 15 cm/sec to the northwest for both the April 25 and the April 28 runs. In making this correction i t was assumed that the dispersion relation held in the absence of current. The mean directions from the wavenumber spectra were used to calculate the component of current in the direction of wave propagation. Thus i f -f"-, was the center frequency of a given band in the absence of current and the component of current along the wavenumber in that band was , then the frequency at which that wavenumber would appear in the presence of the current is f0 , given by so that the apparent energy density w i l l change as well. The relationship between the bandwidth with current present and with no current is the +10 cm/sec and +20 cm/sec curves, as would be expected from the crudely (3.6) where «L is the acceleration due to gravity. This relation is nonlinear (3.7). 66 In practice of course we had energy density as a function of £ a , so that i t was necessary to invert the above relations. This was easily done graphically. Summary of Observations The observed wave fields have power (frequency) spectra which show an equilibrium region in which the energy density is inversely proportional to the f i f t h power of the frequency. Most of the energy present in any frequency interval has the wavenumber which would be expected from the dispersion relation. The waves near the peak of the spectrum come more or less from the direction of maximum fetch. The direction of the waves gradually shifts u n t i l at high frequencies the waves are travelling in more or less the same direction as the wind. 67 CHAPTER IV FREQUENCY DOMAIN STATISTICS This chapter i s concerned with determining whether or not the s t a t i s t i c a l behavior of the observed Fourier c o e f f i c i e n t s of the waves i s consistent with the hypothesis that the wave f i e l d i s a stationary random process. The tests are those of Equations (2.29) - (2.33): i f the complex Fourier c o e f f i c i e n t at frequency CJj i s R j * i then we wish to know whether the means of Rj and Xj are zero, whether and X j are independent, whether the t r i p l e product of R.j or Xj vanishes, and whether the c o e f f i c i e n t s f o r d i f f e r e n t frequency bands are independent. The Populations of C o e f f i c i e n t s The c o e f f i c i e n t s used for the tests were produced by d i v i d i n g each data run into a series of short sections, each of 512 samples, or about ten seconds long. Each section was then transformed to y i e l d 257 complex Fourier c o e f f i c i e n t s , the f i r s t and l a s t of which are pure r e a l , so that the 512 numbers i n temporal space yielded 512 numbers i n frequency space. The frequency separation between adjacent harmonics was thus about 0.10 Hert Since each data run consisted of the simultaneous records from several surface height sensors, the population of c o e f f i c i e n t s for a given frequency had a number of members equal to the number of sensors times the number of sections, or blocks. The i n i t i a l analysis was conducted as i f these were a l independent, although i t was recognized that at low frequencies the records from the d i f f e r e n t sensors were i n fact highly correlated. Generally the analysis was c a r r i e d out on both hanned and unhanned c o e f f i c i e n t s , and only on c o e f f i c i e n t s from frequencies i n which the wave spectrum had s i g n i f i c a n t 68 energy. In order to explain the actual computations, we w i l l introduce the following notation: (4.1) is the Fourier coefficient for the j th harmonic for sensor h for block 5 . Also since we at times w i l l wish to treat RjJ^ and ^ JKJL as i f they were samples from the same population, we let (4.2) and so that Z Tests on Coefficients at Single Frequencies In order to test for zero mean, a l l of theCjJtJl^for a given harmonic number j were treated as independent samples from the same population so that the mean for that harmonic was computed from ' 1 J Z K L n-i. A - i """l K - - - r (4.3) Also, on the hypothesis that the true population had a zero mean, the variance about zero was computed: K L . x % cr3- = - i - H H TL C\Jk2~ °J 2 K L *- «- (4.4), For the hypothetical normally distributed population, the expected standard error of the mean is (4.5). The ratio of the observed mean to the expected standard error > t).e.j IS known to be distributed like Student's _t for l k l _ - l degrees of freedom, so that the significance of an observed mean with respect to the assumed zero mean may be determined. For the number of degrees of freedom available ( > 400 i f the sensors are independent, > 38 i f they are not) the distribution of Student's _t can be approximated by the normal distribution with a loss of accuracy which is insignificant here. The frequencies of occurrence of the quantity for ten intervals of width 0.5 symmetrically arranged about zero were also determined. The resulting distributions were compared with those that would have been expected i f the were normally distributed, in terms of the probability of a given value of determined from a normal distribution. The independence of the real and imaginary parts of the Fourier coefficients was examined by means of the correlation coefficient 70 (4.7), As i s shown i n Weatherburn (1961)^89, i f R i s def med as M lL. n (4.8) where X1V, and Xiv, are samples from uncorrelated populations with means y_ ^ of zero and variances CTJ and <Tt , and 5 - T7 ^- *-4M » then the minimun value of l\ necessary to have a given s i g n i f i c a n c e may be found from (4.9). Here "t4 i s the minimum value of "t with the required s i g n i f i c a n c e from the Student's _t d i s t r i b u t i o n and i s the number of samples, here v< \_ Approximating the Student's _t d i s t r i b u t i o n with the, normal d i s t r i b u t i o n , we f i n d that for (p^R&< K< R,] to be 0.90 "ts must be greater than 1.67, so that i f then to a good approximation for the large numbers of samples a v a i l a b l e JTTT (4.io) Not v/ (4.11). The skewness c o e f f i c i e n t i s given by the average t r i p l e product over the cube of the standard deviation: 1 £ 11 c)kl S- = '" ' (4.12) As shown in Weatherburn (196l) % 57, the expected standard error of the skewness of a sample population from a normally distributed population with zero mean is given by ^ • i ^ = /IT where Nl is the number of members of the sample population, in this case These statistics were computed for both hanned and unhanned coefficients. Each hanned coefficient represents a linear combination of three of the unhanned coefficients. If the unhanned coefficients are normally distributed random numbers with a given variance, then the hanned coefficients w i l l also be normally distributed random numbers, but with a smaller variance. Since a l l of the various statistics used for testing the hypothesis in question have been normalized by the observed variance, the same tests should apply equally well to the hanned and unhanned Fourier coefficients. The only case where any distinction at a l l is required is that of the correlations between coefficients at different frequencies, and that w i l l be discussed at the appropriate time. Some of these statistics are shown in Figures 20 - 23. Values for selected harmonics are shown plotted against harmonic number for each of the four data runs. Looking f i r s t at the plots of ^ ^ and login i t is immediately evident that the hypothesis that the C^i** are drawn from a normal population with zero mean is well substantiated for harmonic numbers above, say, 10, and not so well substantiated below that. For the four day's runs there are 96 fA j for j s D 10, four of which l i e outside the 90% confidence limits, two 72 of these outside the 95% limit, as against the nine or ten expected outside the 90% limit and four or five outside the 95% limit. For l o g 1 0 ~X_*j there are 11 values outside the 90% limits, 8 of which are outside the 95% limit. At lower frequencies the assumption that the observations from the different sensors are independent breaks down, so that the ^ E.j used for normalizing are much too small. Figure 24 shows the coherences among the various data channels for several harmonics for the April 11 run, which is more or less typical. For harmonic number 2 a l l the channels, or sensors, are highly correlated, while for harmonic number 20, the correlation is insignificant. If a l l the sensors were perfectly correlated, this would in effect reduce the number of independent samples to the number of blocks, so that a new expected error of the mean could be calculated from (4.13) Figure 25 shows the results of normalizing V\j with this factor, as well as i o g 1 Q where only 2-L of the variables are assumed independent. The latter, where most of the values are significantly low, reflects the fact that the correlation between the sensors is not perfect so that the number of independent variables is really somewhat greater than I L . The only case M; where , . is significantly different from zero is for the second 5.E. j harmonic for the April 13 run. We may thus conclude that the means of the populations from which ftjjkS. and ijJk.iL are drawn are probably zero. The plots of the correlation coefficient for the four runs a l l show apparently significant deviations from zero for harmonic numbers less than ten. These low frequency values are not significant however i f the lack of FIGURE 20 FOURIER COEFFICIENT STATISTICS vs. HARMONIC NUMBER, APRIL 11 1 l l I I I L4 1 FIGURE 21 FOURIER COEFFICIENT STATISTICS vs. HARMONIC NUMBER, APRIL 13 FIGURE 22 FOURIER COEFFICIENT STATISTICS vs. HARMONIC NUMBER, APRIL 25 I I I I I I I FIGURE 23 FOURIER COEFFICIENT STATISTICS vs. HARMONIC NUMBER, APRIL 28 77 HARMONIC NUMBERS 2 6 10 2 0 2 ( I) 3 (ID 4 (III) 5 (IV) 6 • (V) 7 (VI) 8 (IX) 9 (VIII) 10 (Vl l ) (POSITION) A, A / A • « B / V V - A 2 3 4 5 6 7 8 1 / 0 J . 3 t 5 6 ' 7 8 t » I O 2 3 H 5 6 7 8 < i l O A 3 t 5 S 7 8 f 10 C H A N N E L b C O H E R E N C E B E T W E E N C H A N N E L S a AND b F O R VARIOUS HARMONICS APRIL 11,1967 broken line is 9 0 % probability . level FIGURE 24 COHERENCE AMONG SENSORS FOR VARIOUS HARMONICS 0 h a n n e d * u n h a n n e d FIGURE 25 S T A T I S T I C S FOR THE F I R S T 10 F O U R I E R C O E F F I C I E N T S independence among the sensors is taken into account, as may be seen in Figure 25. A l l of the runs, with the exception of that of April 13, show a few significantly large values of ITj at high frequencies as well, particularly in the case of the unhanned coefficients. There are two possible explanations for this. One is that the effect is actually present in the wave f i e l d as the result of nonlinear effects. If this were the case however the treatment of the coefficients should not have much effect - the hanned coefficients should show significant correlations as well. The other possibility is that the correlations are due to transfer of energy through the transform window from other frequencies. In this case there should be a difference between the hanned and unhanned coefficients as they have markedly different windows. To see how such a transfer leads to a correlation between the real and imaginary parts of a given coefficient, let us consider a sinusoid h it) ~ A Cos to „ t + 6 5iV» t o „ t i JL with the Fourier transform Substituting this in Equation (2.64) and performing the integration, we find the apparent transform , . • f -\C-f-u>„)NM A -v Cjti\ - (AriG) ' l e - 1 ) 80 The r e a l p a r t of t h i s i s fu).\n ftr 7 ft (* I - cos ({-oo.^M&t _ 1 - totU»ui.)W^ w h i l e the imaginary part i s (4.14) A C I - I O S ^ - ^ Q ) Nfot- + t - cpt U » « » » ^ N (4 .15) , The product of the r e a l and imaginary parts w i l l c o n t a i n terms i n B A } plus the terms ii* (-f+ u>.) N&t — 5>KI K+i^NfrV coslW^N&V 1 H I (4.16). Now even i f we l e t A and 6 be independent random functions w i t h zero mean and A1 = (&*• } i . e . be l i k e the r e a l and imaginary parts of a t y p i c a l s t a t i o n a r y wave component, the term i n 6 l + A 1 does not vanish. Thus i t i s p o s s i b l e f o r the observed c o r r e l a t i o n c o e f f i c i e n t to d i f f e r from zero because of the e f f e c t of the transform window even though the c o r r e l a t i o n c o e f f i c i e n t i n the p o p u l a t i o n i s zero. This e f f e c t might be expected t o become 81 increasingly important as the energy levels in the spectrum decrease towards high frequencies. As shown on the various plots, the behavior of the observed skewness of the coefficients S j is in many ways similar to that of the correlation coefficient. It shows apparently significant values at low frequencies which turn out to be merely reflections of the high correlation between sensors. Like the correlation coefficient some significant values of the skewness occur at high frequencies, but unlike the correlation coefficient there is a tendency for large values for both the hanned and unhanned coefficients to occur together, although perhaps with a difference in sign. This suggests the large values of skewness are actually present in the wave components. The most likely cause of this is the second order nonlinearities of Equation (2.9), which w i l l contribute to the skewness of a given coefficient at sixth order. Tests of the Independence of Coefficients at Different Frequencies As a test of the independence of the Fourier coefficients at different frequencies we compute the correlation coefficients U K t r-1"1 ( 4. 1 7 ) where the notation is that of (4.2) and We can use Equation (4.9) to determine the significance of a given value of '"ij (**> "0 against the hypothesis that theC;^*, are uncorrelated provided we can relate r.-{j*.") to the iZ of (4.8). When theCjitdKvi andCjjktJ.t, for the different sensors are completely 82 independent, as at high frequencies, then they may be i d e n t i f i e d with the X; of (4.8). Likewise the C T 0 ; (r^ may be i d e n t i f i e d with the S ; , so that we may take H. - f " j - * 0 > with Ki - U K At very low frequencies, when the C;VA»« are almost the same for a l l the sensors, i . e . for a l l k. with the C j j j ^ and C- independent normal variables, then and L 07j ( . ^ - t_ JT, ^— • A ^ In such a case we i d e n t i f y the C J J I^ and with X, and X t i n (4.8), and the <r6i with the 5; as before, so that R - c { j (.v»-^  <*") , except t h i s time we have N - l _ . F i n a l l y l e t us consider the case where one of the variables i s from a region where the sensors are not independent, i . e . but the other i s from a region where they are. Thus n j ( _ L t=' A - i  with cr e>(^ = C , L and ( T . / C O = fe. ^ J " ^ 83 To r e l a t e t h i s to (4.8) we can again i d e n t i f y LT.JI^ with x 4 andCijC**1) with 5-1 . However i f we wish to equate ^ E T C j j t J l k , with x t , then we must have Thus u i-(4.18) so that we must put N/ i n (4.8) equal to L\TK* To the same degree of approximation as i n Equation (4.11), for the case where the records from the sensors are independent, we may take , 0 % " 7 L K - 1 . ( 4 . 1 9 ) for the case where the sensors are p e r f e c t l y correlated R - L T 4 . / L - J L ( 4 - 2 0 ) and for one c o e f f i c i e n t from frequencies for which the sensors are highly correlated and the other from frequencies for which they are independent, 1^  tr^l JQT~C <4-21>-The c o r r e l a t i o n c o e f f i c i e n t was computed for each of the runs for 1\X,H a n d f ° r R>M * W a n d T ' < ^ r A X with j < i , for both hanned and unhanned c o e f f i c i e n t s . Using the above tests of s i g n i f i c a n c e , with the d i v i d i n g l i n e between c o e f f i c i e n t s from the d i f f e r e n t sensors which are supposedly p e r f e c t l y independent and those which are supposedly p e r f e c t l y correlated somewhat a r b i t r a r i l y taken to be the tenth harmonic, some generalizations can be made. The c o r r e l a t i o n c o e f f i c i e n t s for the unhanned c o e f f i c i e n t s show large areas of s i g n i f i c a n t values, p a r t i c u l a r l y towards high 8 4 frequencies. As these mostly disappear in the hanned coefficients, they are probably associated with window effects. The hanned coefficients in fact show rather fewer significant correlations than would be expected from purely s t a t i s t i c a l considerations. There is an asymmetry between the statistics of the imaginary and real parts. It is extremely marked for the unhanned coefficients, but is probably insignificant for the hanned coefficients. As an example of the results obtained, Figures 26 - 29 show the various correlation coefficients for the April 11 run plotted against the two frequencies and contoured in intervals of — . . '=• . Significant values have been shaded, and the dividing line between the areas in which the different significance levels were used is indicated. The "pebbled" appearance of the plots for the hanned coefficients is due to the alteration of sign of the correlation coefficients produced by the hanning procedure. Conclusions On the basis of the various s t a t i s t i c a l tests applied in this chapter i t would seem that the behavior of the observed Fourier coefficients is indistinguishable from that which would be expected of the coefficients from a stationary random process, with the possible exception that the third moment of the observed coefficients may have a non-zero value at some frequencies. 8C cn tn ru a a rsl X >- _ LU O UJ a: CO to ru cn ai 0.292 0.B81 J.071 1.460 F R E Q U E N C Y ] Hz 1.B50 2 . 2 3 9 FIGURE 2 6 CORRELATION COEFFICIENTS FOR R,Rj AND I i i j UNHANNED, APRIL 11 SIGNIFICANT VALUES SHADED 86 F R E Q U E N C Y j Hz " R T R j " HANNED F I G U R E 2 7 C O R R E L A T I O N C O E F F I C I E N T S F O R ft.Rj A N D I; Xj H A N N E D , A P R I L 1 1 S I G N I F I C A N T V A L U E S S H A D E D 8 7 FIGURE 28 CORRELATION COEFFICIENTS FOR R;Ij UNHANNED, APRIL 11 SIGNIFICANT VALUES SHADED 88 F R E Q U E N C Y j Hz FIGURE 29 CORRELATION C O E F F I C I E N T S FOR R.Ij HANNED, A P R I L 11 S I G N I F I C A N T VALUES SHADED 89 CHAPTER V OBSERVATIONS OF NONLINEAR EFFECTS This chapter deals with two d i f f e r e n t attempts to detect second-order nonlinear e f f e c t s i n the observed wave f i e l d s . The f i r s t method was to t r y to r e l a t e some of the secondary peaks i n the frequency-wavenumber spectra to the expected forced second order waves due to waves at lower frequencies. The second was to compute the bispectrum for the observed waves and to compare i t with a model bispectrum based on the observed frequency and frequency-wavenumber spectra. Second Order N o n l i n e a r i t i e s i n the Frequency-Wavenumber Spectrum Since the waves produced by second order nonlinear interactions among free g r a v i t y waves are forced rather than free o s c i l l a t i o n s and do not obey the usual dispersion r e l a t i o n of Equation (2.4), they might be expected to be distinguishable from the free waves i n the frequency-wavenumber spectrum of a given frequency band. In p a r t i c u l a r , the Stokes second harmonic of a sinusoid of frequency OJ, with wavenumber JL, would have a frequency of 2-OJ, and a wavenumber 2.Jt, . Free g r a v i t y waves at a frequency l u , would however have a wavenumber The d i f f i c u l t y i n such a technique arises i n separating a peak due to such harmonics from a sidelobe of the wavenumber spe c t r a l window of Figure 8. Using the d i r e c t i o n a l information a v a i l a b l e from the lower frequency wave-number spectra i t i s possible to predict the points on the wavenumber spectrum at twice the frequency where the second harmonics should occur. The difference RATIO TEST FOR SECOND HARMONICS, APRIL 11 FIGURE 31 RATIO TEST FOR SECOND HARMONICS, APRIL 13 92 RATIO TEST FOR SECOND HARMONICS, APRIL 25 FIGURE 33 RATIO TEST FOR SECOND HARMONICS, APRIL 28 94 between the harmonics and the sidelobes of the transfer function is that the locations of the sidelobes are fixed relative to the peak energy density, while the location of the harmonics is fixed relative to the wavenumber coordinates. Accordingly the ratio of the energy density at some location fixed relative to the location of the peak energy density to that peak energy density should remain relatively constant for a l l frequency bands i f only the transfer function is involved. If the location chosen happens to coincide with a peak due to the second harmonic in some band this ratio should show an increase in that band. Plots of this energy density ratio against frequency at the expected positions of the second harmonics in each . of the four highest frequency bands are shown in Figures 30 through 33 for each of the four runs. The small crosses on the wavenumber spectra reproduced on these plots show the locations used in the appropriate bands. For example, for the April 11 run the ratio for the 1.17 Hz. band shows no peak (the expected location of the peak is indicated by the small arrow) while that for the 1.27 Hz. band shows a large one. The results for the 1.36 Hz. band are equivocal, but the 1.46 Hz. ratio shows a maximum at the right frequency. (The plots for these last two bands are identical because the locations of the expected second harmonics were in the same position relative to the respective peaks. This is also true of the plots for the 1.17 and 1.36 Hz. bands for the April 13 run.) The main d i f f i c u l t y with this technique arises from the fact that only a few of the discrete wavenumbers for which the wavenumber spectra were computed l i e in the region of the expected harmonics, so that i t is d i f f i c u l t to choose exactly which should be used in forming the ratio. Usually some compromise is required, resulting in the use of the average over two or three 95 locations. This is indicated by the multiple crosses on some of the wave-number spectra. The quality of the results f i n a l l y obtained is obviously not consistent enough to permit quantitative estimates of the percentage of second harmonics present. On the other hand the behavior in many of the bands certainly seems to indicate the presence of such harmonics. The Bispectrum and Second Order Nonlinearities The bispectrum was defined in Equation (2.60) as (5.1) where N is the number of samples in a block or section of data and — t~!' - . As was shown in Equation (2.59) the bispectrum provides a measure of the second order interaction coefficients which were in turn defined in Equation (2.9). Bispectra were computed for the observed surface heights by dividing the record from each sensor into sections of 512 samples corresponding to approximately 10 seconds, and computing the appropriate tr i p l e products of Fourier coefficients for each section. These triple products were then averaged over a l l of the sections to produce a bispectrum for each sensor. The average of the bispectra from a l l the sensors for a given run was. then taken, with the var i a b i l i t y among the sensors providing an indication of the uncertainty in the result. One of the problems encountered when computing bispectra for broadband processes is that they are d i f f i c u l t to interpret intuitively. The procedure f i n a l l y adopted here, as in Hasselmann, Munk and MacDonald (1963), was to produce a computer predicted bispectrum for comparison with that observed. In the f i r s t method tried Equation (2.8) was used to compute the Fourier 96 coefficients resulting from the interactions of a number of sinusoids of various specified frequencies, wavenumbers and amplitudes. These Fourier coefficients were then analysed in the same way as those from the f i e l d data. Various simulated wave fields were used as input to the nonlinear modeling program. Provision was made for allowing the real and imaginary parts of the Fourier coefficients of the input (fi r s t order) waves to vary from block to block as independent normally distributed random numbers with a specified variance or rms amplitude. Some examples of the resulting bispectra when various simple wave fields were used as input to this model are shown to assist in the inter-pretation of the results from the more complicated cases. A single sinusoid with a frequency of 0.59 Hertz and a randomly varying amplitude and phase produced the bispectra shown in Figures 34 and 35. The latter differs from the former in that the coefficients produced by the modeling program were hanned before computation of the bispectrum. Four sinusoids with random amplitude and phase in adjacent frequency bands and a l l propagating in the same direction, with rms amplitudes approximately equal to the rms amplitudes of the observed waves in those frequency bands, produced the bispectrum shown in Figure 36. In this figure the natural symmetry of the bispectrum about the line has been used to allow presentation of two bispectra in a single figure. Below the line of symmetry, i.e. for f, <4U , is shown the bispectrum for the hanned coefficients, while above the line, for f, >-ft , appears the bispectrum for the unhanned coefficients. This technique w i l l be used extensively as a means of comparing the model bispectra with those observed by putting the model in one half of the plot and the observed in the other. 97 Unfortunately, this particular modeling procedure was not very useful for simulating bispectra like those observed, mainly because of computational economics. In order to make the simulated bispectrum converge to that observed i t would have to be based on a s t a t i s t i c a l sample of similar size. When i t is remembered that the observed bispectra represent the averages over thirty or more blocks of data and nine sensors, and that reasonable modeling of the directional spectrum would require five or more random sinusoids in each of thirty frequency bands the magnitude of the required computation becomes apparent. A simpler modeling procedure was thus used for the more complicated spectra. Rewriting Equation (2.59) to take account of (2.10), a, 2 *- v J (5.2) If in addition to i t s other properties ~£.\. is normally distributed, then (5.3). Also we note from (2.40) that Suppose now we take FtV> S(Q 1.MAT (5.4) 98 in in , CO H N CJ c 3 <D r-a CO co T — 1 — r 0.293 SIMULATED B ISPECTRUM f i rs t order wav e = A cos(2n f 0 t ) 1—1—r~ 0.683 3.074 T — 1 — r 1.464 T — 1 — r 1.855 f r equency Hz FIGURE 34 SIMULATED BISPECTRUM FOR SINUSOIDAL SURFACE WAVE 99 SIMULATED B I S P E C T R U M . f i r s t order wave = Acos(2TTf0t) coeff icients hanned 0.293 i • T i i — i — i — i — i — i — ~ r 1.074 J..464 1.855 • . f requency Hz • FIGURE 35-SIMULATED BISPECTRUM FOR SINUSOIDAL SURFACE -WAVE, HANNED 100 0.683 nput* frequencies i — i — i — i — i — r J.855 2.246 1.464 frequency2Hz FIGURE 36 SIMULATED BISPECTRUM FOR FOUR RANDOM SINUSOIDS 3 -2 LIGHT SHADING DENOTES VALUES GREATER THAN 100 cm Hz , HEAVY SHADING DENOTES NEGATIVE VALUES 101 where A is the direction of propagation, proportional to > a n <3 where H F (^ = 1 Then (5.2) becomes \1 s- * +'• +. £ F ( M 5^0 F t W S f 0 , 2 1 IL F C U s t f o F C M S i ^ c ; 1,1 ( 5. 5 ). Since from (2.61) we also have i t is possible to identify the expression between the brackets -^••"^  (5.5) with A t ^ f i L . In computing the model only a limited number of vector wavenumbers could be used in any frequency band. After some experimentation i t was decided to use a Gaussian distribution of energy with direction: - ^ (5.6), This was approximated by eleven wave trains of equal energy whose directions of propagation were adjusted to .make the distribution of energy with direction obey (5.6). Figure 37 shows the resulting directions. In the model the angular deviation & could be varied. Recognizing that the details of this scheme might have a significant effect on the model results two other directional distributions were tried. One of them was that of approximating (5.6) with wave trains equally spaced in direction, but of varying energy. 102 b a n d e n e r g y m u I t i pli e r 0,1 0. -J-0.15 0. 0.1 - r 0. G a u s s i a n m o d e ! a p p r o x i m a t i o n s : equal a m p l i t u d e G a u s s i a n ' e q u a l a n g l e G a u s s i a n u n i f o r m m o d e l - e x +CX FIGURE 37 DIRECTIONAL DISTRIBUTIONS OF ENERGY USED IN MODELING 103 The other was that of using waves equally spaced in angle and of equal amplitude. Both of these are also shown in Figure 37. The observed bispectra were computed from hanned Fourier coefficients. To take account of this the model bispectra were corrected using the followin relation: (5.7) where ^ M ( ^ ; j ^ j l is the model bispectrum for the frequency bands associated with the i th and j th harmonics before correction and ^ > M HC^ ;J^J > ) ^ s t* i e corrected model bispectrum for the same bands. This correction may easily be derived by considering the effect of the hanning convolution on the product of three Fourier coefficients, and by requiring that the integral under the bispectrum be the same in both hanned and unhanned versions. Comparisons of the observed bispectra with the best results obtained by modelling for the April 11 and April 13 runs are shown in Figures 38 and 39. In the end i t became apparent that the factors having .the most effect on the value of the model bispectrum at a given pair of frequencies were the energies in the two frequency bands, with the angular deviation 0\ coming next, followed by the form of the directional distribution. At frequencies above the peak of spectrum the values of the spectra are known f a i r l y accurately, so that the observed and model bispectra could be brought into agreement by adjusting the angular deviation used in the latter. On the othe hand the model assumes that the observed spectra are due to f i r s t order 104 MODEL 0.292 0.B81 J.071 J.4B0 J.850 2.239 f r eq uency2,Hz. FIGURE 38 OBSERVED AND SIMULATED BISPECTRA FOR APRIL 11 DATA 3 -2 CONTOURED IN UNITS OF cm JHz . LIGHT SHADING DENOTES VALUES GREATER THAN 100 cm3Hz~2,, HEAVY SHADING DENOTES NEGATIVE VALUES. 105 M O D E L f r e q u e n c y 2 FIGURE 39 OBSERVED AND SIMULATED BISPECTRA FOR APRIL 13 DATA 3 -2 CONTOURED IN UNITS OF cm Hz . LIGHT SHADING DENOTES VALUES GREATER THAN 100 cm 3Hz" 2, HEAVY SHADING DENOTES NEGATIVE VALUES. 10 6 components only, while in fact they also contain contributions from the products of nonlinear interactions, particularly at frequencies greater than twice the frequency of the peak of the spectrum. Thus the angular deviations obtained by the matching the observed bispectra and the model are probably unrealistic above that frequency. At the frequency of the peak and below the situation is confused by the d i f f i c u l t y of assessing the detailed effects of the spectral window, either hanned or unhanned, on the f i r s t order spectrum. For both of the runs shown i t was necessary to reassign energy from the band immediately below the peak to the peak to obtain reasonable agreement between the model and the observations. No effort was made to extend the model to the zeroth harmonic, and the f i r s t harmonic was treated in a very cavalier way. The power spectra and angular deviations used to produce these two models are shown in Figures 40 and 41. Also shown for comparison are the angular deviations deduced from the wavenumber spectra and as expected from the Miles theory of wave generation, as shown in Gilchrist (1966). For the lower frequencies the mean direction in each frequency band was that found from the wavenumber spectra, while for frequencies higher than those for which wavenumber spectra were computed the mean direction was the same as the wind direction. The models agree with the obseryed bispectra quite well within the range of frequencies from the wave spectrum peak to about twice that frequency, as might have been expected from the previous discussion. Below the frequency of the peak the agreement is not very good, as the observed bispectra show a large negative peak which was impossible to reproduce in the model. At frequencies higher than about twice the peak frequency the model bispectra 150n 1 0 0 S P E C T R U M «model - o b s e r v e d 2,0 Hz 3.0 90° i 6 0 ' 30*1 ANGULAR D E V I A T I O N M i l e s ( 5 . 8 m / s e c wind) m o d e l f r o m w a v e n u m b e r s p e c t r a 1 1 1 1 ; — i r 0 1.0 2.0 Hz 3.0 FIGURE 40 INPUT SPECTRUM AND ANGULAR DEVIATIONS FOR APRIL 11 MODEL 108 S P E C T R U M • m o d e l " O b s e r v e d 3.0 Hz 90% 60°J 3Cf-0 A N G U L A R DEVIAT ION Jyji les ( 5 .8 m / s e c w i n d ) m o d e l „ f r o m w a v e n u m b e r " \ y \ s p e c t r . a 0.0 n r 10 2.0 -I r 3.0 Hz FIGURE 41 INPUT SPECTRUM AND ANGULAR DEVIATIONS FOR APRIL 13 MODEL 109 are larger than those observed, presumably because of the nonlinear content of the observed spectra. It should be noted when making these comparisons that the uncer-tainty of the observed bispectrum as indicated by the variability among the nine sensors is quite large. For example, for the April 11 observations at the peak of the bispectrum the value of the mean over the nine sensors was 194 cur* Hz"^, while their standard deviation about that mean was 154 cm^  Hz"^. Assuming a normal distribution the standard error of the mean would be about 30% of the value of the mean. This value seems to be f a i r l y representative for places where the bispectrum has large values, such as along the line where the frequencies are equal. One the other hand, in some regions of small values the uncertainties are proportionally much larger, as for example at (0.584 Hz, 0.291 Hz) for the April 11 observations, where the mean value of -3.46 cmJHz has a standard error of 12.3 cm Hz" . Examples of the effects of varying the angular deviation and the form of the directional distribution of energy are shown in Figures 42 and 43. In the f i r s t of these the best model for the April 11 run is compared with a model which is identical except that a l l of the angular deviations have been reduced to 2°. in Figure 43 the best model for the April 11 run is compared with a model in which a l l of the parameters are unchanged but the Gaussian directional spectrum is represented by waves of varying energy but equally spaced in direction. Because of d i f f i c u l t i e s in making proper corrections for the effects of the currents no attempt was made to compute model bispectra for the April 25 and April 28 observations. The results obtained indicate that the theory for continuum interactions, 110 2° ANGULAR DEVIATION f requency2 Hz FIGURE 42 EFFECT OF. ANGULAR DEVIATION ON SIMULATED BISPECTRUM I l l E Q U A L A N G L E GAUSSIAN M O D E L requency.H z FIGURE 4 3 EFFECT OF DIRECTIONAL DISTRIBUTION ON SIMULATED BISPECTRUM MODELS REFERRED TO ARE THOSE SHOWN PREVIOUSLY IN FIGURE 3 7 . using the assumptions of stationarity in time and space, is correct, least to the accuracy attained in these observations. i i 3 CHAPTER VI SUMMARY OF CONCLUSIONS The objectives of this study were to show how far the s t a t i s t i c a l assumptions usually made when dealing with the theory of continuum nonlinear interactions were relevant to the case of fetch limited wind generated wave fields, and to observe some consequence of the nonlinearity as a check on the other results. Four wave fields were observed and described in terms of their power spectra and frequency-wavenumber spectra. • Although their directional spectra were found to be strongly influenced by the particular arrangement of fetch at the observing site, their power spectra agreed well with both theory and other observations, at least after correction for the observed currents. This agreement is important from the point of view of establishing the relevance of the results of this work to other f i e l d situations. Observations of the frequency domain statistics for these four wave fields indicated that the assumptions of stationarity and of normal distributions for the Fourier coefficients were correct to good accuracy. Two methods were tried for observing direct evidence of nonlinear effects in the wave f i e l d . One of these, the attempt to detect products of second order interactions in the frequency-wavenumber spectra, failed because of the poor resolution of the array of sensors. The other, the bispectrum, succeeded, with reasonable agreement being achieved between the observed bispectra and bispectra predicted from the power spectra using the results of perturbation theory. None of these results is surprising in the sense of upsetting any previously held ideas, but they are comforting in that they indicate that the estimation of continuum nonlinear effects by means of the perturbation analysis provides answers which may be viewed with confidence. LIST OF REFERENCES Barber, N.F. (196l) The d i r e c t i o n a l resolving power of an array of wave detectors, i n Ocean Wave Spectra; Englewood C l i f f s , N.J.: Prentice H a l l : pp.137-150 Barnett, T.P. and J.C. Wilkerson (1967) On the generation of wind waves as i n f e r r e d from a i r borne measurements of fetch-limited, spectra. J . Marine Research 25 (3), 292-150 Benjamin, T.B. and J.E. F e i r (1967) The d i s i n t e g r a t i o n of wave tra i n s on deep water. Part 1, Theory; J . F l u i d Mechanics Z7_ (3), 417-30 Bingham, C , M.D. Godfrey and J.W. Tukey (1967) Modern techniques of power spectrum estimation. IEEE Trans. Audio and Electroacoustics AU15 (2), 56-66 Dobson, F.W. (1969) Observations of normal pressure on wind-generated se waves. Ph.D. Dis s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia Garrett, J.F. (1969) Some new observations on the equilibrium region of the wind-wave spectrum. J. Marine Research_27 (3), 273-277 G i l c h r i s t , A.W.R. (1966) The d i r e c t i o n a l spectrum of ocean waves: an experimental i n v e s t i g a t i o n of c e r t a i n predictions of the Miles-P h i l l i p s theory of wave generation. J. F l u i d Mechanics _25 (4) pp.795-816 Hasselmann, K. (1962) On the nonlinear energy transfer i n a gravity-wave spectrum, Part 1 General Theory. J . F l u i d Mechanics _12 481-500 Hasselmann, K. (1963) On the nonlinear energy transfer i n a gravity-wave spectrum, Part 2. J . F l u i d Mechanics 15 273-281: Part 3. Ibid.15 pp. 385-398 Hasselmann, K., W. Munk and G. MacDonald (1963) Bispectra of ocean waves i n Time Series Analysis; New York: John Wiley: pp.125-139 Havelock, T.H. (1940) The pressure of water waves upon a fixed obstacle. Proc. Roy. Soc. A 175, 409-421 IEEE Transactions on Audio and E l e c t r o a c o u s t i c s . Special issue on Fast Fourier Transform and i t s a p p l i c a t i o n to d i g i t a l f i l t e r i n g and sp e c t r a l a n a l y s i s . June 1967, v o l . AU15, no. 2 Kinsman, B. (I960) Surface waves at short fetches and low wind speeds -a f i e l d study. Vols. 1, 2, 3. Chesapeake Bay I n s t i t u t e Technical Report 19, Ref 60-1, 581 pp. 116 Longuet-Higgins, M.S. (1969a) On wave breaking and the equilibrium spectrum of wind-generated waves. Proc. Roy. Soc. A 310, 151-159 Longuet-Higgins, M.S. (1969b) A nonlinear mechanism for the generation of sea waves. Proc. Roy. Soc. A 311, 371-384 Phillips, O.M. (1958) The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mechanics 4, 426-484 Phillips, O.M. (I960) On the dynamics of unsteady gravity waves of fi n i t e amplitude. Part 1. J. Fluid Mechanics _9, 193-217 Phillips, O.M. (1966) The dynamics of the upper ocean; Cambridge; Cambridge University Press; 277 pps. Snyder, R.L. and C.S. Cox (1966) A f i e l d study of the wind generation of ocean waves. J. Marine Research_24 (2), 141-178 Weatherburn, CE. (1961) A f i r s t course in mathematical statisti c s ; Cambridge; Cambridge University Press; 277 pps. Yaglom, A.M. (1962) An introduction to the theory of stationary random functions. Trans. R.A. Silverman; Englewood C l i f f s , N.J.; Prentice Hall; 235 pps. 117 APPENDIX A EXPERIMENTAL TECHNIQUE Physical arrangement at site The measurements were taken at the Spanish Banks site of the Institute of Oceanography. This consists of a platform supported by pilings which are driven into a large tid a l fl a t on the south west shore of English Bay at Vancouver, B.C. A map of the location has been shown in Figure 1 on page 28. Located on the platform i t s e l f is a small hut which houses recording equipment and personnel. A gasoline powered motor generator supplied 110 volt a.c. power to the electronics. The wave array was suspended from a frame placed 9.5 meters east of the platform and connected to the platform deck by a catwalk. On the highest tides the water reaches a depth of nearly 5 meters at this site while at the lowest the sand bottom is completely uncovered. The photographs in Figure 44 give a good idea of the general arrangement at the time the measurements were made. In Figure 44a the water is only a few inches deep and the array frame on the right of the picture has been lowered to approximately the level used for making observations. A small hand winch was used to raise and lower the array and guy lines were available for levelling i t . The sensor electronics were housed in the plywood boxes mounted on the array frame. The bands of black tape on the array supports are spaced at intervals of one foot. The apparent t i l t of the array supports and the platform is due to camera angle and is fortunately not real. Figure 44b shows the situation when the water is approximately at the level used for observations, and with the array in the raised position. Although the primary reason for raising the array was to 118 hut wind d i rec t ion vane i i i i w i n d s p e e d m a s t a r r a y suppor ts (b ac kg round) F i g u r e 4 4 a eas t F i g u r e 4 4 b FIGURE 44 PHOTOGRAPHS OF OBSERVING SITE protect the sensors from damage by drifting debris, i t also allowed easy access to the sensor electronics and to the probes themselves. Figure 44b was taken looking i n the direction from which the wind was blowing during the observations. The shoreline visible through the array is about seven kilometers distant. As may be seen from the photographs the platform supports and array supports are potentially a source of backscattering from waves passing through the array. As long as the wave crests were below the horizontal cross beam the most important reflectors would be the array supports, the pilings, and the diagonal cross braces. Data was not taken i f the waves were hitting the horizontal beam. The geometrical relationship between the array and the scatterers has been shown in Figure 2, page 29. The reflection from the cylindrical supports and pilings is easily calculated from potential flow theory (e.g. Havelock, 1940). Suppose the only motions present are the incident wave and the reflection and suppose further that the incident wave is a plane wave proceeding in the -x direction. Then the potential can be written as $ = $ r = A e e + e. e. <j>r where - A e! Expanding <J}; as an i n f i n i t e sum of Bessel f u n c t i o n s i n polar c y l i n d r i c a l coordinates w i t h the centre of the o b s t a c l e taken as the o r i g i n , 4>r as an i n f i n i t e sum of Hankel func t i o n s of the second k i n d , and i n t r o d u c i n g the boundary c o n d i t i o n that the normal v e l o c i t y must v a n i s h on the surface of the c y l i n d e r , we f i n d that Hz & K ^ r l + I ZL i f i To ( J ^ cos A© 4=1 * Here &. i s the ra d i u s of the c y l i n d e r , r i s the distance f r o m . i t s center, and i = v-1 . The q u a n t i t y of i n t e r e s t i s a c t u a l l y the surface e l e v a t i o n , but since Using asymptotic expressions f o r "T^ ' ( M and Hji 00 as x —* O t h i s ;ives f o r > f e a L — W O , i . e . f o r waves very long compared to the s i z e of the s c a t t e r e r so that f o r Jta.<:< 1 , 121 + ! ^ T< CM cos e _ To cfe^ Two q u a n t i t i e s were c a l c u l a t e d d i g i t a l l y , ^ <^>r^ and \^S^ <<]);"> v ((>; > where (3L> denotes I n t h e d i g i t a l c o m p u t a t i o n the sums were t r u n c a t e d when the i n d e x jl was e q u a l t o t h e l a r g e s t i n t e g e r l e s s t h a n J t r *• S~ , or i n t h e case o f c a l c u l a t i n g the c o e f f i c i e n t s bjj_ f o r 5. l e s s t h a n The r e f l e c t i o n s due t o t h e d i a g o n a l c r o s s beams were c a l c u l a t e d on the b a s i s o f s e v e r a l a s s u m p t i o n s . I t was assumed t h a t the d i a g o n a l beams c o u l d be a p p r o x i m a t e d by v e r t i c a l beams w i t h t h e same w a t e r l i n e d i m e n s i o n s . A l s o i t was assumed t h a t o n l y the case o f a p p r o x i m a t e l y normal i n c i d e n c e was o f i n t e r e s t . I n t h i s case the r e f l e c t e d wave r e q u i r e d by t h e boundary c o n d i t i o n t h a t n ormal v e l o c i t i e s must v a n i s h a t t h e s u r f a c e o f t h e o b s t a c l e behaves as i f i t were a p l a n e wave e x p e r i e n c i n g F r e s n e l d i f f r a c t i o n by a s l i t t he s i z e o f the beam. I t was t h u s p o s s i b l e t o use t a b u l a t e d v a l u e s o f the F r e s n e l i n t e g r a l s t o compute t h e r a t i o o f t h e r e f l e c t e d p o t e n t i a l t o the i n c i d e n t p o t e n t i a l a t t h e p o s i t i o n s o f v a r i o u s s e n s o r s . T a b l e IV l i s t s t h e r e s u l t s o f t h e s e c a l c u l a t i o n s f o r t h e s e n s o r s a t the 122 extremes of the array, (shown in Figure 2) for several frequencies. As may be seen in the table the reflected amplitudes may be quite large, particularly toward higher frequencies. In assessing the effect these reflections w i l l have on the observations some account must be taken of the fact that wind generated waves are not long crested sinusoids and that the coherence between surface elevations at two points dies out quite rapidly as the points are separated. If the reflected wave is coherent with the incident wave at a sensor some distance from the reflector then the reflected and incident waves must be vectorially added before squaring to find the energy. On the other hand i f they are not coherent the energies as determined from the squares of the respective amplitudes are added. A coherent reflection in phase with the incident wave and with an amplitude of 10% of that of the incident wave adds 21% to the energy, while an incoherent reflection with the same amplitude adds only 1%. Figure 45 shows for several frequencies how coherence (rigorously defined in Equation AI.6) varied with separation along (east-west) and across (north-south) the wave array on a typical day. The distance from the diagonal cross beams to the nearest sensor (IX) is 5.4 meters. For an east-west separation of 5.4 meters the figure suggests the coherence w i l l have dropped to insignificant values for waves of frequencies above 0.85 Hertz. The distance from the southern array support to the nearest sensor (IV) is 1.0 meters. For a north-south separation of 1.0 meters the coherence w i l l be insignificant for a l l frequencies above 1.0 Hertz. Table V gives the expected effect of the reflections on the spectrum. For 0.5 Hertz the reflected waves from a l l obstacles have been assumed coherent and have been vectorially added to the incident waves. At 1.0 Hertz the 123 TABLE IV RATIO OF REFLECTED AND INCIDENT AMPLITUDES FOR VARIOUS REFLECTORS Sensor Position: I IV V IX Frequency Obstacle Reflected Amplitude/incident Amplitude 0.5 Hz. North array support . 006 .003 .003 .002 South array support .003 .007 .003 .002 North pile .034 .033 .03 .040 South pile .032 .034 .03 .038 Diagonal beams .02 .03 .02 .033 1.0 Hz. North array support .04 .021 .028 .019 South array support .023 .052 .028 .019 North pile .085 .088 .085 .11 South pile .084 .087 .084 .11 Diagonal beams .067 . 066 .068 .12 1.5 Hz. North array support .12 .085 .096 .076 South array support .089 .15 .096 .076 North pile .103 .12 .11 . 16 South pile .114 .10 .11 .14 Diagonal beams .16 .17 .16 .29 3.0 Hz. North array support .143 .095 .11 .091 South array support^ .104 .18 .11 .091 North pile .11 .11 .11 .14 South pile .11 .11 .10 .14 Diagonal beams .48 .57 .62 .69 e a s t - w e s t s e p a r a t i o n , m e t e r s 0 . 0 nor th-south s e p a r a t i o n m e t e r s 1 . 2 C O H E R E N C E vs SEPARATION APRIL 11,1967 FIGURE 45 COHERENCE vs. SEPARATION FOR VARIOUS FREQUENCIES, APRIL 11 125 TABLE V EFFECT OF REFLECTIONS ON SPECTRUM Ratio of reflected power to incident power Frequency, Hz. Sensor: I IV V IX 0.5 .023 .036 .020 .029 1.0 .038 .016 .006 .039 1.5 .07 .08 .07 .14 3.0 .29 .39 .43 .53 126 reflections from the array supports have been assumed coherent. At 1.5 Hertz and 3.0 Hertz a l l reflections have been treated as incoherent. Observed spectra plotted as f S CO for the four sensors used in the reflection calculations are shown for one of the days in Figure 46. The effects of the calculated reflections on a spectrum inversely proportional to the f i f t h power of frequency are also shown for comparison. Surface Height Sensors The surface height sensors used were of the capacitance wire type, in which the actual sensor consisted of a 1mm diameter (24 AWG) Teflon insulated copper wire passing vertically through the water surface. Figure 47 is a block diagram of the system and Figure 48 shows the actual circuits involved. The variable capacitor formed with the center conductor of the wire as one plate and the salt water as the other was incorporated in the tank circuit of a Clapp type LC oscillator. Variations in the height of the water on the wire thus caused variations in the oscillator frequency around a center frequency of 265 kHz. The outputs from the oscillators, which were located on the array frame near the capacitance wires, were then passed through buffer amplifiers and up about 40 feet of coaxial cable (RG 58AU) to a set of mixers in the hut containing the recording and monitoring apparatus. In the mixers the signals were separately combined with the signal from a beat frequency oscillator identical in construction and circuitry to the sensor oscillators, except that the probe was replaced by a trimmer capacitor. This capacitor and the trimmers in each oscillator were adjusted so that the difference between the BFO frequency and the mean frequency from each probe oscillator was approx-imately 3300 Hz. This was the zero modulation frequency of the recording 127 •20 . f 5 S ( f ) - cm^ H z 4 -10 APRIL 11,1967 0.5 frequency, Hz 1.0 1.5 2.0 2.5 3.0 J — i i i i i i i i i i i i i i i i i i i i i i i • i • ! < • ! r16 CALCULATED SPECTRA c m 2 H z 4 8 incident +• scattered inci dent 0.5 frequency, Hz 1.0 1.5 2.0 J i i i i i i i i i i i -I I I L Y IX. 2.5 3.0 i i i i J i i i FIGURE 46 OBSERVED SPECTRA AND PREDICTED EFFECTS OF REFLECTIONS LOCATIONS OF SENSORS REFERRED TO WITH ROMAN NUMERALS ARE SHOWN IN FIGURE 2. SENSING..AND RECORDING SYSTEM .005 2N292 (> .y/.. 1 5 K 005 ><?—Q OUTPUT l500pF .03 4 .7K V W 9 INPUT CAPACITANCE IN jJLF RESISTANCE IN Q UNLESS OTHEWISE INDICATED FIGURE 48 SENSOR SYSTEM CIRCUITS 130 system used. The choice of this technique was based on two considerations. First, local experience suggests that capacitance wire sensors require f a i r l y high frequency components in the voltage actually imposed on the wire to avoid changes in sensitivity apparently due to deposition of some contaminant onto the wire from the water surface (see Gilchrist, 1966). Second, i t was planned originally to analyse the data using analog techniques in which case i t would be d i f f i c u l t to linearize the output signal to compensate for nonlinearities introduced by the sensing electronics. The effect of the fact that for simple oscillators frequency is not a linear function of the capacitance had to be minimized. This was done by making the per cent changes in capacitance and hence the per cent modulation of the probe oscillators quite small. In the circuit used the frequency of oscillation is given by •f o where f is frequency, \_ and C are inductance and capacitance and \C is a constant 2 x 10 sec -2 Taking f„ and C„ to be the frequency and capacitance corresponding to the mean water level, a Taylor's series expansion in C. gives where A C = C - Co and A C <^ 1 Suppose now A C = A H COS C o t where A is the sensitivity of the probe, 1.6 pf/cm, and H is the amplitude of the water surface motion. Then 131 f - f . = - A H COS tot* 8rr ^ C 0 *-f AlUl *• A l H W w O f L ! } The ratio of the second harmonic to the fundamental is then ( I _ _ 1 _ _ J ^ 3iTr*{* L C 2 - 1 C„ Inserting values appropriate to the circuits used, this becomes approximately -3.2 x IO"4 H . In comparison, for a simple Stokes' wave on deep water with a frequency of 0.5 Hz. the ratio of second harmonic to fundamental is The signal oscillator frequency was 265.00 kHz i 1.350 kHz, the BFO frequency was 268.375 kHz and the frequency of the recorded signal was 3.375 t 1.350 kHz. Signal Handling The signals were recorded on an Ampex CP-100 14 channel magnetic tape recorder at a tape speed of 1 7/8 inches per second. The frequency modulated output from the mixers was recorded directly on the tape, and was later reproduced using Ampex FM reproduce amplifiers designed for t 40% modulation. The voltage output of the FM reproduce amplifiers as determined with an Ampex TC-10 FM Calibration Unit can be f a i r l y well approximated by For this tape speed -ftt = 3.375 Hz , A = 1.092 x 10~3V/Hz 132 and at the worst / 5 = 1.35 x 10"3 v o l t " 1 . The sensitivity of the sensor oscillators was typically about 36 Hz/cm, so that for a sinusoidal variation of the surface elevation with amplitude H cm V = 3.1 3 * i p " 2 - M to* + l . o * * l c T 6 ( ] £ v ^ w 2 « t ) The ratio of second harmonic due to tape recorder nonlinearities to fundamental is thus about 5.3 x 10~^ U which is two orders of magnitude smaller than for a Stokes wave. A l l of the results to be discussed were obtained from di g i t a l analysis of the data. The di g i t a l data were obtained by sampling the analog data using an analog to di g i t a l converter designed and built at IOUBC. In the mode of operation used in this work the nine channels of analog data were sampled in sequence with a 45 microsecond delay between adjacent channels. These 'cross channel sweeps' were repeated typically 50 times a second. Thus the interval between samples was about four hundred times the time lag between adjacent channels and about forty-five times the maximum time delay between channels. The analog data were converted into a 10 bit binary word with a resolution of 10 mi l l i v o l t s . Taking into account the gains of the amplifiers incorporated between the tape recorder and the converter, this gives a surface height resolution of about 0.7 mm. Since the tape recorder output contained energy at frequencies higher than the folding frequency associated with this sampling rate, the analog signals were passed through 'low pass' f i l t e r s before digitizing. The f i l t e r s used were designed so the phase shifts introduced were linear with frequency; the effect is simply that of a time delay which leaves the waveforms in the pass band unchanged. The circuit used and it s transfer characteristic are shown 133 in Figure 49. Nine of these f i l t e r s were constructed and carefully phase matched. The output from the analog to di g i t a l converter was fed into a Control Data Corporation 8092 Teleprogrammer at the U.B.C. Computing Centre, which wrote i t on an IBM compatible di g i t a l magnetic tape, using a special format. These tapes were then used as input for the data processing. Data Processing The d i g i t a l data produced by the analog to dig i t a l converter has been processed in a number of different ways. Some of these, such as the tabulation of the observed distributions of surface elevations and estimation of the statistics associated with these distributions were done in the most obvious and straightforward way and need not be described. On the other hand an elaborate and flexible system was evolved for analyses in the frequency domain. As this system has been used greatly by others at the Institute of Oceanography and elsewhere i t perhaps merits a brief description. The frequency domain analysis system may logically be separated into two distinct parts. The f i r s t of these is the transformation of the time series to the frequency domain and the second consists of the operations performed on the frequency domain data to produce spectra, etc. These are separated physically as well as logically and comprise several computer programs. A block diagram of the system is shown in Figure 50. The di g i t a l tape from the A-D converter is written in a peculiar format so a l l input to the system is done through a set of so-called 'OCEAN' subroutines written by J.R. Wilson of this Institute, which are especially designed to handle this format. If another source of input is to be used an equivalent set of subroutines must be supplied. 1 3 4 - 1 log, 0 f 0 — 0— x — x at tenuat ion dB K 5 log Ac •o 1 0 0 K •—w - 1 0 . 3 0 6 / J F 15 -1 0 -+- J — i — i i i i i. 6 Hz 6 0 • 2 ° 3 0 -p h a s e pegrees 2 5 6 0 + 1 • 3 0 9 0 -L 3 5 o i i r I 1 1 1 r 5 \ 1(5) frequency, Hz \ ^ lag radians 1 2 . 5 ^ Hz o s L'2 FIGURE 49 CHARACTERISTICS OF DIGITIZING FILTERS 135. a n a l o g t a p e a - d c o n v e r t e r o t h e r t a p e s i v O C E A N ' su b-c a r d s > rou t i nes N T A P E V E R I F Y ' s F T O R ' f o u r i e r t r ans fo rm fl igital^ ' t a p e . m o m e n t s , distributions, etc. c o e f f i c i e n t v S C O R ' s p e c t r a c r o s s -spec t ra V B I S P ' b i s p e c t r a x C O D I S T ' c o e f f i c i e n t s t a t i s t i c s X W N S P ' w a v e n u m b e r s p e c t r a FIGURE 50 BLOCK DIAGRAM OF ANALYSIS SYSTEM 136 r e a d control c a r d s i n i t i a l set up i r e a d 1 block of data sort + I s to re = z i — f o u r i e r t r a n s f o r m c o e f f i c i e n t s r e p l a c e data p r i n t out (op t iona l ) I h a nn i nq (op t iona l ; I w r i t e c o e f f i c ient t a p e no-] l a s t b lock? T y e s L n o - l a s t f i l e ? yes _ J _ end s t o r a g e a r r a y s 1 6 7 8 9 10 1024 .words 'FTOR' BLOCK DIAGRAM FIGURE 51 BLOCK DIAGRAM OF 'FTOR' 137 The transformation to the frequency domain is done by the program called FTOR. Figure 51 shows a simplified block diagram of this program. The i n i t i a l stage of the processing consists of reading a block of data from the d i g i t a l tape into the computer memory. The data is separated in memory according to the channel of origin. If only one channel is being processed then the number of points in the data block may be equal to any power of 2, up to 2^ or 8192. If more channels are used, then 2ra samples from each channel can be accommodated, provided always that k x 2"1 is not greater than 10240 points, where k is the number of channels and is not greater than 10. This restriction was imposed by the amount of core storage available on the IBM 7044 computer on which the i n i t i a l programming was done. The data from each channel is then transformed using a 'fast' Fourier transform algorithm (see IEEE Transactions on Audio and Electroacoustics, June 1967) available from the SHARE library as PK FORT, SDA 3465. The 2(m~^ + 1 complex Fourier coefficients derived from each 2 m samples of real data replace the original data in the computer memory. These coefficients are next written on a tape in such a way that the coefficients at the same frequency from the different channels are written sequentially, much the same as in the original data for which samples; taken at the same time from the various channels were written sequentially. Because of the potential effects of the spectral window, as discussed in Chapter II and later in this section, provision is made for the hanning of the coefficients before they are written on tape. When a l l the coefficients from one block of data have been written out, another block of data is read and the process is repeated. This goes on until either a predetermined number of blocks has been done or the input data is exhausted. In the latter case the last block usually has less than the 2 m data points required and is discarded. A typical wave record of 138 30,000 samples from each of nine channels would, i f analysed in blocks of 512 samples each, yield about 58 blocks and take about 460 seconds to convert and write out using the IBM 7044. With the IBM System 360 Model 67 presently at the U.B.C. Computing Center the same job takes about one third as long in terms of central processor time. The tape containing the coefficients is used as input to various programs for computing frequency domain st a t i s t i c s . In this connection i t is important to realize that the coefficients themselves represent a multivariate time series with one sample from each block. Channel number, time and frequency may a l l be considered as independent variables. Many of the computations have already been described with varying degrees of detail in Chapters II and IV. To cl a r i f y the exact methods used the computation of the spectra and cross spectra should be further described. Let us take (Al.l) to be the complex Fourier coefficient for the In. th harmonic in the A th block. The harmonic index k. ranges from 0 to K = w / t , where NJ is the number of samples in the block, and the block index Jl ranges between 1 and the total number of blocks L . Since we believe we are dealing with purely random data we assume for the time being that L. (Chapter IV dealt with the distribution of Jl^e. a n ^ i t ! J mean and other statistics.) The energy density at a frequency ( = ~- is estimated by 139 (A1.2) where is the time interval between samples and is thus the 'duration'. This yields spectral estimates at an inconveniently large number of frequencies, so they are often grouped in bands: ' ' *=' (A1.3) where +u = The cospectrum between data from channels 1 and 2 is estimated from A c and the quadrature spectrum from (A1.4) 4= « 1 = 1 The coherence and phase are then computed from (A1.5) (A1.6) and 140 (A1.7) We would like to know how well the estimated spectra represent the true spectra of the process. The measurement and analysis procedure can introduce errors of two kinds: those associated with the particular method of calculating the spectra, usually referred to as the 'spectral window1 which have been discussed in Chapters II and IV, and those relating to the fact that only a f i n i t e sample of the process is available. The uncertainty due to having only a limited sample has in the past usually been estimated by means of arguments based on the "X distribution and the assumed stationarity of the process. One of the advantages of the system presently under discussion is that the sampling uncertainties may be estimated directly from the data. We, for example, routinely determine the variance over the blocks of the spectral estimates at each frequency If we assume, or determine, a distribution for +• &\jt_ then we may set confidence limits for S«. tC^ . In practice the (T^St^A^ = ^ is printed out, along with the quantity The latter indicates any trend present in the spectral estimates and hence is an indication of the stationarity of the process. Limits are drawn on the plotted output corresponding to the standard error of the mean of i«i-f), 141 based on an assumed normal distribution for the | fifa \ . The number of blocks is usually sufficient to give reasonable agreement between the *Y_l and normal distributions (nonzero mean). The calculation of a l l spectra and cross spectra from the Fourier coefficients for a nine channel wave record of 58 blocks of 512 samples each took about 250 seconds on the IBM 7044 and slightly less than half as long on the IBM 360-67. A simplified block diagram of SCOR, the program for computing the spectra and cross spectra is shown in Figure 52. The distribution of energy from a f i n i t e length of a sinusoid of any frequency can be calculated. Consider the complex sinusoid and its Fourier transform when sampled for the period O - Y o <2-142 r e a d cont ro l c a r d s i n i t ia l ize p o s i t ion t ape r e a d block h e a d i n g i __ » r e a d r e c o r d c o m p u t e s p e c t r a + c r o s s s p e c t r a i y e s no- l a s t b l o c k 7 y e s c o m p u t e a v e r a g e s c o r r e c t phases . ( o p t i o n a l ) p r i n t out T p lo t ( v a r i o u s opt ions) n o - l a s t f i l e ? y e s end :SCOR ' B L O C K DIAGRAM FIGURE 52 BLOCK DIAGRAM OF 'SCOR' 143 where ^ Co0-to COo-<-0 and Its spectrum over a frequency band of width ;p becomes The cospectrum is given by 7 y|_ Ccoo-^yr ] and the quadrature spectrum by If we now consider the case where the input signal is -<p with the power.spectrum Then the spectrum estimated from a short sample of \J(*") w i l l be given by . t o -<p A similar result holds for the cospectrum and quadrature spectrum. From this i t can be seen that the effect of the 'spectral window' :>«AX\ Try. i u _ w N ) l r r U 144 X 1 - c o s C X-oo^ T is to allow energy to 'leak' from one frequency band to another. The envelope of the window depends inversely on the square of the frequency difference and the square of the length of the sampling interval. To apply this to Fourier series representation of f i n i t e lengths of sampled data we need only replace *V with N where W is the number of samples and k f the time between samples, and i*> with where ' NM-K. is an integer less than ^ +- i . Any energy present at frequencies equal to where -f is not an integer w i l l thus leak to other frequencies. The possible effects of this should not be under-estimated. For example the leakage from a large low frequency d r i f t super-imposed on a fluctuating signal of very small amplitude might completely overshadow the spectrum of the fluctuating signal and produce an apparent spectrum inversely proportional to frequency squared. There are several ways of guarding against this. The method incorporated as an option in this analysis system is to smooth the Fourier coefficients before computing the spectra, as is suggested in Bingham, Godfrey and Tukey (1967). A convolution over three coefficients with weights -0.25, 0.50, -0.25 was applied immediately after taking the Fourier transform. This particular set of weights is sometimes referred to as 'hanning'. In analysing the effect of this we should f i r s t note, in our previous notation, that while w« is arbitrary, to - where K is an integer. Hence the radian frequencies of the harmonics adjacent to that with frequency <o are ur ~ = 1 0 ~ ;p and c o * = HELUn^ = t o + HL . Thus (."«,-to"^r - ( o o 0 - u , y r + 2.TT 145 and C Wo— to*" ~} f = ( o j o - i o ^ V - 2 rrr . On this basis RHANNEO , _ . . U J ~ U I 0 U ) - (Uo and Mir** \_ 4-tr-L _ t < o 0 - < o y - r* - "} Hence the spectra and cross spectra due to a sinusoid of frequency <o0 w i l l be the same as those obtained without smoothing but multiplied by the factor r V-The envelope of the spectral window is now proportional to ( ^ u > c > — f a for large UJ0-UJ as opposed to (fc>0 - for unsmoothed coefficients. The disadvantage to smoothing the coefficients is that very narrow peaks are broadened, as may be seen by considering the case of a pure sinusoid with an integral number of cycles in the block length. Before smoothing the entire energy of this signal would have appeared in a single harmonic, but after smoothing i t is spread over three harmonics. Calibration Since one of the objects of the experiments was to detect the small nonlinearities in the waves i t was necessary to understand the sensing system as well as possible. Consequently i t was f e l t that some sort of dynamic 146 c a l i b r a t i o n method should be t r i e d i n addition to the usual method of c a l i b r a t i n g the sensor by holding i t at various known immersions. One technique considered was that of o s c i l l a t i n g the probe v e r t i c a l l y through the water surface i n a way s i m i l a r to that used by Kinsman (I960). However th i s was rejected because close observation of other wire probes i n the f i e l d had led us to f e e l that the bow wave and wake due to the o r b i t a l v e l o c i t i e s of the waves might be nearly as important as the e f f e c t s of surface';tension. The i d e a l c a l i b r a t o r would have been one capable of producing purely sinusoidal waves of various amplitudes and frequencies i n a tank. In addition to the d i f f i c u l t i e s discussed by Benjamin and F e i r (1967), and the problem of devising a sensor to c a l i b r a t e the c a l i b r a t o r , such a device was too expensive. Thus we h i t upon another scheme. This was to move the probe o r b i t a l l y through a s t i l l water surface, i . e . i n such a way that the wire remained v e r t i c a l while a l l points on i t described c i r c l e s . The r e l a t i v e v e l o c i t i e s at the surface remain the same as between a purely si n u s o i d a l wave and a stationary probe. A diagram of the system i s shown i n Figure 53. The apparatus f i n a l l y constructed was capable of moving the probe with fixed amplitudes of 10, 20, and 30 cm and with frequencies of 0.5, 1.0, and 2.0 Hertz. This machine was mounted on a framework over a ten foot diameter tank behind the I n s t i t u t e of Oceanography. V i b r a t i o n of the supporting structure prevented p r a c t i c a l use of the 2.0 Hertz speed. The surface of the tank was covered with a f l o a t i n g plywood l i d with a large hole for the probe and probe support. This hole was bordered by- screen wire beaches to minimize r e f l e c t i o n s of the waves caused by the support and the probe. In addition a v e r t i c a l b a f f l e was placed between the support and the probe to 1 4 7 s e n s o r w i r e \ beach ^=^>— lid A b a r r i e r •m-\ w a t e r \ It i chaT? T ^ s n a f t s b e a c h lid support fairing path of % B' FIGURE 53 CALIBRATOR 148. 0 -2H T"4-'N CD O -6H •20 c m p-p cal i b r a t i o n P r 3 c a l i b r a t o r b a c k g r o u n d / \ ' - - ^ v A „ J \ 3 4 •' £ f r e q u e n c y , H z FIGURE 54 CALIBRATION SPECTRUM 149 prevent the waves from the support from reaching the probe. The support was faired to reduce its wavemaking. The result was quite satisfactory as the surface motion in the area containing the probe appeared to the eye to be less than a millimeter. A spectrum of the record obtained from a stationary sensor suspended in the test area while the calibrator and support were in motion is shown in Figure 57. Also shown for comparison is a spectrum obtained from a calibration run with the sensor moving for the same frequency and amplitude. The sensitivity of a single sensor as measured on this device and from static calibrations was reproducible within a few per cent over periods of several months, although i t varied somewhat among the sensors. In practice the values obtained from the dynamic calibrations were adjusted to remove systematic differences between the variances of the surface elevations as measured in the f i e l d with the various sensors. For example the variance of the surface elevation when averaged over a l l the data might appear smaller when measured with a particular sensor than the variance averaged over a l l the sensors and data. The value used for the sensitivity for that particular sensor was then adjusted to bring its variance into agreement with the average. The same value of the sensitivity was of course used for a l l the data runs. Table VI shows the sensitivities measured with the calibrator and the values after adjustment to remove the systematic differences. Some of the differences removed by adjustment were probably due to sampling errors rather than real variations in sensitivity. This was suggested by the fact that the variance among the sensors about their mean for any one run after the sensitivity adjustment was somewhat less than would be expected from sampling theory, while before the adjustment i t was somewhat greater than would be expected. 150 TABLE VI SENSITIVITY OF SENSORS TR output (mv)  Sensor sensitivity = s i n e w a v e a m p l i t u d e ( m m ) Probe Number Peak to 60cm l/2Hz peak amplitude 40cm l/2Hz 20-in calibrator cm l/2Hz 20cm 1 Hz Average sensitivity Adjusted for variances 1 5.05 4.86 5.03 4.98 4.55 2 4.47 4.47 4.33 3 4.34 4.27 4.25 4.29 4.23 4 4.68 4.75 4.75 4.75 4.73 4.83 5 4.11 4.36 4.40 4.18 4.26 4.32 6 4.39 4.36 4.33 4.36 4.49 8 4.87 4.79 4.75 4.80 5.07 17 4.56 4.43 4.23 4.40 4.35 18 4.68 4.60 4.66 4.64 4.60 Av. 4.61 4.56 4.52 4.53 4.53 151 Sensitivities obtained by the dynamic and static calibration methods agreed well, although i t is f e l t the dynamic technique is more accurate. The main objective in developing the dynamic calibration method was the evaluation of the inherent nonlinearities in the sensing system. This proved to be much more d i f f i c u l t than the measurement of the sensitivity. Suppose that the relationship between V the voltage output of a sensor and \ the surface elevation may be represented by a power series V ~- ±\ - f 3 * r I 1 + • • • -where A is the sensitivity discussed above and |3 is a coefficient which we wish to determine. If the surface elevation is a pure sinusoid such as is produced by the calibrator, with an amplitude W } then the sensor output to second order in W should be V - ff^tt2- _ ctU cos <oV + ( 3 c ^ i Z COS + X As may be seen from the sample calibration spectrum in Figure 55, there is in fact significant energy at higher harmonics than the second, but as the experiment was mainly concerned with detecting the second harmonics of the waves 3^ is the only coefficient which has been measured. If we consider the power spectrum of V , the integral under the peak at the input frequency w w i l l be —^-— , while the integral under the peak at 2u> w i l l be I- g-— . This provides one means of estimating p : where I n is the integral under the spectral peak at to . Another way of estimating |3 makes use of the bispectrum, which has been discussed 152 extensively i n Chapter V. In t h i s p a r t i c u l a r case the value of the bispectrum at t w » w ) w i l l be given by so that (3 may be found f r rom e - ~TTF~ where X, i s as above. The estimates of 0 from the bispectrum agreed with that from the i n t e g r a l to about 15%. The estimate based on the spectrum was almost always larger than the estimate based on the bispectrum, probably because the i n t e g r a l under the s p e c t r a l peak at 1<«> , which was usually small, included both system noise and leakage through the spectral window. Hence the b i s p e c t r a l estimate was usually given more weight. Table VII shows the values f i n a l l y accepted for |3 In order to assess the importance of the p a r t i c u l a r numerical values consider the case where the surface el e v a t i o n has the form, to second order, as i t would for a Stokes wave on deep water. Then the apparent surface ele v a t i o n as recorded by the sensor would be 1 *PP ~ *I ~ a - C o 4 w * v a?-u>s l o+( )fe + In Che f i e l d data the peak of the spectrum occurred at about 0 .6 Hz. or at a wavenumber of about 1.2x10"^ cm"-'-. Taking as a t y p i c a l s e n s i t i v i t y 4.5xl0"2 v/cm we can define a c r i t i c a l |3 , (3c_ , as that for which the sensor's n o n l i n e a r i t i e s are as large as those of the waves. We f i n d TABLE VII NONLINEARITY OF SENSORS /3 ( v o l t - 1 ) Calibrator: Frequency Hz. 0.5 0.5 1.0 Peak to peak amplitude, cm. 40 20 20 Probe number 1 0.015 0.070 0.116 2 0.042 3 0.032 0.060 0.063 5 0.016 0.075 0.140 6 0.025 0.050 8 0.031 0.064 0.024 17 0.028 0.064 0.027 18 0.032 0.058 0.070 Av. 0.028 0.065 0.070 154 I3- = £ o. 1.7 The average in Table VII is about one fourth of . There are some features of the calibration data which are not explained by the hypothesis of a simple quadratic response for the sensors and which lead to reservations concerning the direct application to the f i e l d data of /J as determined from the calibrator. It was found to be almost impossible to achieve a smooth regular sinusoidal output from the sensors using the calibrator. One recurrent problem was the clipping of the wave form near the point of maximum immersion. This seemed to get worse with time during any given calibration run. Scrupulous cleaning and oiling of the probe wire usually alleviated the problem to the point where the clipping was no longer detectable on the chart recordings, but i t was never entirely eliminated and would appear again unexpectedly. Although the mechanism was never understood i t was f e l t i t must be related to a buildup of contaminants on the probe near the point of maximum immersion. However covering the surface of the water with a film of motor o i l did not seem to affect the performance of the system In any case i t is thought this problem is less severe under fi e l d conditions because of the random nature of the probe immersion in the actual wave fi e l d which contrasts with the fact that the point of maximum immersion in the calibrator was repeated within a millimeter every cycle. This is supported by the chart recordings of the fi e l d data which show no evident clipping. The other major problem was the periodic occurrence of small bumps on the record, much as would be expected from an imperfection in the probe insulation. In some cases such imperfections could actually be found, but in 1 5 5 many others they could not. This too could often be practically eliminated by cleaning and oiling the wire. There is no easy way to determine whether this happened in the f i e l d as the characteristics would look much the same as very small waves. In addition to these purely practical d i f f i c u l t i e s i t should be remembered that the simulation of orbital motion of the water particles by orbitally moving the sensor is an imperfect model of a gravity wave in ways which may be important to the interactions at the point where the wire penetrates the water surface. In particular in transforming to a coordinate system at rest with repect to the water particles in the waves we have transformed into a coordinate system which is accelerated with respect to the earth. Thus the force on a small drop of water clinging to the wire is proportional to gravity in the f i e l d , but in the calibrator depends on the acceleration of the wire as well. The small capillary waves which are always present in the wind wave f i e l d also probably affect the interaction between the surface and the wire since the accelerations they impose probably help the mean water level to get over 'sticky spots' on the wire. Dr. R.W.Stewart has pointed out that this would be similar to the action of the bias frequency used in magnetic tape recording. In addition to these speculations there is some definite evidence that the behavior of the system in the calibrator and in the f i e l d must be different. One indicator of the nonlinearity of a weakly nonlinear system is the skewness of the distribution of the observed values. In particular i f we consider the case of a pure sinusoid *\ - H w\" detected with a quadratic sensor: 1 5 6 so that we find that and the skewness coefficient On the other hand i f the same sensor is used to detect a random signal "S with a normal distribution with v - o etc., we find that The f i e l d data and calibration data were broken up into a number of short sections and the standard deviation and skewness calculated for each section. Figures 55 and 56 show the joint distributions of these st a t i s t i c s . (Note that in the plot skewness decreases to the right.) In Figure 55 there are three main groupings of values. The cluster near zero skewness and large standard deviation is from the 40 cm peak-to-peak amplitude calibrations, while the cluster at smaller standard deviation but larger positive skewness is from the 20 cm peak-to-peak calibrations. The scatter of points at large negative skewness and small standard deviation comes from calibrations in which clipping occurred near maximum immersion, and w i l l be disregarded hereafter. The most striking feature of this figure is that the skewness decreases with increasing standard deviation, which would not be expected for a simple quadratic sensor, and suggests that the apparent second harmonic might be of fixed amplitude. Figure 56 is a composite of the distributions from a l l the data runs.. One immediately apparent difference between the two distributions is that the f i e l d data is unimodal while the distribution from the 10 cm amplitude calibrations has two peaks, one from clipped and the other from undipped data. This suggests that clipping was not a problem in the f i e l d . Unlike that of the calibrations, the skewness of the field data seems to increase slightly with standard deviation, and the wind waves are somewhat less skew generally than would be expected from an extrapolation of the least squares f i t to the calibration distributions. This is in spite of the fact that the calibrations correspond to pure sines and the f i e l d data to random signals so that the same degree of quadratic nonlinearity in the sensors should have produced skewnesses four times larger for the wind waves than for the calibrations, as was shown above. 158 .0 S K E W N E S S 0 -.5 -1.0 / 1 / / / 3 .6/ 5 2 7 26 7 8 ( 2 / 4 1 / / 1 3 2 • 1 2 iz 8 1 2 3 )6 / 21 22 2 1 1 1 1 2 1 / 1 12 5 5 3 / 1 1 5 1 3 2 1 / 1 2 1 / = 0. 0.4 10 SD 2.5 S T A N D A R D D E V I A T I O N V O L T S 2.0 1.5 1.0 0.5 0.0 F I G U R E 5 5 J O I N T D I S T R I B U T I O N OF SKEWNESS AND STANDARD D E V I A T I O N FROM C A L I B R A T I O N 159 1.0 S K E W N E S S 0 -.5 -1.0 SK : 0 0 8 1 + 0.1 2 7 s r 1 1 t \ I \ 3 2 1 1 I 5 1 1 1 1 1 2 1 3 9 7 9 3 2-1 2 7 7 14 7 4 1 5 2 ^ 2,7 ¥ 1 2 5 !& 6 1 1. 1 1 5 MO, 1 2 5 4 2 2 1 2 7 6 4 5 \ \ 10, 8 2 4 1 1 \ 1 1 \ "2.5 S T A N D A R D D E V I A T I O N V O L T S 2.0 1.5 1.0 0.5 .0.0 FIGURE 56 JOINT DISTRIBUTION OF SKEWNESS AND STANDARD DEVIATION FROM FIELD DATA 160 These factors lead to the feeling that the estimates of the sensor nonlinearities should be regarded as indicative rather than definitive and that they are in fact probably somewhat larger than those experienced in the f i e l d . The relationship between the average of the spectra for the calibration with 20 cm peak-to-peak amplitude and a frequency of 0.5 Hertz and the observed wave spectra is explored in Figures 57, 58 and 59, which show how various ways of normalizing the spectra affect the relative importance of particular features. Measurement Technique for Wavenumber Spectra In Chapter II we have shown how a two dimensional spatial Fourier trans-form of the complex cross spectra measured over a large grid of separations yields information about the distribution of energy among various wavenumbers (Equation 2.52). At that time we considered only the ideal case where cross spectra were available for the whole grid of possible separations. We wish now to consider the more r e a l i s t i c case where only a limited number of cross spectra are available. This is most easily done by treating the cross spectra from the sensor array with i t s limited number of separations as a sample from the complete array of complex cross spectra. This may be represented as the product of the infinite array and a sampling function: Here the sampling function C ^ \ for <Aj L3 available from the array = O for other <*,,|3 The observed wavenumber spectrum is the spatial discrete Fourier transform of the observed cross spectrum, 161 1 1 1 1 -0.5 CO' 0.5 0.8 LOGio FREQUENCY FIGURE 57 COMPARISON OF CALIBRATION SPECTRUM WITH MEAN SPECTRA OF FIELD DATA 162 1 1 1 • 1 -0 .5 0.0 0.5 0.8 L O G i o ( F R E Q U E N C Y X S C A L I N G CONSTANT) FIGURE 58 CALIBRATION SPECTRUM AND MEAN SPECTRA OF FIELD DATA NORMALIZED TO BRING PEAKS TO SAME FREQUENCY 163 -0.5 0.0 0.5 0.8 LOGio (FREQUENCY x S C A L N G CONSTANT) FIGURE 59 CALIBRATION SPECTRUM AND MEAN SPECTRA OF FIELD DATA NORMALIZED IN FREQUENCY AND ENERGY where V i (-^ ,5;^  is given by (2.52), and S lr 4 Since is real, ^ is the wavenumber spectral window associated with a particular sampling function and hence with a particular array. If we consider an array with T separation numbers *• ,(3^  j j = i j i , . . . - T then from the definition of the sampling function ^_ ; IM- J j v 0 ( 3 ^ ^ - \ 0 If we also take note of the fact that each pair of probes in an array gives a pair of reciprocal separations ( 5 > if) and l - o ^ - ^ we find further that where the separations d f t (3f j p - ±a 1 p include only the 165; positive separations greater than zero from a given array. Using this expression we can calculate the window for any given array and hence assess i t s suitability for a given purpose. One feature of TL_ which is readily apparent is that i t w i l l be periodic in wavenumber with a period in Jtx equal to - i - ^ " and in k~ equal to . This is similar to the folding of a frequency spectrum about the Nyquist frequency. The central problem in array design is to achieve satisfactory resolution with an economical number of sensors. For a good introduction to the art the reader should see Barber (196l). A reasonable amount of background information was available for designing the array for this experiment from frequency spectra collected at the site and from the directional spectrum measured by Gilchrist (1966). Using this information an array using nine sensors was fi n a l l y selected, with maximum resolution in the range 1.0 < \k\ £~ 10.0. The arrangement of the sensors and the resulting separations are shown in Figure 60. The wavenumber window.'associated with this array is shown in Figure 6 l . One way of visualizing this wavenumber window, or transfer function, is to imagine the result of observing a purely sinusoidal wave of frequency fo and wavenumber i i B with the array. The real wavenumber spectrum v*J Uoj^of this wave would be a single sharp "spike" at Ai„ . The observed wavenumber spectrum would however consist of the transfer function Tj£ centered on t6.0 . If the normalization were done properly the sum (integral) under both the real and observed wavenumber spectra would give the same value. If the wavelength of the sinusoid were f a i r l y long compared to the size of the array jk would be small and would l i e near the origin in a plot of the wavenumber spectrum. When blurred by the transfer function, the peak as viewed from the origin would subtend a large angle. Thus for waves long compared to the array the directional resolution of the array is poor. On the other hand i f the wave-length of the sinusoid were comparable to the distance between the sensors the periodic nature of would mean that there might appear to be two peaks on the plot of the wavenumber spectrum, introducing an ambiguity about (spatial 'alaising'). Under some circumstances the poor directional resolution for long waves may be partly overcome. As shown in Figure 24, page 77, the coherence between widely separated probes is high for low frequencies. This means that the relative phase between various pairs of probes may be measured with some precision. This phase may then be plotted against separation and contours of equal phase drawn. In cases where the energy in a frequency band is concentrated over a narrow range of directions the phase contours w i l l be parallel straight lines perpendicular to the direction of propagation of the waves. Since the rate of change of phase along the propagation direction may also be determined i t is thus possible to infer the wavenumber. Of course in cases where the energy is distributed over a wide range of directions the phase contours w i l l be too complicated to interpret. In addition to the wavenumber spectral window i t is necessary to remember the effects of the cross-spectral window introduced by the fi n i t e lengths of data used to calculate the cross-spectra, and by the smoothing of the Fourier coefficients i f done. If the frequency of the sinusoid considered above were between two harmonics then energy from i t would appear at several frequencies in the cross-spectra. Since this energy would always be associated with the spatial wavenumber of the original sinusoid, the wavenumber spectra for several frequency bands would show energy at that wavenumber. Thus the 167 J f o o t a r r a y N s e p a r a t i o n s • • » - 5 A x 5 5 & y , f ee t FIGURE 60 ARRANGEMENT OF SENSORS AND RESULTING SEPARATIONS 168 FIGURE 61 WAVENUMBER SPECTRAL WINDOW FOR HARMONIC SINE 169 effect of the cross spectral window is to cause the wavenumber spectral window to be broader than suggested by Tv alone. Another point worth noting is that since the wavenumbers were computed using a discrete spatial Fourier transform, values have only been obtained for a grid of spatial harmonics. The spacing of these harmonics is 1 ^ l * * * At low frequencies and small wavenumbers i t often happens that none of the spatial harmonics is particularly near the wavenumber of the energy maximum. On the computer contoured plots this gives the impression of energy at wavenumbers other than those which would be expected from the dispersion relation. An attempt to model the effective wavenumber spectral window for waves with non-harmonic wavenumbers was made. A broadband transfer function "1 8j«. which was defined as the resultant wave number spectrum of a sum of five waves with wavenumbers of [o, o \ t cT\ , (_ O f l j O ^ a n c * ( O j ~ ^ has been shown in Figure 8 on page 42. 170 APPENDIX B POTENTIAL FLOW PERTURBATION ANALYSIS This appendix gives a brief review of the potential flow theory for gravity waves using the notation used in Chapter II, which closely follows that of Hasselmann (1962). We assume we are dealing with the irrotational motion of an inviscid ideal f l u i d . We represent this motion by means of a velocity potential 4> such that v 4 = «£ where £ is the vector velocity of a flu i d particle. In the case of a horizontally unbounded incompressible fl u i d of constant depth r\ in a rectangular coordinate system with the acceleration due to gravity directed along the -1 axis the potential must satisfy the.'following relations: V 1 4 = o ( B . l ) , which expresses the conservation of mass, ll=->* (B.2), the lower kinematic boundary condition, i l (B.3), the surface kinematic boundary condition, where ^ is the 4 coordinate of the free surface and ^ denotes the horizontal gradient operator (' »i Jl ^ ' a n d l = o 15 = o (B.5), 171 which is the surface dynamic boundary condition, where K is a constant. The origin of the coordinate system is chosen so that the plane I - o coincides with the mean water level. It has been assumed that the viscous effects associated with horizontal velocities near the bottom are negligible. Also we w i l l take the pressure at the free surface to be a constant and absorb % in (B.4) into K , thus neglecting the effects of surface tension and the processes of wave generation acting across the surface. When expanded in a Taylor series about £ - o (B.3) becomes Similarly (B.4) becomes We wish to write <|> and as perturbation series <P" - , <F + * 3+ + • • - (B.7) 1 * .\ + (B. 8 ) where the prescript denotes the order of the perturbation, proportional to the wave slope. Equations (B.l) and (B.2) are linear in a l l perturbation orders (B.6), = 0 172 One solution to these is (B.9), where hj - +-"3 ^ , and the Fourier sum is used rather than the Fourier integral both for compactness and for analogy with the analysis procedure. Substituting the perturbation series into Equations (B .5) and (B .6) we find for the f i r s t order and o For the second order we have i f and = O (B.10) (B.ll) (B.12) 4 s o (B.13), We now eliminate ,^  from (B.10) and (B.ll) with the operation which yields ^ + (B.14). We next substitute for ( ^ from (B.9), after putting = \ 'U*' " (B.15) 173 where s is a sign parameter with the conditions -V- and — . Since § is to be real we have - C 1 ^ ^ ^ * where the asterisk indicates the complex conjugate. The result of the substitution into (B.14) is that M. S - O (B.16) which w i l l be satisfied when Equation (B.ll) then gives (B.17), (B.18) where u > f c and jjj are related by (B.17). Since the coordinate system has been chosen to make $ vanish ,K w i l l vanish also. If we represent (B.19) then we have ^ - z z „ - z ; e (B.20). In a similar way we eliminate from (B.12) and (B.13) by the operation which, after manipulation, yields 174 (B.21) a l l evaluated at *z = o Substitution of the f i r s t order terms using the relations given in (B.10) and (B.ll) gives, after more manipulation (B.22) where t<j>* has been used for $ If from (B.9) we take A I T ' 2 i k l a L then the left hand side of (B.22) becomes We may solve the resulting equation independently for each | i ( with the result that 3 175 A where ^ ^ I, Jz,,^ £ C s. ^ , + S I W L } V . This is almost identical with Hasselmann's (1962) Equation (4.7). Taking this result and the solutions for ,<J7 and , substituting into (B.13) and collecting terms we obtain t> - I Z I T { > L V JU;,. S, S U ^ S ( U ) , ( c o 0 T . _ ( S H O J + . S J W ^ ^ ( , s , ( o , -t -s^ j f V v X l _ a kt-ki. * ~ (B.24) Although i t may seem unlikely that such an elaborate relation could yield any familiar results, i f we take \ ( l L K - U > + ^ - ' ( ( J l K - t o f ^ t- e. 176 so that and and substitute i t into (B.24) letting the depth increase so that -t&ul^jU-W —» 4- , we find This is of course the second order contribution to the Stokes progressive wave of permanent form. 

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