NOAA Technical Report ERL 408-GLERL 14 

Spectral Growth and Nonlinear 

Characteristics of Wind Waves 

in Lake Ontario 

Paul Chi Liu 

Great Lakes Experimental Research Laboratory 

Ann Arbor, Michigan 

November 1979 

U. S. DEPARTMENT OF COMMERCE 

Philip M. Klutznik 

National Oceanic and Atmospheric Administration 

Richard A. Frank, Administrator 

Environmental Research Laboratories 

Boulder. Colorado 

Wilmot Hess, Director 
NOTICE 

Mention of a commercial company or product does not constitute an 

endorsement by NOAA Environmental Research Laboratories. Use 

for publicity or advertising purposes of information from this publication concerning proprietary products or the tests of such products is 

not authorized. 

This report is based on a dissertation submitted in partial fulfillment 

of the requirements for the degree of Doctor of Philosophy (Oceanic 

Science) at the University of Michigan, 1977. 
CONTENTS 

Page 

SymbolsUsed ........................................................ iv 

Abstract ............................................................ 1 

1.Introduction ....................................................... 1 

2 . Theoretical Considerations ............................................ 2 

3 . Observational Considerations .......................................... 3 

4 . Data Acquisition and Processing ........................................ 5 

5 . Spectral Computations ............................................... 6 

6 . Results and Discussion ............................................... 7 

6.1TheEpisodes .................................................... 7 

6.2 The Unispectra .................................................. 8 

6.3 Temporal Growth of Unispectral Components .......................... 9 

6.4 The Source Function-Empirical and Theoretical ........................ 11 

6.5 The Bispectra ................................................... 16 

6.6 Temporal Growth of Bispectral Components ........................... 16 

6.7 The Trispectra .................................................. 17 

6.8 Temporal Growth of Trispectral Components ........................... 18 

6.9 Further Remarks on Bispectra and Trispectra ........................... 19 

7 . Summary and Concluding Remarks ..................................... 23 

8.References ......................................................... 36 

Appendix A . Relationships Between Higher-Order Covariances and 

Higher-Orderspectra ....................................... 39 

Appendix B Testing . for Stationarity ...................................... 43 

Appendix C . Application of Barnett's Parameterization of Nonlinear 

SourcePunction ............................................ 45 

Appendix D . Trispectra During the Episodes on 30 September 1972 

and70ctober1972 ......................................... 47 

iii 
SYMBOLS USED 

A(k,X;w, 79 

Fourier coefficient of <(x.X;t,T) used in 

appendix A 

index counter 

pressure on the free surface resulting 

from air motion; also used as a counter 

A(t,o) for data points a deterministic modulating function, 

which is unity for a stationary process position vector 

D a parameter used in Barnett's parameterization of nonlinear energy transfer 

that represents the part receiving 

energy from other components 

nth order covariance function 

nth order spectrum; S without subscript 

n represents a second-order spectrum 

total energy total time; also representing a coupling 

function in the Boltzmann integral in 

appendix C 

space vector in Cartesian coordinates 

linear frequency = w/2n 

mean frequency 

acceleration due to gravity a space parameter used in appendix A; 

also used as a Fourier transform, 

X(w), for data r,(t) 

source function; also used in Barnett's 

parameterization to represent the part 

that transfers energy to other components 

Cartesian coordinate 

complex random function for surface r 

used in the Fourier-Stieltjes representation 

a counting symbol used in stationarity 

tests 

(-1)"' in complex representation, also 

used as an index counter 

Cartesian coordinate, directed vertically upward 

index counter free surface displacement 

wave number vector with magnitude 

k = Ikl; k also is used as an index 

counter 

wave direction 

3.14159 . . . . 

counter for data points density of water 

index counter time 

L counter for data points velocity potential 

M counter for data points complex random function for 4 used in 

the Fourier-Stieltjes representation N counter for data points 

radian wave frequency N, N(k,) action density at wave number ki 

used in the Boltzmann integral in appendix C 
SPECTRAL GROWTI3 AND NONLINEAR CHARACTERISTICS 

OF WIND WAVES IN LAKE ONTARIO 

Paul Chi Liu 

ABSTRACT. Recent studies have shown that the growth processes of wind waves are primarily 

associated with the nonlinear energy flux due to wave-wave interactions. A detailed empirical examination of these interactions uses calculated unispectra, bispectra, and trispectra of continuously recorded 

wave data during three episodes of growing waves. While the unispectmm provides information on the 

energy content of the frequency components, the bispectmm and trispectmm generally provide information on the interactive relations between two- and three-frequency components respectively. These 

higher-order interactive relations can be considered characterizations of nonlinear interactions. The 

results indicate that the peak-energy frequency transfers more energy to the lower frequency components 

than to the higher ones, which is confirmation that unispectral peaks shift progressively toward lower frequencies during wave growth. 

1. INTRODUCTION 

The study of wind-generated waves has experienced significant and extensive development during 

the last 20 years or so. Ursell's (1956) review, in 

which he found the state of our knowledge of windwave generation profoundly unsatisfactory with respect to both theory and observation, has often been 

credited with providing the major stimulation for 

modern studies. After two decades and vast theoretical and experimental efforts, a recent review by Barnett and Kenyon (1975) observed that 

At the time Ursell reviewed the field, the 

body of theoretical work exceeded that of 

the experimental work, but both were in an 

unsatisfactory state. Today the same ratio 

holds in that the theoretical ideas are still 

ahead of the experimental testing. In particular, field observations relevant to wave 

generation and dissipation in the oceans in 

1955 were nearly nonexistent. Today they 

are simply very scarce. 

Theorv often leads ex~eriment in science. In the 

study of wind waves, however, the difference is so 

large that experiment cannot interact with theory 

very efficiently. Theoretical analyses result in mathematical complexity, which prevents exact and practicable solutions; at the same time a lack of basic 

knowledge of the actual wave processes limits the 

ability to develop new models. This report presents a 

detailed empirical examination of the temporal evolution processes of the energy spectrum of windgenerated surface waves. Our measurements and 

analyses, presented here, help satisfy the need for 

basic inforkation about wave processes. 

Because several recent studies (Hasselmann et 

al., 1973; Longuet-Higgins, 1976; Fox, 1976) stress 

the importance of nonlinear energy transfer in the 

wave-growth processes, we have examined these implications empirically. Since the nonlinearity comes 

from the higher-order terms in the equations of 

motion, the first step in evaluating the nonlinearity 

of a set of wave data recorded from a single station is 

to analyze the higher-order moments of the process. 

In practice, this is equivalent to performing bispectral and trispectral analyses of the data. 

A bispectrum is the two-dimensional Fourier 

transform of the third-order covariance function of 

the data. A trispectrum is the three-dimensional 

Fourier transform of the fourth-order covariance 

function of the data. As two- and three-dimensional 

Fourier transform of the corresponding covariance 

functions are very cuhnbersome, we have found a 

way to calculate bisptra and trispectra by using a 

fast Fourier transform algorithm directly on the wave 

data. (The details will be discussed in section 5.) 

Physically, just as the unispectra provide information on the energy codtent of the frequency components, the bispectra and trispectra provide information on the interactive relations between two-frequency components and between three-frequeqcy 

components, respectively. We consider these higherorder relations to be estimates or characterizations of 

nonlinear interactions. 

Hasselmann, Murik, and McDonald (1962) and 

Garrett (1970) have diemonstrated that calculations 

of observed bispectra of ocean waves correlate reasonably well with theoretically derived bispectra. 

These studies, however, are not extensive enough to 

provide much insight into the detailed behavior of 

the physical processes. 

Using continuous wave data recorded in Lake 

Ontario, this report examines, identifies, and resolves the temporal characteristics of linear and nonlinear interactions during wave growth. Our demonstration that the mean energy of the waves is closely 
related to the unispectra, bispectra, and trispectra of 

the data provides some analytical background to our 

basically intuitive approach. 

2. THEORETICAL CONSIDERATIONS 

All theoretical studies of surface waves start 

from basic equations of fluid dynamics with varied 

idealizations or assumptions for obtaining the solutions. We assume irrotational motion of a horizontally unbounded incompressible fluid with infinite depth and a free surface at z = { (x, t), where x 

= (x, y) and z are Cartesian coordinates and the zaxis is directed vertically upward; then a velocity 

potential 4 (x, z, t) exists and the motion is governed 

by the Laplace equation 

Further neglect: ilg the surface tension, the kinetic and 

dynamic boundary conditions at the free surface 

z = { are given, respectively, by 

and 

where p represents the pressure on the free surface resulting from the air motion, e the water density, and 

g the acceleration due to gravity. Now if we assume 

that the random surface dis~lacement is statisticallv 

stationary with respect to both space and time, we 

can use the Fourier-Stieltjes representation (Phillips, 

1966) as 

Under the same assumptions and the further assumption that 4-0 as z--a, the solution to (1) can be 

represented by 

Here d@(k,w) and dY(k,o) are complex random functions of the horizontal wave number vector k and frequency w with k = Ikl. Analogously, the pressure 

on the surface resulting from air motion can be represented by 

The representations (4), (S), and (6) are quite general 

and appropriate for studying stationary random 

wave processes. Most of the studies in the literature 

consider processes that are stationary only with respect to space and thus reduce the equations of 

motion into differential equations with time derivatives only. Examples are Phillips (1960), Hasselmann 

(19621, and Benney (1962). Assuming the process is 

stationary with respect to both space and time, however, further reduces the equations of motion into 

algebraic operations. Substituting (4), (S), and (6) 

into (2) and (3); expanding the factor ekr in (5) as a 

power series; and using (4) again in the series, we 

obtain 

+ jk, jw, jk,, j,,, - k'ak" kfd@(k',w') 

x dY(k",wn ) dY(k - k' - k", w - w' - a" ) 

and 

x dY(k - k' - k", u - W' - u") 

Equations (7) and (B), retaining terms to the third 

order, can be solved for dY(k,o) and d@(k,w) if the 

representation dP(k,w) is given. From the point of 

view of the empirical study, the higher order terms in 

(7) and (8) clearly point to the need for study of 

higher order spectra. Most of the nonlinear studies in 

the literature assume the processes are undisturbed 

by air motion; hence dP(k,w) = 0 in (8). Linearized 

analyses by Phillips (1957) and Miles (1957) considered dP(k,w) to be primarily associated with the 

turbulent wind field and to consist of two parts: the 

part produced by the turbulent eddies in the wind 

and in random phase with the wave field, and the 

part directly induced by and phase-locked with the 

wave field. A combination of the two mechanisms 

(Phillips, 1966) shows that the growth of wave 
energy is initially linear with time under the turbulent 

pressure alone and subsequently becomes exponential because of the induced pressure acting on the 

growing waves. A more realistic representation for 

dP(k,o) is not available. 

The Fourier-Stieltjes transforms dY(k,w) and 

d@(k,w), solvable from (7) and (8), are closely related 

to the distribution of energy per unit among the components of the wave field. The mean energy of the 

waves per unit projected surface area is 

where the bars denote ensemble averages. It can be 

shown (e.g., Phillips, 1961) that 

hence, substituting (4) and (5) into (lo), we obtain 

The relation between d@(k,w) and dY(k,o), correct to 

the first order, is given by 

Furthermore, since we assume stationarity with respect to both time and space, we expect non-zero 

contributions to the integral in (11) to occur only 

when both k = 0 and w = 0; thus (11) becomes 

where dY*(k,w) is the complex conjugate of dY(k,w). 

From appendix A, we see that the averages 

dY(k,w)dY(kl,wl) . . . dY*(k,,w,) relate to the nth 

order spectrum. Thus, to study the complete energy 

distribution, it is necessary to examine the higher 

order spectra. 

3. OBSERVATIONAL CONSIDERATIONS 

The results presented in section 2 and appendix 

A are derived for the characterization of the whole 

lake surface. To correlate the results with actual 

wave observations, usually made from a wave gage 

installed at a single selected location, the equations 

are integrated and normalized over all directions and 

wave numbers; this explicitly depends on time and 

wave frequency only. We can rewrite equations A.8 

and A.9 as 

and 

Equation (13) now becomes 

where S2(w), S3(w,wl), and S4(w,w1,wZ) are the unispectrum, bispectrum, and trispectrum, respectively. 

For a completely stationary process with respect 

to both time and space, E is constant in (16). However, our intent is to examine the time rate of change 

of the mean energy during a growing wave process 

that is in general nonstationary. Because in the nonstationary case E changes with time, its application 

seems to be somewhat contradictory. To pursue our 

interest within the framework of (16), we use a basic 
assumption of local stationarity in which the process 

is assumed to be stationary and E constant only 

within a given segment of data, a segment long 

enough to provide sufficient degrees of freedom to 

lead to reasonable spectral estimates. Although E and 

hence the spectra are constant within the given segment, they invariably change from segment to segment. Therefore if we calculate S2, S3, and S, over 

consecutive overlapping segments during wave 

growth, we can indirectly examine the temporal 

growth behavior of the energy spectra. 

The above scheme is demonstrated graphically 

by figure 1. The discrete time series ~(t,), t,= nAt, n 

= 1, 2, . . . , N represents an episode of growing 

wave data NAt in total time length with a sampling 

rate of l/At. We subdivide ~(t,) into M overlapping 

segments {,(tp), tp = PAt, P = 1, 2, . . ., L, rn = 1, 2, 

. . ., M. Each {,(tp) is LAt in time length. The data P 

= K to P = L of {,(t,) overlap the data P = 1 to P = L 

-K + 1of {m+l(tP).Thu~wehave N = (M-1)K + 

L. While the process ~(t,) is not stationary, we assume each {,,(tp) is locally stationary and hence all 

considerations we have presented pertaining to {(t) 

can be applied to {,,(!At), P = 1, 2, . . ., L. 

Since we represented {(t) as a continuous 

process, 

To use the discrete time series, we write 

rat(! At) = j2,/AteiwPAtdYAt(w), (18) 

where 

can be considered as complex amplitudes at frequencies w + 2rP/At, in view of the fact that eidAt is 

periodic at frequency w + 2xlAt. The relationship 

between the continuous nth order spectrum S,(wl, 

. . ., LO,) and the discrete nth order spectrum Sn,At(~l, 

. . . , a,) has been given by Brillinger and Rosenblatt 

(1967) as 

with 

w,=w,+. ..+w,-1, 

and 

Thus using the Nyquist frequency of */At radians 

per second or 1/(2At) Hz in sampling to avoid aliasing problems, we can effectively represent the continuous process with a discrete time series in our subsequent analysis. 

Some further remarks concerning our stationarity assumption can be stated here: 

Figure 1.-Graphical demonstration of data segments. 

4 
(1) The use of the local stationarity assumption is 

not new; Kolmogoroff (1941) used local homogeneity 

in his turbulence studies. Most wave growth studies 

in the literature use this assumption implicitly. To 

evaluate the approximate validity of this assumption, 

appendix B gives some nonparametric stationarity 

tests on the data used in this study. The results show 

that the assumption is generally within the acceptable 

range of the tests and can be considered valid for our 

purpose. 

(2) We are not attempting explicitly to calculate E 

in (16). Rather we use (16) to show the significance of 

the spectral terms in connection with mean energy 

and thus concentrate our interests on the temporal 

growths of the frequency components of S2(u), 

S3(w,w1), and S4(w,ul,u2) and their physical implications. 

(3) A different approach to analyzing a non-stationary process, perhaps more rigorous statistically, 

was developed by Priestly (1965). He introduced a 

representation of 

where A(t,w) is a deterministic modulating function 

that approaches unity when the process approaches 

stationarity. This representation could have been 

used in our analyses. However, since A(t,u) is not a 

known function and its application inevitably requires further assumptions and complications, we 

did not use it. It is of interest to note that Priestly's 

model leads to the conclusion that "the evolutionary 

spectrum at each instant of time may be estimated 

from a single realization of a process." This is exactly 

what the local stationarity assumption implies. 

(4) We define local stationarity here in its literal 

sense. Silverman (1957) introduced the concept of a 

locally stationary random process with a locally stationary covariance that can be written as the product 

of a stationary covariance and a nonnegative function. Since we allow our local covariance to vary 

from segment to segment and the variation is generally smooth and gradual, it seems analogously possible that a nonnegative function could depend on 

time segments and that a stationary covariance can 

be deduced from our consecutive local covariances. 

As we expect the property to apply to third- and 

fourth-order covariances also, our assumption carries a different sense than Silverman's rigorous 

process. 

4. DATA ACQUISITION AND 

PROCESSING 

The data used in this study were recorded in 

Lake Ontario from 1 April 1972 to 31 March 1973, 

the International Field Year for the Great Lakes 

(IFYGL). Seven Waverider Buoys and a large number 

of Physical Data Collection System (PDCS) buoys 

LAKE ONTARIO 

Figure 2.-Location of wind and wave gages in Lake Ontario. 

were deployed for the IFYGL programs. The wave 

data in this study were recorded from the two Waveriders designated as 0s-1 and 0s-2 (figure 2); the 

corresponding wind data used here were recorded 

from PDCS buoy 11. 

The Waveriders were deployed in 150 m of 

water and freely moored to a chain sinker with a 

mooring line approximately twice the depth of the 

water. The Waverider Buoy, manufactured by Datawell, Holland, is of spherical shape 1 m in diameter 

and weighs about 100 kg. It contains two main components: an accelerometer and a transmitter. The accelerometer, mounted on a pendulous system, measures the vertical component of acceleration as the 

buoy moves with the waves. Two electronic integrators in cascade then transform the output into a 

voltage that represents the vertical displacement of 

the buoy. This voltage controls the frequency of an 

audio oscillator, which in turn modulates a crystalcontrolled transmitter that transmits the signal by 

telemetry to a shore receiver. The telemetered wave 

data were recorded continuously on analog magnetic 

tapes. The analog wave data tapes were subsequently 

processed through a computer digitization and editing system (Liu and Robbins, 1974) to obtain final 

data tapes, which were digitized at a sampling rate of 

approximately three per second. This sampling rate is 

more than sufficient to avoid an aliasing problem, 

since the buoy response is such as to adequately 

damp waves having a frequency of > 1.0 Hz. On the 

other hand the Waverider's frequency range is given 

as between 0.065 Hz and 0.50 Hz; hence, in the actual computations we use a sampling period of At = 

2/3 s to yield a Nyquist frequency of 0.75 Hz for the 

computed spectra. Referring to the scheme discussed 

in the last section, we use the following numbers in 

the analysis: 

Consequently each selected episode is 177.5 minutes 

long. The episodes are subdivided into 64 equal segments of 20 minutes each, with an overlap of 17.5 

minutes from one segment to the next. 
5. SPECTRAL COMPUTATIONS 

In the previous discussion we have been using 

general nth order spectra. For the actual applications, 

however, we shall concentrate only on n = 2, 3, and 

4 for the unispectrum, bispectrum, and trispectrum, 

respectively, defined as 

(22) The corresponding transforms of the symmetry relations are 

dY(w)dY(wz)dY*(wl + ~2) SAW) = S2(-4, (31) 

S3(~1r~2) = dwldw2 

S3(w1,w2) = S3(~2,~1) = S3(w2,-w1 -w2) 

= j W, j W,R3(~1r 72)e-i(W1T1+ w272'd~1d72, (23) = S~(-WI - 02,~~) 

= S3(~l,-wl -~2) = S3(-0i - WZ,OI), (32) 

and 

and 

x d~ld72d73, (24) = S4(-~1- 02 - ~3r~3r~2) 

= S4(w2,-w1- a2 - ~3~~3) 

where = S4(w2,w3,-w1- W2 - '03) 

= S4(~3,-~1- W2 - ~3~~2) 

R2(7) = {(t){(t + 7), (25) = S4(w3,wZ,-w1 - 02 - ~3) 

= S4(-~1 - 02 - ~3r~lr~3) 

~3(71,72) = {(t)t(t + 71){(t + 72)t (26) =s~(-wI- wz - ~3r~3r~l) 

= S4(~1,-~1 -WZ - ~3rw.3) 

and = S4(~1,"3,-~1- w2 - w3) 

= S4(~3,-~1 - a2 - ~3r~1) 

R4(~1r~2,~3) = c(t){(t + ~l)C(t + 72)b(f + 73).(27) = S4(-~1 ~4(~3,~1,-~1 - ~2 - - ~3r~lr~2) 02 - 03) 

The above definitions lead to the following symmetry relations: 

and 

Because of these symmetries, we need only to estimate the S,'s, n = 2, 3, and 4, within a fundamental 

-) (29) region. The fundamental region for unispectrum 

S2(w) is the line segment 0 I w I ON; for bispectrum 

S3(~lrc&) is the triangle defined by 0 I w2 I wt, and 

0 I wl I WN; and for trispectrum S4(wlr w2, w3) is the 

tetrahedron defined by 0 I w3 I w2, 0 I w2 5 wl, 

and 0 5 wl I WN, with WN = 2n/(2At) representing 

the Nyquist frequency. 

Comparing the definitions (22), (23), and (24) 

with (A.19) in appendix A, we see that for a finite 

segment of a time series the coefficients of its Fourier 

transform X(w) can be used as approximations for the 

theoretical values dY(w) and hence for the appropriate products of Fourier coefficients for estimating 
corresponding spectral densities. We used the following approach generally similar to those discussed by 

Haubrich (1965) and Hinch and Clay (1968). 

Starting with f(PAt), P = I, 2, . . ., L, we subdivide the series into P non-overlapping groups each 

of length K, such that for p = 0,1, . . ., P - 1 and k = 1,2, . . ., Kwe have 

and the K complex Fourier coefficients for each group 

are given by 

We can then sum and obtain the average estimates by 

and 

Equations (34)-(37) represent the basic approach in 

spectral computations used in this report. The approach is quite efficient and feasible, especially with 

the fast Fourier transform algorithm (Cooley and 

Tukey, 1965) available as a computing subroutine. In 

the actual computations, we use P = 30 for each 

data segment of L = 1800 to obtain smoothed spectral estimates with 60 degrees of freedom and a 95 

percent confidence interval between 1.48 and 0.72. 

6. RESULTS AND DISCUSSION 

6.1 The Episodes 

From the ample supply of wave data recorded in 

Lake Ontario during IFYGL, we selected three episodes for our study: 

9 August 1972 EST 1215-1515, 

30 September 1972 EST 0540-0840, 

' 7 October 1972 EST 0000-0300. 

The August and October episodes were recorded 

from Waverider 0s-2 shown in figure 2; the September episode was recorded from Waverider 0s-1. The 

wind conditions during these periods, recorded at 10- 

minute intervals at 4 m above the lake surface on 

PDCS buoy 11, are shown in figures 3, 4, and 5. The 

group of short and straight lines plotted on the figures under wind direction and wind speed indicate 

the locations in time of the 64 overlapping segments 

of wave data analyzed for each episode. Each segment is 20 minutes long and has a 17.5-minute overlap with the next segment. Although each of the episodes represents a growing, nonstationary wave 

field, the 20-minute segments are assumed to be 

locally stationary, and thus we can calculate unispectrum, bispectrum, and trispectrum for each segment 

using equations (35), (36), and (37), respectively. 

The three episodes are representative of growing 

wave conditions in the Great Lakes. Because we are 

interested in the early stage of wave growth, none of 

the episodes is under severe storm conditions. The 9 

August episode started at a wind speed of 8 m s-' and 

increased to more than 11 m s-I in 3 hours. The 

9 August 1972 

Figure 3.-Wind conditions for the episode of 9 Ailgust 1972. The 

series of short lines in the lower part indicates the locations in 

time of the 64 overlapping segments of wave data analyzed. 
30 September 1972 7 October 1972 

Figure 4.-Wind conditions for the episode of 30 September 1972. Figure 5.-Wind conditions for the episode of 7 October 1972. The 

The series of short lines in the lower part indicates the locations series of short lines in the lower part indicates the locations in 

in time of the 64 overlapping segments of wave data analyzed. time of the 64 overlapping segments of wave data analyzed. 

westerly wind direction provided long and approximately constant fetches during the episode. The 30 

September episode included the passage of a steep 

low-pressure center directly over eastern Lake Ontario, and wind speeds increased from 4 m s-' to 

more than 14 m s-' within an hour, while wind directions were changing rapidly from south through west 

to north during the early part of the episode. The 7 

October episode was milder with wind speed increased from 2.5 m s -' to 8.5 m s -' under southwesterly direction. The total wave energy under 

these three wind fields was growing steadily in each 

of the three cases. 

6.2 ,,the Unispectra 

We first plot the computed unispectral density 

versus frequency versus time for three episodes as 

shown in figures 6, 7, and 8. These three-dimensional 

perspective figures present a clear overview of the 

spectral growth of the episodes. Several basic characteristics can be observed from these figures: 

(1) The growth activity is mainly concentrated 

over the low-frequency side; the high-frequency side 

does not change much during the episode. 

(2) During the growth period, the peaks of the 

spectra tend to shift toward lower frequencies. 

(3) The growth rate varies for different frequency 

components. 

(4) Once a frequency component grows to be the 

spectral peak, it reaches a relative equilibrium and its 

growth rate tends to diminish. 

(5) A comparison of these local spectral growth 

episodes with their respective wind conditions shown 

in figures 3, 4, and 5 indicates that dominant growth 

of the spectra happens during increasing wind 

speeds. 

The above features are generally known; similar 

results were obtained in laboratory studies, e., 

Jacobson and Colonel1 (1972) and more recently by 

Wu et al. (1979). 
Figure 6.-Perspective view of the unispectrum during the episode Figure 7.-Perspective view of the unispectrum during the episode 

of 9 August 1972. of 30 September 1972. 

With respect to the well-known shifting of specA 

tral peaks during wave growth, Hasselmann et al. 

(1973) suggested that nonlinear energy transfer plays 

a dominant role in the process. They concluded that 

the spectral form results from the self-stabilizing 

property of the nonlinear interactions that continually readjust the energy distribution within the 

spectrum. We shall further explore these implications. 

6.3 Temporal Growth of Unispectral 

Components 

To examine the time-dependent behavior of the 

frequency components individually, we first smooth 

them in time by hanning and then plot them semilogarithmically. The components have different time 

dependences, but they can be grouped into three frequency ranges. The results are shown in figure 9 for 

the 9 August episode, in figure 10 for the 30 September episode, and in figure 11 for the 7 October 

episode. 

(1) Figures 9(a), 10(a), and ll(a) show the low-frequency group that contains components with frequencies less than all the peak-energy frequencies 

during the episode. The components in this group 

can be characterized by their sensitivity to the wind 

field. Their growth seems to follow directly with the 

increases in wind speed. The approximate linearity 

during the growth as shown in the logarithmic plot 

indicates the exponential growth that Miles' theory 

predicts. 

(2) Figures 9(c), 10(c), and ll(c) show the high-frequency group that contains components with frequencies beyond the peak-energy frequencies during 

the episode. The components in this group represent 

the portion of the spectrum that is usually considered 

Figure 8.-Perspective view of the unispectrum during the episode 

of 7 October 1972. 

to be the equilibrium range. They tend to parallel 

each other in time and are insensitive to either increasing wind speed or time duration. Some anomalous behavior shown in figure 10(c) during the early 

stage may be due to the fact that wind direction during this short time interval was changing continuously and thus further complicated the whole 

picture. 

(3) Figures 9(b), 10(b), and ll(b) show the components, lying between groups 1 and 2, that have been 

peak frequency during the episode. The behavior of 

components in this group is complicated since they 

have mixed properties of both of the first two groups. 

This group contains a large part of the total spectral 

energy. 
Minutes from beginning of episode 

Figure 9a.-Growth of low-frequency unispectral components during the episode of 9 August 1972. 

4 

A 

N 

E. 

\ 

N 

A 

E 

0 

V 

n 

-2- - 3 

c 

8 

J 

1 

0 

Figure 9b. -Growth of middle-frequency unispectral components during the episode of 9 August 1972. 

7 

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0 30 60 90 120 150 

Minutes from beginning of episode 
6666666666666L666 6a666666666666666666 

...................... 00000000060 

XXXXXXX XXX 

xxxxxxxxxxxx+~~~OO **++*** YXXX 

XXXXXXXXXXXX++tt++-+,+++++ ***********tttt$*******.t.tt 

Minutes from beginning of episode 

Figure 9c.-Growth of high-freqqency unispectral components during the episode of 9 August 1972. 

6.4 The Source FunctionEmpirical and Theoretical 

One of the main problems in developing wave 

growth models is determining the source function G 

in the energy balance equation (Gelci et al., 1956; 

Hasselmann, 1960): 

whek E is the energy density, the second and third 

terms on the left side represent the advection and refraction of the processes, respectively; the dots denote the time derivatives. The source function G is 

the total rate of change of the energy representing all 

processes of generation, dissipation, and linear and 

nonlinear interactions. Most wave studies in recent 

years have aimed at assessing some aspects of the 

source function. Perhaps the most significant one is 

the Joint North Sea Wave Project (JONSWAP) in 

which Hasselmann et al. (1973) found that the nonlinear energy transfer, computed from integrating the 

theoretical Boltzmann integral (appendix C), is important for explaining the form of the wave spectrum 

during wave growth. 

With the available tem~oral functions of the frequency components, we estimated aE/at empirically 

for each spectrum S2(w) by fitting a smooth cubic 

spline curve through the components and calculating 

their time derivatives. For waves at a single location, 

if both the advection and refraction terms can be 

ignored in equation (38), the empirical aS2(w)/at also 

represents the empirical source function. (Given 

duration t, fetch L, and group velocity v, advection 

can be neglected if t 4 L/v. This condition is generally satisfied in this study since duration is small for 

increasing wind speed and fetch is large during the 

episodes.) 

Hasselmann et al. (1973) also calculated empirical source functions by parameterization of (38). 

Their method includes the advection term. However, 

they give only one average source function for each 

episode. Our method, on the other hand, provides 

consecutive source functions throughout the episode 

and thus more detailed information. 

With these empirical source functions available, 

the next task is to correlate them with the theoretical 

nonlinear source functions. The numerical integration of the Boltzmann integral by Sell and Hasselmann (1972) requires 30 minutes of computer time 

for each spectrum and, at this stage, is rather impractical. In appendix C we show that Barnett's (1966) 

parameterization of Hasselmann's earlier calculations 

(1963a,b) leads to essentially the same conclusions as 

those obtained by Sell and Hasselmann. Therefore 

we choose to use the low-frequency part of the results 

calculated from Barnett's parameterized scheme to 

represent the theoretical nonlinear source function in . 

.... 

-.- 
1111111111111111111111111111111111111111 111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 10a.-Growth of low-frequency unispectral components during the episode of 30 September 1972. 

Figure 10b.-Growth of middle-frequency unispectral components during the episode of 30 September 

1972. 
1111111111111111111111111111111111111111111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 

- 

"'3- z. 

\ 

PI 

A 

E 

0 

Y 

n 

-2- 

Y 

$ 

, 

0 

Figure 10c.-Growth of high-frequency unispectral components during the episode of 30 September 1972. 

- 

............. ,..~"~~~~~~~~~~~8~~~mm~~..gm~~~~~~~O

0~000000 000 0002200000 08000660 

~OOOOOOOOOPtOOOOOO xxx!?!?BO 

g!xxx .::, xx. XXX ..xxx. ::I!I::xx::t+++++*:!fffii XXXXX+ *+ + ******* 

....** .."H ..I! itiij::iii~~**tt~~~iZ~i~d~~~

....... & .A!+ 0 

a&u&bs" 

6k8:8 x 

f tt0'9 X - X X X X 

- - 

-1 111111111111111111 1111111111111111111111111111111111111IIIIIIIIL 

0 30 60 90 120 150 

J 

Minutes from beginning of episode 

Figure 1la.-Growth of low-frequency unispectral components during the episode of 7 October 1972. 
-1 1111111111111111111111111111111111111JllllllllllllJllllIIIIIIIII 

0 30 60 90 120 150 

Minutes from beginning of episode 

cn 

0 

4 

1 - - 

0 - - 

Figure 1lc.-Growth of high-frequency unispectral components during the episode of 7 October 1972. 

-1 1111111111111111111111111111111111111111111111111111111111111111- 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 1lb.-Growth of middle-frequency unispectral components during the episode of 7 October 1972. 
0 01 02 03 04 0 5 0 6 07 -0 5 

Frequency w12n Hz kM-05 Frequency ol2n llz Frequency o12n Hz Frequency w12n Hz 

Figure 12.-Unispectra (-1 and empirical (-) and Barnett's (---4 Figure 13.-Unispectra (-) and empirical (-) and Barnett's (----3 

source functions during the episode of 9 August 1972. The num- source functions during the episode of 30 September 1972. The 

bers 2.5, 32.5, . . ., and 152.5 are minutes from beginning of numbers 2.5, 32.5, . . ., and 152.5 are minutes from beginning 

episode. of episode. 

our subsequent comparisons. (More recently, Wu et 

al. (1979) also used Barnett's parametric equation in 

calculating nonlinear energy transfer and found good 

comparisons in the low and intermediate frequency 

region of a wave spectrum.) We ignore the high frequency part of the parameterized results because it 

was based on Neumann's (1953) assumption that 

energy density is proportional to wb6 at the high frequency side rather than to w-=, which most recent 

studies have confirmed as correct. The results for the 

three episodes are shown in figures 12, 13, and 14. 

Because of gradual changes in the process, we present 

only six spectra and source functions for each episode 

at 30 minutes apart to show the essential features. In 

the figures we plot the logarithm of unispectra log 

S2(w), and the empirical and calculated a[log 

S,(w)]/at all with respect to the frequency of w/27r. 

The distinction between the empirical and calculated 

"theoretical" source functions is quite evident. The 

theoretical source function consistently has large 

positive lobes at frequencies below the peak-energy 

frequency. The empirical source function develops 

similar positive lobes only toward latter stages of the 

episode when the growth activity is intense. At the 

beginning of the episode when the growth is generally slow or when the waves are well developed and 

further growth is slowing down, the empirical source 

function tends to be much less pronounced in its 

lobes, positive and negative. 

This behavior of the empirical and theoretical 

source functions is not particularly surprising since 

the theoretical results are dependent on the shape of 

the spectrum whereas the empirical results are dependent on temporal changes in the spectrum. This 

also provides a possible explanation for the dominance of nonlinear interactions during growth. If we 

assume that the empirical results represent the complete source function and the theoretical results only 

the nonlinear interactions, their differences then represent the other processes, such as wave breaking and 
Frequency o12n Hz Frequency o12n Hz 

Figure 14.-Unispectra t-) and empirical (-) and Barnett's (-----9 

source functions during the episode of 7 October 1972. The 

numbers 2.5, 32.5, . . ., and 152.5 are minutes from beginning 

of episode. 

dissipation. At the beginning of the episode, when 

growth is generally slow or when the waves are well 

developed and growth is also slowing down, the nonlinear interactions must be balanced by some equally 

important dissipation process in order to yield the 

smaller total source function. During rapid growth, 

on the other hand, both the nonlinear interactions 

and the total source functions have large positive 

peaks, which implies that dissipation is less significant and that most of the growth is through nonlinear interaction. 

6.5 The Bispectra 

There are three published bispectral studies on 

wind-generated waves: Hasselmann, Munk, and 

McDonald (1962), and Garrett (1970) compared four 

bispectra computed from actual wave data with 

those theoretically derived and found satisfactory 

agreement, and Houmb (1974) presented three 

examples of computed wave bispectra. In the present 

study we are interested in the higher order interactions during wave growth. As a first step we have 

computed 64 consecutive bispectra for each of the 

three growth episodes. Because the temporal variations are gradual, we present only six bispectra from 

each episode at 30 minutes apart as representative of 

the whole process. Figures 15, 16, and 17 show the 

sample bispectra for episodes 9 August, 30 September, and 7 October. 

Since the bispectrum S3(~tr~Z), computed from 

equation (36), is complex, it is convenient to express 

it as a bispectral amplitude rather than as co-bispectra and quadrature-bispectra separately. Figures 

15, 16, and 17 show the contours of the logarithms of 

bispectral amplitude plotted in the fundamental region of 0 < w2 < W, and 0 < w, I WN. 

Several main features can be seen in these 

figures: 

(1) The bispectrum has a hill at (wp,w,), where up is 

the frequency of the spectral peak in the corresponding unispectrum. 

(2) Each bispectrum also shows two ridges approximately parallel to the two frequency axes and 

sloping down from the bispectral hill toward both 

higher and lower frequencies. 

(3) The magnitude of the bispectral amplitude increases and the hills and ridges migrate toward lower 

frequencies during wave growth. 

(4) These obsenred characteristics are qualitatively 

similar to those depicted by Hasselmann, Munk, and 

McDonald's (1962) deep-water wave co-bispectrum. 

It is not readily discernible that these features result from a nonlinear Drocess. However. from the 

qualitative resemblanc; to Hasselmann, Munk, and 

McDonald's results and the fact that their results 

agree with those derived from a theoretical nonlinear 

process, we may conjecture that these features are 

characteristic of a weakly nonlinear wave growth 

process and hence proceed to examine their temporal 

behavior. 

6.6 Temporal Growth of 

Bispectral Components 

Since the' predominant bispectral interactions 

occur when the frequency of the spectral peak in the 

corrpsponding unispectrum interacts with itself and 

these interactions migrate toward lower frequencies 

during wave growth, it is sufficient that we examine 

the time-dependent behavior of those bispectral components oi the 45" line whose frequencies interact 

with themselves. We proceed in a manner similar to 

that for unispectral components by temporally 

smoothing them with hanning and then plotting them 

semi-logarithmically as a function of time. The results are quite similar to those unispectral components. weagain separate them into three groups for 

the three episodes (figtlres 18, 19, and 20). 
Frequency w 12a (Hz) Frequency w 127 (Hz) 

Fi~~rr 15. -Hisp~ctra during the cpisudc uf August 1972. The nurnberq 2.5, 32.5, . . ., and 152.5 arc 

minutes from beginn~ng ot episode. 

Several points should be noted here: 

(I I The figures are plotted with log[S,(w,w)] versus 

linear time as before but with a smaller range. Consequently the exponential growth rates shown in figures 18. 29, and 20 that appear comparable with 

those shown in figures ?, 10, and 11 are acttraIlv 

about four times smaller. 

(2) We have seen that in unispectral growth the 

spectral peak moves to a lower frequency during 

wave growth. The bispectsa provide some indication 

of phenomena that can be attributed to this transfer, 

namely, that the interactinns of ~RP peak frequency 

with the next lower Crequency grow consistently 

stronger relative to the interactions with the next 

higher frequency during the latter part of the ~rowth. 

Examples are shown in figures 21, 22, and 23. As 

growth continues, the component of the next lower 

frequency eventually bccornes the spectra1 peak. 

(3) It the bispectral amplitudes represent the ~cand-order nonlinear intcrackions in the wave growth 

Drocess. as we have coniectur~d. then otIr results 

show that the nonlinear interactions should also have 

functional relationships with respect to time. This 1s 

not unexpected. Since the theoretical nonlinear 

source term is a function of the unispectrum, it must 

vary with time if the ~~nispectrum does. 

6.7 The Trispectra 

RsilFinger and Rosenblat t (1967) presented one 

set of calculated trispectra for daily sunspot numbers. Their work rcprcsents the only trispectral calculations available in the literature. They did not, however, provide an interpretation of their results. Trispectral analysis has remained rdatively unexplored. 

By extending the calculation scheme Haubrich 

(19651 used in bispectral calculations one step further 

and using equation (371, we have been able to calculate a trispcctrum for each of the 64 data segments for 

each episode. It is not immediately clear how to 

visualize and pres~nt the trisp~ctra effectlveFy. It is 

even more cumbersome to try to present them in 

time. Since the fundamental region for the trispectrum is the tetrahedron defined by 0 I wl 5 WZ, 0 5 

CIIL 5 w~, and 0 i w1 5 w,~, we have the trispectral 

density S,(wl,wl,w,) in addition to the three trequency axe5 to form a four-dimensional manifold imbedded in three-dimensional Euclidean geometry. By 
Frequency w 12a (Hz) Frequency w 127 (Hz) 

Fi~~rr 15. -Hisp~ctra during the cpisudc uf August 1972. The nurnberq 2.5, 32.5, . . ., and 152.5 arc 

minutes from beginn~ng ot episode. 

Several points should be noted here: 

(I I The figures are plotted with log[S,(w,w)] versus 

linear time as before but with a smaller range. Consequently the exponential growth rates shown in figures 18. 29, and 20 that appear comparable with 

those shown in figures ?, 10, and 11 are acttraIlv 

about four times smaller. 

(2) We have seen that in unispectral growth the 

spectral peak moves to a lower frequency during 

wave growth. The bispectsa provide some indication 

of phenomena that can be attributed to this transfer, 

namely, that the interactinns of ~RP peak frequency 

with the next lower Crequency grow consistently 

stronger relative to the interactions with the next 

higher frequency during the latter part of the ~rowth. 

Examples are shown in figures 21, 22, and 23. As 

growth continues, the component of the next lower 

frequency eventually bccornes the spectra1 peak. 

(3) It the bispectral amplitudes represent the ~cand-order nonlinear intcrackions in the wave growth 

Drocess. as we have coniectur~d. then otIr results 

show that the nonlinear interactions should also have 

functional relationships with respect to time. This 1s 

not unexpected. Since the theoretical nonlinear 

source term is a function of the unispectrum, it must 

vary with time if the ~~nispectrum does. 

6.7 The Trispectra 

RsilFinger and Rosenblat t (1967) presented one 

set of calculated trispectra for daily sunspot numbers. Their work rcprcsents the only trispectral calculations available in the literature. They did not, however, provide an interpretation of their results. Trispectral analysis has remained rdatively unexplored. 

By extending the calculation scheme Haubrich 

(19651 used in bispectral calculations one step further 

and using equation (371, we have been able to calculate a trispcctrum for each of the 64 data segments for 

each episode. It is not immediately clear how to 

visualize and pres~nt the trisp~ctra effectlveFy. It is 

even more cumbersome to try to present them in 

time. Since the fundamental region for the trispectrum is the tetrahedron defined by 0 I wl 5 WZ, 0 5 

CIIL 5 w~, and 0 i w1 5 w,~, we have the trispectral 

density S,(wl,wl,w,) in addition to the three trequency axe5 to form a four-dimensional manifold imbedded in three-dimensional Euclidean geometry. By 
Frequency o J 27 (Hz) Frequency w J 2% (Hz) 

Figure 17.-Bispectra durin~ the episode of 7 October 1972. The numbers 2.5, 32.5, . . ., and 752.5 are 

minute5 from beginning ul t.pi.sudc. 

time. The results, shown in figures 25, 26, and 27, are 

again quite similar to the unispectral and bispectral 

components. Since we have only the lower frequency 

components, grouping is not necessary. The results 

show again that the third-order interactions have 

functional relationships with respect to time during 

the wave growth process. 

In figures 28, 29, and 30, we concentrate on the 

peak-energy trequency achieved at the laster stage of 

the episode by plotting the temporal growth of the 

frequency interacting with itself to the third order, 

the frequency interacting with itself and the next 

lower frequency, and the frequency interacting with 

the second-order interaction of the next higher frequency. These figures show quite clearly that the 

shifting of the peak-energy frequency toward the 

lower frequency is due to the stronger interactions of 

the peak-energy frequency with the lower frequency 

and that the interaction provides energy transfer 

from the peak-energy frequency to the lower frequency. This confirms that the higher-order nonlinear interactions are responsible for the shifting of 

spectral peaks during wave growth. 

6.9 Further Remarks on 

Bispectra and Trispectra 

What have we gained from bispectral and trispectral analysis? First, we have developed methods 

for computing and presenting bispectra and trispectra from data. This experience will be useful in 

the luture. Second, and more important, we have 

presented evidence that during the later intensive 

part of wave growth, the interactions of the peakenergy frequency component with the next lower frequency component grow consistently stronger compared with the interactions with the next higher frequency component, and the next lower frequency 

subseqtrently becomes the peak-energy frequency. 

This result, which provides a clear expIanation for 

the well-known fact of shifting spectral peaks toward 

lower frequencies, can be demonstrated onIy by bispectral and trispectral analysis. 

Benjamin and Feir (1967) showed theoretically 

and experimentally that for deep-water waves with a 

fundamental frequency w, because of coupling 

through the nonlinear boundary conditions at the 
8' 

2 

6 lllllllllllllllll!llllllllllllllllllllll 111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 18a.-Growth of low-frequency bispectral components during the episode of 9 August 1972. 

Figure 1Sb.-Growth of middle-frequency bispectral components during the episode of 9 August 1972. 

N A 

N 

5 

0 

A 14 Eo. c. - 

A 

N 

3- 

C 

210 

c9 

E 

m 

0 

A 

6 

0) 

0 

2 

6~~~~~~~~~~~~~~~~~~~~~~~1~~~~~~~~~~~~1

0 30 60 90 120 150 

Minutes from beginning of episode 

00°000000 xH~;f~OOXxXXXx 

xxxXXfO +++++++++ x x 

*****++++++++++ xXX~~~XX** XxxlXXX ******* +++++++++++++++++++++xx ******aatO x" .Z&k", ************;;;;;+ +++++ XX*XXXXXX * 

i.xxxxx XXXX mO@ 0000000000000000000 

00000000000000000 

- 

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Figure 18c.-Growth of high-frequency bispectral components during the episode of 9 August 1972. 

0 30 60 90 120 150 

Minutes from beginning of episode 
6 1111111111111111111111111111111111111111111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 19a.-Growth of low-frequency bispectral components during the episode of 30 September 1972. 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 19b.-Growth of middle-frequency bispectral components during the episode of 30 September 

1972. 

6l1111111111111111111111111111111111111111111111111111lli1ll'rllll 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 19c.-Growth of high-frequency bispectral components during the episode of 30 September 1972. 
X X XXXXX 

Minutes from beginning of episode 

Figure 20a.-Growth of low-frequency bispectral components during the episode of 7 October 1972. 

Figure 20b.-Growth of middle-frequency bispectral components during the episode of 7 October 1972. 

t 

\ m 

14 

0 

Y - 

A PI 

3, 3- 

310 

m 

0 

0 

4 

6 

1111111111111111111111111111111111111111111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 

+ + **** +++.'*:xy&+,:;;;$$$++ bhhhhhh6h&h,,6** 

- ,, ,ti..t**:triffrh.;tf meeeeeeeeee 

. . t ..:r~:rr~il~~,,+iiitt:~::::~~~ ......... . .... at:.! 

. ...~....8t.ae . . . . . . . hh6 ***+++++ 666 ** ++ 

hhh.hhhhhhh ah ** + * + 

-**** *Jet:++++ - **tt 

++++, +++++++++ 

1111111111111111111111111111111111111111l11111111111111111111111 

Figure 20c.-Growth of high-frequency bispectral components during the episode of 7 October 1972. 

0 30 60 . 90 120 150 

Minutes from beginning of episode 
~'~~~~~~~~~~~~~~~~~~1~~~~111111111111111111111111111

0 30 60 90 120 .. 150 

Minutes from beginning of episode 

61~~~~~~~~~~~~~~~~

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 21.-Growth of bispectral components during the episode of 9 August 1972. Above: 0.175 Hz vs. 

0.150,0.175, and 0.200 Hz; Below: 0.200 Hz vs. 0.175,0.200, and 0.225 Hz. 

free surface, energy is transferred to the side-band 

frequencies w (1 k {). Our results appear to resemble, to some extent, this side-band energy transfer theory. We may conjecture that, analogously, the 

stronger interaction with the lower side-band frequency component than with the higher side-band 

frequency component is also due to the complicated 

couplings of the nonlinear boundary conditions at 

the free surface. Furthermore, Longuet-Higgins's 

(1976) recent theory, indicating that the transfer of 

energy tends to reduce any symmetry in the spectrum, can also be qualitatively attributed to these unequal side-band interactions. 

7. SUMMARY AND 

CONCLUDING REMARKS 

In this report we set out to empirically examine 

the temporal growth pyocesses of wind-generated 

waves using data recorded in Lake Ontario. We have 

examined unispectra, bispectra, .and trispectra computed consecutively for the three selected episodes. 

The scheme used for our study is based on the assumed property of local stationarity. This assumption allows us to apply successfully the analysis 

method developed for stationary processes to study 

nonstationaq wave growth processes. 
0 30 60 90 120 150 

Minutes from beginning of episode 

a88882g28882$2z 

PQ~xx~~~~~ XXXXXX 

~~~~~~~~~~~~~IIII~I~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 22.-Growth of bispectral components during the episode of 30 September 1972. Ahooe: 0.175 Hz 

vs. 0.150,0.175, and 0.200 Hz; Below: 0.200 Hz vs. 0.175, 0.200, and 0.225 Hz. 

Results can be summarized as follows: 

(1) Wind waves grow fastest during increasing 

wind speeds. 

(2) Each unispectral component has a different 

temporal growth rate. The components can be 

grouped into three frequency groups: the low-frequency components, which consistently grow under 

increasing wind speeds; the high-frequency components, which are insensitive to wind speed and time; 

and the mid-frequency components, which contain 

those attaining spectral peaks during the episode and 

which contain mixed properties of both high- and 

low-frequency component groups. 

(3) Bispectral and trispectral components generally 

demonstrate temporal growth behavior similar to 

that of unispectral components with smaller growth 

rates. 

(4) From comparing empirical source functions 

with parameterized theoretical nonlinear source 

functions, we found that at the beginning and ending 

stages of wave growth the growth rate is low and the 

empirical source function is smaller; the larger nonlinear source function under the spectral peak must 

be balanced by significant dissipation processes. During intensive wave growth, where nonlinear and empirical source functions are both large, dissipation 
0 30 60 90 120 150 

Minutes from beginning of episode 

4 

6 1111111111111111111111111111111111111111111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 23.-Growth of bispectral components during the episode of 7 October 1972. Above: 0.225 Hz vs. 

0.200, 0.225, and 0.250 Hz; Below: 0.250 Hz vs. 0.225,0.250, and 0.275 Hz. 

becomes less significant. terest of exploring the empirical aspects. Since the de- 

(5) During the latter part of wave growth, the in- tailed process of wave growth is still far from being 

teractions of the peak-energy frequency component completely understood, we hope our results will prowith the next lower frequency component grow con- vide some insight that can be useful for further 

sistently stronger than the interactions with the next understanding of it. 

higher frequency component, and the next lower frequency subsequently becomes the peak-energy frequency. This result, as demonstrated clearly from an 

examination of the temporal growth of bispectral and 

trispectral components, provides an explanation for 

the well-known fact of unispectral peaks shifting 

toward lower Frequencies during wave growth. 

These results are consistent with our primary in- 
Frequency wE 27r (Hz) 

'0 0.1 OP 03 04 05 06 07 

Frequency m/ 2w (Hz) 

Fipyre 24a. -Trispectra during the episode of 9 August 1972, 2.5 minutes from beginning of episode. 
Y;</ o 1 0.175 ! I ,r 3 

I "'I I 11 i 

0.150 15 2 

j- I 

Frequency w/2a (Hz) 

Figure 24h.-Trispectra during the episode of 9 August 1972,32.5 rnin~ttrs from beginning of episode. 
1 Frequency ol2r (Hz) -=. --. 

Frequency o J27r (Hz) 

Fi~llre 24c.-Triqpectra di~rinc the epi5ude of 4 A~tgust 1972, 62.5 minutes from beginning nf episode. 
I -> ,, 

1 0.000 . L 

I 

03~ i p4 I-! c.>F ::y 1 ::-* 0 1 

fW FT- P -. - . F r: % ",. 9 u L- 1.) ' - , ' ,2? : 6 0 ; 

I 

I Frequency w/ Zn (Hz) I 

81 _L.,.LL : -- 

i)i, ,; 

Frequency w / 2a (Hz) 

Fijir~rc 24d. - Trispectra during the ppisodr of 9 AURIIS~ 1972, 92 5 minutes from bcninniny: of episode. 
Frequency o / 2a (Hz) 

Figure 24e.-Trispectra during the episode of 9 Auguqt 1972,122.5 minutes from be~innin~ of episode. 
f"'II I ' 'I' ' T 'I 

Frequency co J Prr (Hz) 

Figure 24f.-Trispectra durin~ the episode of 9 August 1972,152.5 minutes from beginning of episode. 
0000000 

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0 0000006~ ++++++++++ X I XXX 

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0.1 25 - 

0 

c. 

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z. 

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z 

Eo Y 

n 

A 

0 

3, .- 

3 v 

0 

2 

6lll~~ll~llllllllllIllllllllllllll~lllIllllll~l~lll~llll~III~III11I 

0 30 60 90 120 150 

Minutes from beginning of episode 

- 

+ + + ++ + + 

XXX XX xx 

OBOOOOO 

Figure 25.-Growth of trispectral components during the episode of 9 August 1972. 

26 

0 

CI 

N 

I, 

\ 22 

z 

E Y 

n 

A 

C) a 18- 

++ 00000 OOaOo - 00@=@0 ++++++ - +++++$go xxxxx OOoO +:::+""" + X 0 ~OOO~~~ 

X I X + ooO XXxxxxxx 

0 O 

rxx + 0000000 

x + ad 000~00 ..... ++ 0 0 xX ++ 0 0 ...... - 

3, X + 0 0 

7 X 

X .... + 

Y 

3 X 0 ............ 0 X + 0 X 

E X X 

..... $ '4xx 

X - 

2 XXXXX X +++ 

- 

o,/2r (w2~2wlo3/2~ 

Hz 

0.100 

0.125 

0.150 

0.175 

0.200 

0.225 

6-11111~1~11111~1111l~111~1111~~1111111111111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 26.-Growth of trispectral components during the episode of 30 September 1972. 

¤ 

o 

m 

X 

+ 
Minutes from beginning of episode 

Figure 27.-Growth of trispectral components during the episode of 7 October 1972. 

Figure 28a.-Growth of trispectral components during the episode of 9 August 1972: Third-order interactions among components 0.175, 0.200, and 0.255 Hz. 

6 l111111111111111111111111111111111111111111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 
llll?,?,l?,bbdg 

xxxxxxxxxx X X 0 X 

1111111111111111111111111111111111111111111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 28b.-Growth of trispectral components during the episode of 9 August 1972: Third-order interactions among components 0.200,0.225, and 0.250 Hz. 

6 1111111111111111111111111111111111111111111111111111111IIIIIIIII 

0 30 60 90 120 150 

Minutes from beginning of episode I 

Figure 29a.-Growth of trispectral components during the episode of 30 September 1972: Third-order 

interactions among components 0.175,0.200, and 0.225 Hz. 
6 Illlllllllllllllll1111111111111111111111111111111111111111111111 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 29b.-Growth of trispectral components during the episode of 30 September 1972: Third-order 

interactions among components 0.200,0.225, and 0.250 Hz. 

6 

0 30 60 90 120 150 

Minutes from beginning of episode 

Figure 30a.-Growth of trispectral components during the episode of 7 October 1972: Third-order interactions among components 0.175,0.200, and 0.225 Hz. 
>oO 00 X XXXX 

OOOoO 

Minutes from beginning of episode 

Figure 30b.-Growth of trispectral components during the episode of 7 October 1972: Third-order interactions among components 0.200,0.225, and 0.250 Hz. 

8. REFERENCES 

Barnett, T. P. (1966): On the generation, dissipation and 

prediction of ocean wind waves. Ph.D. thesis, Scripps 

Institution of Oceanography, San Diego, Calif., 190 pp. 

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in the study of wind waves. Rep. Prog. Phys. 38:667- 

729. 

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197 pp. 

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wave trains on deep water. 1. Fluid Mech. 27:417-430. 

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1. Fluid Mech. 14:577-584. 

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Roy. Soc. Lond. (Ser. A) 289:301-320. 

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data from a single ocean wave gage. Report of the Johns 

Hopkins Univ. Applied Physics Lab., Laurel, Maryland, 

188 pp. 

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and interpretation of kth order spectra. In Spectral 

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Cooley, J. W., and J. W. Tukey (1965): An algorithm for 

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storm surge on Lake Huron. In Proc. 15th Conf. Great 

Lakes Res., Int. Assoc. for Great Lakes Research, 565- 

581. 

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British Columbia, Vancouver, Canada, 177 pp. 

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in a gravity-wave spectrum. Part 2: Conservation 

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in a gravity-wave spectrum. Part 3: Computation of the 

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M. Rosenblatt, Wiley, New York, 125-139. 

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D. E. Cartwright, K. Enke, J. A. Ewing, H. Gienapp, 

D. E. Hasselmann, P. Kruseman, A. Merrburg, 

P. Miller, D. J. Olbers, K. Richter, W. Sell, and H. 

Walden (1973): Measurements of wind-wave growth 

and swell decay during the Joint North Sea Wave Project 

(JONSWAP). Deut. Hydrogr. Z., Suppl. A, 8:12. 
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Reynolds numbers. C. R. Acad. Sci. U.S.S.R. 30:299- 

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GLERL. In Proc. Internat. Symp. on Ocean Wave Measurement and Analysis (Am. Soc. of Civil Engineers) 

1:64-73. 

Longuet-Higgins, M. S. (1976): On the nonlinear transfer 

of energy in the peak of a gravity wave spectrum. Proc. 

Roy. Soc. Lond. (Ser. A) 347:311-328. 

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shear flows. 1. Fluid Mech. 3: 185-204. 

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of Res. Inst. for Appl. Mech. 16:251-264. 

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Memo. No. 43. 

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gravity waves of finite amplitude. Part I. 1. Fluid Mech. 

9:193-217. 

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gravity waves of finite amplitude. Part 11.1. Fluid Mech. 

11:143-155. 

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Cambridge University Press, London, England, 261 pp. 

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237. 

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Miss. 

Sell, W., and K. Hasselmann (1972): Computations of 

nonlinear energy transfer for JONSWAP and empirical 

wind wave spectra. Report of Inst. Geophys., University 

of Hamburg, Hamburg, Germany, 248 pp. 

Silverman, R. A. (1957): Locally stationary random processes. IRE Trans. Inf. Theory 3:182-187. 

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white-cap dissipation of wind-generated waves. Dyn. of 

Atmos. and Oceans 3:55-78. 

Appendix A: Relationships Between Higher Order Covariances and Higher Order Spectra 

In order to define a general higher order spectrum in terms of the Fourier components of the sur- +i E (kj • xi - witj) - (k,, . xn - writ,,) 

x e face displacement, we follow Batchelor's (1953) basic 

approach by considering a stationary random funcC' I 

tion of both time and space defined by ,+id& lm x lm - I) (e-id;~l - 1 ) 

+ixlm . . . {(x,t) for -X 5 xi I X 

and -TI ts T 

{(x,X;t, T) = (A.1) 

for Ixil > X 

0 and It1 > T 

which satisfies the condition for a Fourier integral to 

exist. The corresponding Fourier coefficient is 

1 x dxldx2 . . .dx,(dt)". A(k,X;w, T) = - jxjt{(x,X;t, T)e+i(k.x-wt)dxdt. (2~)~ 

(A.2) Now for stationarity with respect to both space and 

time, the right side can be written as 

From Weiner's generalized harmonic analysis, it can 

be shown that as the parameters X and T extend to 1 

00, the limit exists as -I . . . j {(x + rl,t + T~) . . . (2~)~" 

We can write 

k+dk,w+dw dY(k,w) = [Y(k,w)lk,w 

() dxdt, (A.4) 

x e 

which is the inverse relation of 

e+idk~mr~rn - 1 ) (e-id:7~ - 

Now we can obtain the ensemble average of an nth x n . . . (m=l +jr1rn -171 order product of dY(k,w) as 
Since the stationary process is inde&nt of both x n-1 

and t, the above relation is identically zero udess -' E w,rj 

Xe i=l d~,...d~,-~, 

k, - kl - k2- . . . - k,-i = 0 and 

and 

w,-w1-w2-. . . -a,-1 =o. 

' E wjr, 

Furthermore we let dk's and dw's approach zero to x e i=1 dul . . . Lfw,-l. 

obtain 

dY(kl,wl)dY(kz,wz) . . . dY(kn-l,an-l)dY*(kn,wn) For n = 2, we have the following familiar pair of 

dkldk2 . . . dk,-ldwldw2 . . . classic spectral analyses: 

1 

-- 1 S2(w) = - jR2(~)e-iwrd~, - 2?r (A.13) 

(2?r)4(n-l) j. . . jRn(r1,rzr. . - rn-l,71r72,. . .7n-1) 

and 

and w, = q + w2 . . . + 0,-1 

and 

Detailed discussions corresponding to the formal 

definitions (A.ll) and (A.12), as well as conditions 

for existence and convergence of estimates, are 

(A.8) beyond the scope of our present study and can be 

found in Brillinger and Rosenblat t (1967). 

In practical applications, we may define 

Let the Fourier transform of S;(t) be X(w); then we dY(klrw1)dY(k2,w2) . . . dY(kn-l,wn-l)dY*(kn,o,) have dkldk2. . . dk,-ld~ld~2. . . 

if k,#kl+...+k,-l 

and on # wl +. . . + an-1. 

and 

3;-(t) = j,X(w)eiwtdw. (A.17) 

Now for the nth order covariance we have 

The last expression in (A.8) gives the general definition of the nth order energy spectrum, where R,,(T~ . . ., T,,-~) = {(t){(t + 71 . . . {(t + ~,,-1) 

1 = lim T j,S;(t) j,, . . . j X(w1). . . X(~,,-I) "'11-1 T- m 

ei[(wl+. . .+~,,.~)t+w~7~+ . . .'",,-l7,,-ll 

1 For observations at single points, we may integrate = lim - T j.1 . . . j X(wl). . . x(wl,-l) 

and normalize the above equations over all directions T- m '"17-1 

and wave numbers to obtain 

ei(ulrl+. . .+u,,.,r,,.,) 
Comparing (A.18) with integrated (A.12), we have 

formally 

Sn(wl, . . .r ~n-1) 

Thus, in practical applications we can use the 

product of Fourier transforms of h(t), with the aid of 

the fast Fourier transform algorithm, to facilitate the 

calculations of higher order spectra. 

Appendix B: Testing for Stationarity 

In this report we have assumed local stationarity 

to study the temporal growth of spectral components 

that are in general nonstationary. To test the assumption of local stationarity, we adopted a testing 

method similar to that used bv Bierkaas (1976). The 

scheme divides each data set iito P equal-length subsegments; calculates the <, CZ, r3, and c4 for each 

sub-segment; and then applies two nonparametric 

tests, the run test, and the reverse arrangement test 

(Bendat and Piersol, 1966) to the data set to examine 

its stationarity. 

In the run test, the test parameter for each interval is compared to the median value of the test 

parameter for all intervals. If the test parameter is 

greater than or equal to the median value, +1 is assigned to the interval; otherwise, -1 is assigned. A sequence of consecutive +l's or -1's is called a run, 

and the number of runs N, in the P intervals is determined. N, gives an indication as to whether or not results are independent random observations of the 

same random variable. The reverse arrangement test 

is most useful in detecting monotonic trends in the 

time series. Consider the test parameter Ai, with i = 

1, 2, . . ., P. If Ai > Aj for i < j, the pair of parameters is called a reverse arrangement. 

The total number of reverse arrangements, Nu, is 

defined as follows: 

Let 

Table B.1. Stationarity test results 

Then 

P 

and 

Based on P and the level of significance, the acceptance ranges for N, and N, can be calculated 

(Bendat and Piersol, 1966) and used as a basis to accept or reject the stationarity assumption. 

Accordingly, in our analysis eight separate tests 

were performed for each data set. The results, presented in terms of the percentage of data failing the 

tests for the three episodes studied, are shown in 

table B.1. At a level of simificance .. a = 0.10. the 

overall failure rate was 29 percent; the rate reduced 

to 19 percent for a = 0.02. Every data set passed at 

least two of the eizht tests verformed. A total of 27 

percent passed all eight test;. The parameter l3 posed 

the best results; c4, on the other hand, posed the 

worst results. Although, ideally, absolute stationarity requires that the data pass all the tests, we 

find that for a basically nonstationary process these 

results generally tend to enhance the acceptability of 

our assumption of local stationarity. Thus, without 

pursuing more complicated physical and statistical 

nonstationary analysis, we can employ available stationary analysis methods to study an otherwise nonstationary process. 

Percentage of Data Failed 

August 9 September 30 October 7 All Data 

Level of a=0.10 a=0.02 a=0.10 a=0.02 a=0.10 cr=0.02 a=0.10 a=0.02 

Significance 

Run Test 

Reverse Arrangement Test F 0 0 2 0 2 0 1 0 

- P 28 17 53 30 50 33 44 27 

- I3 8 2 5 0 0 0 4 0 

r' 34 20 53 38 48 33 45 30 

All Parameters 29 19 34 21 24 16 29 19 

Appendix C: Application of Barnett's Parameterization of a Nonlinear Source Function 

Theoretical results have shown that the nonlinear energy transfer of gravity waves can be expressed by the Boltzmann integral as 

where N, = N(k,) = S(k,)/w, with w, = (gki)lI2 and S- 

(k,) is the twodimensional wave spectrum with respect to wave numbers lk,l = k,. The delta functions 

express conditions for resonance between waves. The 

coupling coefficient T(kl,k2,k3,k4) is a very complicated function and its precise form does not provide 

any physical interpretation. While the Boltzmann 

integral has been applied to other fields of study, 

Hasselmann (1962, 1963a,b) first derived equation 

(C.1) for gravity waves. Other derivations have been 

given by Benney and Saffman (1966) and Watson 

and West (1975). Recently Longuet-Higgins (1976) 

presented a simplified model and showed that 

T(kl,k2,k3,k4) is equal to 47r when the four wave 

numbers kl, k2, k3, and k, are nearly equal. This result implies that the exchange of energy within the 

peak of the spectrum is of dominant importance. No 

~actical application of this simplified model has 

been developed, however. 

Attempts to evaluate the integral (C.l) have 

been made by Hasselmann (1963b) and more recently 

by Sell and Hasselmann (1972). The latter computations, typically 30 minutes per spectrum on a 

CDC6600 computer, lead to interpretations of the 

JONSWAP measurements that strongly suggest that 

the nonlinear transfers of energy play an essential 

role in the development of the wave spectrum, particularly in the growth of the wave energy at low 

frequencies. 

Because of the complexity of the direct calculations of (C.l), Barnett (1966), using a Neumann spectrum, obtained a parameterization of (C.1) as part of 

a wave prediction stheme based on the earlier Hasselmann (1963a,b) results. Mitsuyasu (1968) applied 

this parameterization to his studies on nonlinear 

energy transfer in wave spectrum and obtained fair 

agreement between calculated and observed nonlinear source functions. Resio and Vincent (1976) 

satisfactorily applied Barnett's prediction scheme for 

hindcasting waves in the Great Lakes. 

Barnett's parameterization was based on Hasselmann's (1962) theoretical results, which demonstrated that the wave-wave interaction of a spectral 

component consists of an active part that transfers its 

energy to other components and a passive part that 

receives energy from other components. Thus, 

Frequency in Hz 

Figure C.1.-A white spectrum and its computed source functions. 

with 

x exp [-4 (I - py + o.(?)~] , 

for f > 0.42 fo, and 19 - sol < " ; 2 

G = 0, otherwise; 

and 

- - 

for f > 0.53 fo; 

D = 0, otherwise, 
- 

-Sell 6. Hasselman 

Barnett 

P 

00 2 4 .6 6 10 1.2 1.4 16 

Frequency in Hz 

Figure C.2.-Pierson-Moskowitz spectrum and its computed 

source functions. 

where E, fo, and O0 are total energy, mean frequency, 

and mean direction, respectively, defined by 

E = I IS(f, O)dfdO, (C.5) 

1 

fo =XI I fS(f,O)dfdo, (C.6) 

and 

In the above equations, 0 is the wave direction and f 

= w/27r is the linear wave frequency. 

We applied the above parameterization by 

further assuming a directional spreading fador of 

(8/37r) cos40 to compare some of the results given in 

Sell and Hasselmann (1972) that led to the major conclusions based on JONSWAP measurements. The results for a fairly white spectrum, a Pierson-Moskowitz spectrum, and a mean JONSWAP spectrum are 

000012 - - Sell & Hasselman 

I - Barnett 

S .000010 - .- 

C) 

- .000004 - 

-.000006. 1 11))) 

.40 50 60 70 60 90 100 110 

30 Frequency in Hz 

Figure C.3.-JONSWAP spectrum and its computed source 

functions. 

presented in figures C.l, C.2, and C.3, respectively. 

The JONSWAP study (Hasselmann et al., 1973) concluded that, as the sharpness increases from figure 

C.l to figure C.3, the evolution of a sharp peak is primarily controlled by the nonlinear energy transfer 

evidenced by the shifting of the positive lobe in Sell 

and Hasselmann's calculations toward lower frequencies. While not necessarily profound, the same 

conclusion can also be drawn from the shifting of the 

positive lobe as a result of Barnett's parameterization. Therefore, with an overwhelming savings in 

effort and computer time, Barnett's parameterization 

can be used approximately for the examination of 

theoretical nonlinear energy transfers, particularly 

with respect to the location of the positive lobes of 

the parameterized nonlinear source function. 
Appendix D: Trispectra During the Episodes 30 September 1972 and 7 October 1972 

We present here the calcuIated trispectra for the tral sheets located 30 minutes apart, corresponding in 

episodes of 30 September and 7 October as figures time to the bispectra shown in figures 16 and 7 7. 

D.1 and D.2. Each figurc contains six sets of trispecFrequency o/2a (Hz) 

Figure IJ.la. - Trispectra durin~ thc cpi,ridc ~>f 30 Seplember 1472, 2.5 minute4 frnm beginning of 

episode. 
Frequency Y / 2s (Hz) 

Figure n.lb.--Trispectra during the episode of 30 September 1972, 32.5 minutes frum be~inning of 

episode. 
I 'I' ' I ' " I 

i - --- 0" -~7d F [ &J 

I 

01'' ' - - 

I 

# 

Frequency ar J27r (Hz) 

' i 

Frequency w / 2~ (Hz) 

Figure D.Ic.- Trispectra during the epicode of 30 September 1972, 62.5 minutes from be~inning of 

episode. 
Frequency w f 27 (Hz) 

Figure D.ld.-Trispectra during the episode of 30 September 1972, 92.5 minutes [rum beginning of 

episode. 
Frequency wS2r (Hx) 

Si~ure D.le.-Tri~pectra during the epi50de of 30 September 1972, 122.5 minute5 from beginning ot 

epiqode. 
Frequency u/ 2w (Hz) 

Figure D.lf.-Trispectm during the episode of 30 September 1972, 152.5 minutes from beainnina of 

episode. 
Frequency w / 2n (Hz) 

Figure D.Za.-Trispectra during the episode of ?October 1972, 2.5 minutes from be~inning of episode. 
Frequency w / 27r (Hz) 

Figure D.Zb.-Trispectra during the episode of 7 October 1972,32.5 minutes from beginning of episode. 
Frequency w /2n (Hz) 

Fik~irc l3.2~. -Tri5pcctra during the cpisodc of 7 Ortoher 1972, h2 5 minntrs from he~innin~ of epfsode. 
Frequency m J 21 {Hz) 

F~RUI~ D.2d.-Tri~pectra during the episde of 7 October 1972, $2.5 minuter from beginning of episode. 
00 I1 I'll 'I1 'I' I 

0.3 

0.150 t 

1 

Frequency w / 2n (Hz) 

Figure D.2e.-Tri~pectra during the episode of T October 1972,122.5 minutes from beg inn in^ of episode. 
Frequency w / 2s (Hz) 

Fi~urel3.2f.-Trispctm during the episode of 7Octoher 1972,152.5 minutes from beg inn in^ of episnd~. 

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