MODULMiLc!:" OF '[.T. C;..:7...\TErD WAVE.
.."'! CL''VITY ,.V'ES
By
ALL. ';: i;.. RE".'E, .JR .
A Pl TS .1.. V 1.'.! : . j1 T .
O. ., . ., i .. ; . ' .: ; .. .
7; P.\FrT.I: '. J; L: ,I .L'::: I *:. *. .; i. :T:, ;C
DE'E: ;. ;" Y' :T'.',"1, O' G11. .;'.) i;
"1i 9
UNIVIT'.RSIT ;7 7ri. ,O .
ACKlJOWILEDG, ENTS
1 wish to express my appreciation to Professor Omar H. Shcedin for
providing tle motivation from which this study grew and the means to
carry it on to a conclusion.
I owe many thanks al'o to other members of the Coastal a'.d Oceo.o
graphic E'ngineerirg Laboratory staff. In particular I want to thank
Mrs. Melody G.T.dy for painstakingly transforming the handwritten copy or
this paper into a typed rough draft, and Mrs. Lillean Pieter for pro
ducing reproduicidbl e J di.awings.
In addition the highly professonal approach of Mrs. Elaine Mabry
has trade the final manuscript of superior quality.
HM sincereC,t apUrecI t!ion is due my wife, Margo, for her remar.kablc
ability to encourage and motivate me over the period during which this
di..scrrt ticn 'as written.
This work was sponsored by NOAA Space Oceanography Prograrn I: i'..r
Crant NG2972 anu by the Jet Propulsion laboratoryy under Contract
954030.
TABLE OF CONTENTS
Page
Acknowledgments . . . . . . . . .. . . . . ii
List of Figures . . . . . . . . . . . . .
Abstract . . . . . . . . ... . . . . . . ix
Chapter I. Introduction . . . . . . . .. . . . j
A. The Need for the Study. of High Frequency ..Waves . . .. 1
B. Description and Scope of the Present Work . . . .. 4
Chapter II. Investigations of Wav'es and Associated Energy
Transfers . . . . . . . . . . . 5
A. Transfer of Mechanical Energy to Wa.vet Through the
MilesPhillips mechanism . . . . . . . . . 5
B. Inteiracticns Amo.ig Components of a Wave System . . . 15
1. Tick Second Order Perturbation of the Wave Spectrum .. 16
2. Benjamin and Feir Side band interactions . . . . 1
3. Phillips Resonant Interactiorn Anong Gravity Uaves . 22
4. McGoldrick Resonant Interaction Among CapillaryGra.vicy
Wa.ves . . . . . . . . ... . . . 26
5. Hasselmann Fifth Order Nonlinear Inrtrchange . . .. 28
6. Valenzuela CapillaryGravity Wave Resonant InLterction 30
C. Experimental Studies of High Frequency ;.ves . . . .. 32
Chapter III. Measurement of Wave Slope . . . . . . .. 39
A. Measurement of Slope Versus Height . . . . . .. 39
B. The WindWave Facility . . . . . . . . .. 44
C. LaserOptical System for Measuring Slope . . . . .. 47
D. Supporting iHeasurements . . . . . . . . 66
E. Experimental Conditions . . . . . . . ... 70
F. Digital Data Acquisition and Reduction . . . . .. 74
Chapter IV. Experimental Results . . . . . . . .. 83
A. Time Series Data . . . . . . . . .. . 83
B. First Order Spectra . . . . . . . .. .. . 99
C. Spectral Modulations . . . . . . . . .. . 104
D. Amplitude Modulation of the Spectrum . . . . . .. Ill
TABLE OF CONTENTS (centinnied)
Page
Chapter V. Theories of Interaction Between Short Waves and
Long Waves . . . . . . . . . . . 117
A. The TwoScale Model of LonguetHiggins and Stewart . . 118
B. The Solution of the TwoScale Wave Model by Direct
Integration of the Wave Energy Equation. . . . . .122
C. The Solution of the TwoScale Wave Model by Perturbation
of the Wave Energy Equation .. . . . . . . .125
D. The Modulation of Short Wave Spectra by Long Waves .... .128
Chapter VI. Comparison Between Experimental and Theoretical
Results . . . . . . . . . . . 137
A. Determination of Spectral Modulation by the Modeling
Methods . . . . . . . . . . .... 37
B. Model Based on the Wave Fnergy Equation . . . . ..
C. Model Based on the IHamiltonian Forliulation . . . . 14
Chapter VII. Conclusions . . . . . . . . . . 18
Appendix 1. The Boltzmann Transport Equation . . . . . 3
Appendix II. The WaveWave Interaction Integral . . . . .
List of References . . . . . . . . ... .. . . 192
Biographical Sketch . . . . . . . . ... . .. 196
LIST OF FIGURES
Figure Page
1 Plan and side views of the windwave tank model . . .. 45
2 Wave generator section and test section of the windwave
model . . . . . . * . . 46
3 Orientation diagram for laser beam refraction at the
air water interface . . . .... . . . . . 48
4 The conversion of deflection angle to wave slope . . . 51
5 Schematic view of the receiver of the optical wave slope
measurement system . .. . . . . . . . . 52
6 Maximum deflection and slope angles measurable as a
function of distance from the objective lens to the
local water level . . . . . . . . . . 54
7 Analog conversion of the photodiode output signals to
orthogonal axes of deflection . . . . . . .. 56
8 Error introduced into the slope measurement due to the
finite size of the laser beam . . . .... . . 58
9 Static response of the instrument receiver to deflection
and azimuth changes . . . . .... . . . . 60
10 X channel instrument calibration curve . . . . . 61
11 Schematic cross section of the wave tank at the instrument
installation site . . . . . .... . . . . 62
12 The slope measurement receiver in operating position in
the wave tank . . . . . .... . . . . . 64
13 Side view of the installed relationship between the system
light source and receiver . . . .. . . . . 65
14 The wave gauge calibration for experiment 3 ..... . 67
15 Three local wind velocity profiles. . . . . . . 69
16 Wave record segmenting scheme used to produce short wave
slope epics for phase averaging . . . . . .... .. 76
LTST OF FIGURES (continued)
Figure
17
18
Visualization of a typical tr
Phase averaged slope energy s
face ot the long wave crest
19 Wave slope and height time re
tests T13 and T1 . . .
20 Wave slope and height time re
tests T13 and T16 . . .
21 Wave slrpe and height time re
test T17 . . . . .
22 Wa.ve slope and height time re
test T18 . . . . .
23 Wave slope and height time re
test T19 . . . . .
2L Wave slop.: and height time re
test T O2 . . . . .
25 Wave slope and height time re
tcr T2] . . . . .
26 Average total wave slope spec
27 Average total wave slope spec
end removal sequence . .
pectral estimate at the front
Scod f r ep rimn 3 and
cords for experiment F3 and
cords. . . . . . . .and
'cords for experiment E3 and
cords. . . . . . . .peret E3 and
'cords for experiment E3 and
'cords for experiment E3 and
'cords for experiment E3 and
. . . . . . . .
28 Average total wave slope spectra . . . . . . .
29 Phase averaged short wave slope spectra from the crest and
trough regions of the long wave profile for experiment E3
and test T15 . . . . . . . . . . . .
30 Phase averaged short wave slope spectra from the crest and
trough regions of the long wave profile for experiment E3
and test T16 . . . . . . . . . . . .
31 Example of the short wave energy modulation for selected
free wave frequencies from experiment E3 and test T15.
Advection is not considered . . . . . . . .
32 Example of the short wave energy modulation for selected
free wave frequencies from experiment E3 and test T16.
Advection is not considered . . . . . . . .
Page
79
81
85
87
89
91
93
95
97
100
101
102
106
108
112
113
LIST OF FIGURES (continued)
Figure Page
33 Example of the short wave energy modulation for selected
free wave frequencies from experiment E3 and test T15.
The advection correction is applied . . . . ... 114
34 Example of the short wave energy modulation for selected
free wave frequencies from experiment E3 and test T16.
The advection correction is applied . . . . ... .115
35 Comparison for experiment E3 and test T16 between
the experimental modulation results and the theoretical
prediction based on the hydrodynamic energy equation
with the cyclic perturbation . . . . . . ... 142
36 Comparison for experiment E3 and test T19 between
the vxperimencal modulation results and the theoretical
prediction based on the hydrodynamic energy equation
with the cyclic perturbation . . . . . . ... 143
37 Eight point amplitude modulation comparison . . ... .148
38 Magnitudes and phases of the short wave slope energy
amplitude modulation for the conditions of experiment E3
and test T14 . . . . . . . . ... . . . 150
39 Magnitudes and phases of the short wave slope energy
amplitude modulation for the conditions of experiment E3
and test T15 . . . . . . . . .. . . . 152
40 Magnitudes and phases of the short wave slope energy
amplitude modulation for the conditions of experiment E3
and test T16 . . . . . . . . ... . . . 154
41 Magnitudes and phases of the short wave slope energy
amplitude modulation for the conditions of experiment E3
and test T17 . . . . . . . . ... . . . 156
42 Magnitudes and phases of the short wave slope energy
amplitude modulation for the conditions of experiment E3
and test T18 . . . . . . . . ... .. .. 158
43 Magnitudes and phases of tlhe short wave slope energy
amplitude modulation for the conditions of experiment E3
and test T19 . . . . . . . . ... ... .. 160
44 Magnitudes and phases of the short wave slope energy
amplitude modulation for the conditions of experiment E3
and test T20 . . . . . . . . ... . . . 162
vii
LIST OF FIGURES (contLiued)
Page
Figure
45 Magnitudes and phases of
amplitude modulation for
and test T21 . . .
46 Magnitudes and phases of
amplitude modulation for
5.00 z . . . .
47 Magnitudes and phases of
amplitude modulation for
6.25 z . . . .
48 !agnitudcs and phases of
amplitude modulation for
9.38 z . . . .
49 Magnitudes and phase of
amplitude modulation for
10.00 Hz . . . .
50 Magnitudes and phases of
amplitude modulation for
12.50 Hz . . . .
51 Magnitudes and phases of
amplitude modulation for
15.63 lz . . . .
52 Magnitude, and phases of
amplitude modulation for
20.00 Hz . . . .
the short wave slope energy
the conditions of experiment E3
the short wave slope energy
the free wave frequency of
the short wave slope energy
the free wave frequency of
the short wave slope energy
the free wave frequency of
the short wave slope energy
the free wave frequency of
the short wave slope energy
the free wave frequency of
the short wave slope energy
the free wave frequency of
the short wave slope energy
the free wave frequency of
viii
164
166
168
170
S172
. 174
. 176
178
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MODULATION OF WIND GENERATED WAVES
BY LONG GRAVITY WAVES
By
Allan M. Reece, Jr.
December, 1976
Chairman: Omar H. Shemdin
Major Department: Civil Engineering
The cyclic short wave variations, phase related to the long wave
profile, that occur during active generation of the short wave field by
wind are investigated. measurements consisting of wave slope rime
series are made in a laboratory environment allowing the independent
generation of two scales of motion. The large scale, long wave motion
is developed mechanically with a 2.0 s period and 0.06 radian maximum
slope. The snall scale, short wave motion occupies a broader spectral
reion and is generated primarily by the action of the overlyine air
flow. The wind speeds used are referenced as .4 n/s to 10.0 n/s. The
s'iort waves of particular interest are chosen to be those of 5.00 to
20.00 Hz due to the rapid responses expected as a result of strong air
sea and hydroeynamnic coupling in that range.
Wave sil. )r is measured locally, and continuously in time with a
de;ice utilizing the phenmer.non of tFtical refraction at the air sea
interface as the basis for the detection of slope angle. It is found
thar within the experimental Ltonds short wave slope energy exhibits a
cyclic variation along the long wave profile that is related to that
profile and characterized as a modulation. The observed variation is
separable conceptually and practically into effects having two origins.
The first effect is the shift in short wave frequency brought about by
bodily advection of the short wave profile by flow underlying it associ
ated with the long wave. The shifted frequency is found to be adequately
predicted by the linear combination, of the small scale free wave fre
quency and the advection effect given by the product of short wave
.vvenumber and underlying velocity. The modulation of frequency is
found to be an increasing function of free short wave frequency. Fre
quency shifts vary from 40O to 1261 of the free wave frequency, increas
ing with free w.ve frequency and reference wind speed. For the long
wave investigated the peak to peak variation about the mean advected
frequency is 585 of that frequency for all wind speeds.
The second effect is that of small scale slope amplitude modulation
brought about by straining against the long wave orbital flow and sub
sequent relaxation through wavewave interaction toward the mean value
of tlhe energy. The peak to peak energy excursion of a variance element
in the wave slope spectrum is considered after removal of the frequency
modulation end is commonly found to be 100. of the mean value of the
clergy. The magnitude of the excursion becomes smaller as short wave
frequency increases, and larger as wind speed increases. The experi
mentally determined phase of the energy excursions, relative to the long
wave profile, place maximum values 450 to 1800 ahead of the long wave
crest.
The behavior of the energy content of a variance element in the
short wave portion of the spectrum is characterized as a relaxation.
The relaxation behavior, due to wavewave coupling, is introduced into
the Boltzmann transport equation to describe the evolution in space of
the short wave slope energy along the long wave profile. The following
assumptions are employed in the analysis: (1) The wave system is two
dimensional. (2) The short wave energy exists in a steady state over
the large scale of motion. (3) The local variations in short wave
energy are cyclic and related to the long wave horizontal orbital
velocity. (4) The long wave is sinusoidal. (5) The local short wave
frequency is given by the correct free wave frequency plus the linear
modification due to the underlying flow. (6) Atmospheric input produces
exponential wave growth. (7) Damping by nonconservative forces is
negligible. (8) Wavewave coupling among the short waves produces a
relaxation type of energy drain from the disturbed energy values of a
variance element. The analysis yields results that fit the experimental
values well when the relaxation factor is taken to be a constant value
of 6.3. If the relaxation factor is allowed to vary from 1.57 to 15.7,
for any particular test and frequency the agreement between experiment
and theory for the amplitude modulation can generally be made exact.
CHAPTER I
INTRODUCTION
I.A. The .:eed for the Study of Hi.th Frequency Uaves
An uin.crtailding of how shore waves eolve during their life span
in time is essential to the advancement of two principle areas of
oceanographic research. The field of remote sensing of oceanic param
eters is presently undergoing w, period of intense development. In many
instances remote sensing is a function of short wayv: activity. From a
more theoretical standpoint sIort. ;av'.s i. rLicail. reolatc.! to the
study of the development of .*in ocea'njl wave ectrum.
The large extent of the ocean, combined with its ..>her i:.1tici
table nature, has made it difficult to obtain a gret density c; d ca
concerning its dynamic state. As a result, the predictions of temporal
and spatial distributions of surface waves have always contained consid
erable uncertainty. Pcccntly the tools developzd f'or rcr.,mote C:tnsing
operations have been applied to pro'blcms of a ieorph\sical nature.
Photographic, nonvisual optical, and mr.icrcn.c ir.srruL.ets ar ;einr 
used for water .ave detection. Groups asociaLte 'iLh the technology, o
space flight: are searching for new applicaLions for space verhic!?.s.
Combining r.he advanced sensing capabilities of remote sensors with he
ubiquitous platform provided by a space vehicle, could provide a remar':
ably efficient technique for monitoring the motion and local environi inc
of the ocean surface. Of special interest cnrrentl.: is the role air
borne and spaceborne radar will fulfill in the remonc sending of
i
atmospheric and oceanic parameters. Radiation of radar frequencies
interacts with water waves of short wavelength through the mechanism of
Bragg scattering (Crombie, 1955). If the response of the short waves to
the wind and wave conditions were known, the wind and sea state could be
inferred from the radar return. This information has obvious scien
tific, military, and commercial importance.
Short wave activity undoubtedly plays a major role in the develop
ment of an ocean wave spectrum. Stewart (1961) indicated that wave
motion is probably the first line in the path of energy flux from the
air flow to the total water motion. Using data compiled from ocean wave
growth measurements, he demonstrated that the minimum contribution to
the total drag on the air flow due to the direct flux of energy to the
wave motion is at least 20%. This figure is probably quite low because
it did not consider the whole spectrum that existed, nor the dissipation
and flow of energy from the waves to the mean motion that occurred.
Stewart argued that since the flow of air over the interface is aero
dynamically rough, or nearly so, the energy of the air flow that passes
to the water must go by way of the correlation product of the local
pressure and vertical surface velocity. The work done by normal stress
es produces only irrotational motions. A wave motion must result, which
will, however, augment the surface drift current.
Short waves are suspected of being responsible for a conduction of
energy to longer waves by several mechanisms. Wave interaction theories
predict energy flows among groups of short waves, some of which are
longer than others. These theories will be mentioned in Chapter II.
Short waves can also create an energy flux to much longer waves by their
support of a variable wind stress. LonquetHiggins (1969a) demonstrated
that a fluctuating tangential stress at the free surface is dynamically
equivalent to a normal stress fluctuation lagging the tangential stress
b\ 100. This results from a change in boundary layer thickness brought
about by the fluctuation in the tangential stress. In particular the
stress he worked with had a sinusoidal variation along the surface of a
sinLisoidal long wave. Wu (1968) found exparimientally that, at low wind
speeds at least, surface roughness is related to short wave size and
distribution. If the short waves were Lo vary cyclicly along a sinusoi
dal lcng wave profile in such a mannner that the wind shear reached a
maximum near the long wave crest, a normal stre's maximum would appear
on the rear face of the long wave crest. The correlation product of
this pressure with the local veitical surface velocity would lead to an
additional net inflo'.. of energy to the long wave motion.
The response of short waves to the wind and sea is not entirely
clear. Pierson (1975) expressed the idea that wind speed dependence in
the overall mean spectra of short waves does exi:st. However, the idea
is not heuviiy supported by experimental evidence (Phillips, 1969). On
th other hand, evidence strongly indicates that the short wave motion
is not steady in a local mean sense, but may vary cyclicly with a longer
wave com:'.;onent and intermittently with coupling to the airflow. Fluc
t.'tions in the short wave energy related to the long wave motion were
r.ticed e.arl in the study of waves, and were pointed out for the first
time by : c.asell (1844). He noticed sceepening of the short waves in the
neighborhood of the long wave crests. Dorman and '*iol]oChristensen
(].972) made measurements of wave response to wind gusts, indicating that
there '.ere bursts of momentum exchange between air and water motions
that exceeded the mean by a factor of 103 during generation.
I.B. Description and Scope of the Present ork
The thrust of the present study is toward the investigation of
cyclic short wave energy changes, phase related to the long wave, that
occur during active generation of the short wave field by wind. To
accomplish this objective, experimental measurements of short wave slope
time series ace made in a laboratory environment where the basic long
wave parameters can be controlled as desired and the wind speeds are
accurately reproducible. An instrument system, operating on the prin
ciple of optical refraction at the air water interface, detects the
slope of the interface at a point without disrupting the flow. The
slope time series obtained in this manner are digitally analyzed so that
the energy present within a specified variance element of the slope
spectrum can be observed as a function of reference wind speed and phase
location along the long wave profile.
The short wave energy variations along Lhe long wave, determined i.
this manner, are used to verify a reasonable scheme for prediction of
Iite energy variations based on the Boltzmann transport equation of
Hamiltonian mechanics. Tne short wave frequency range identified as
being of particular interest is the band containing all frequencies from
5.00 to 20.0U Hz. Waves in this region tend to be very responsive.
That is, they. are strongly coupled to the air flow so they exhibit rapid
growth ratps, and they are strongly coupled to the water motion so they
exhibit short interaction times. Variance elements within this range
have energy densities that are observed to vary as a function of long
wave phase.
CHAPTER II
INVESTIGATIONS OF WAVES
ANE ASSOCIATED ENERGY TRANSFERS
II.A. Transfer of Mechanical Energy' to Waves Through the
Mile&Phillips Mechanism
While casually observing the motion of the ocean surface, one can
easily overlook tne fact that what is occurring is actually the coupled
mocion of two media in the region of their common boundary the air sea
Interface. The motions are exchanging energy in both directions across
the interface. During a wave generation situation, the motion of the
waLer Is t.volving in space and time because the net flux of energy
across the interface is from the air flow to the water flow. Energy is
passed from the air to tlhe water through the action of normal and tangen
tial tre:s.es (Kinsman, 1965). Pressure fluctuations at the inteface
are capable of passing on energy in such a manner that irrotational
motioni results. Irrotational analyses have been sho.'n to yield consid
erablt sirplifications oi and good approximations to the water wave
problem. For these' reasons analytical approaches to the generation of
water waves have concentrated on energy flow associated with pressure
fluctuations.
Pressure fluctuations are available in the air flow under all
conditions to create and to feed the water wave motion. Eddies, passing
in the air flow, create local unsteadiness in the pressure regardless of
the shape of the water surface. Once the water surface has attained a
wave shape it forces a modification of the air flow streamlines, gener
ating pressure fluctuations. So there are two types of identifiable
pressure fluctuations. Based on these different pressure fluctuations,
two rather different mechanisms for wave generation have been proposed.
These two mechanisms were proposed concurrently, finally combined into
one model, and have formed, since the later 1950's, the primary theo
retical basis for the prediction of wave generation (Phillips, 1962).
The first mechanism proposes a resonance action between the pos
sible surface wave modes and the eddies convected along by the mean air
flow (Phillips, 1957). The equations describing the water motion are
linearized, and simplified by assuming inviscid, irrotational flow. The
equation of motion for the system is the dynamic free surface boundary
condition including surface pressure and surface tension, evaluated at
the mean surface level
94 p T
tp p
+ gn + V nr (21)
at Pw Pw
where
S= velocity potential of water motion,
p = water surface displacement,
p = surface pressure,
T = surface tension,
and
p = water density.
Fourier transformation nicely converts the equation, (21), to a differ
ential equation in terms of the transformed variables. The variables
are written in equations (22), (23), and (24) in terms of their
generalized Fourier transforms in space. The equations are
n(x,t) = A(k,t) ei ( k xdk (22)
p(x,c) = j P(kt) ei(k )dk (23)
and
*(x,z,t) = e 9 dk (24)
k
O
where a prime indicates a time derivative. The resultant differential
equation, is in a fomn describing a system in forced oscillation
k
A"(k,t) + A(k,t) P(k,t) (25)
w
where o is given by the familiar dispersion relation for free surface
waves with surface tension included,
Tk3
o = gk + .(26)
p
The solution of the equation yields, ii, the focm of a convolution inte
gral, the time history of the amplitude of a wave component of wave
number, k, as it responds to an atmospheric pressure forcing function
that is quite independent of the water motion. The solution is
ik(t ) io(t T)
A( t) P(kT) e e ] dT (27)
w 0
where r is the lag variable. The asymptotic form of the solution for
the energy spectrum of the wave displacement, t(k,t), is developed from
the mean product of A(k,t) and A*(k',t). Kinsman (1965) formed the mean
product with the use of equation (27), which yields
k2 t t
(k,t) =  II(k,T,T') sin o(t T) sin o(t T') dTdr'
pa 2
po 6 d
w 0 0
after integration over k' space, and where II(k,T,T') 6(k k') is
defined as P(k,T) P*(k',T). If the pressure spectrum, II(k,T,T') is
considered stationary it becomes a function of time separation, T T'
rather than time. Then the integral above transforms the pressure
spectrum into the frequency domain. In terms of the energy spectrum of
the pressure fluctuations, F(k,a), the wave energy spectrum is given as
k2t
(k;t) = F(k,o) ,(28)
4p a
w
which shows the wave energy developing linearly in time. The response
of the water surface depends on the magnitude of the pressure fluctua
tion and the time over which interactions are allowed to occur. For any
given angle of wave propagation, a, relative to the direction of the
mean wind, waves that have the phase velocity given by
c(k) = U cos a (29)
where U is the mean wind velocity, have the longest interaction times
and largest responses. Every wave of wavenumber, k, would have a pre
ferred direction, a. If the input, F(k,a), were a narrow band function
the water wave spectrum would show a local energy maximum at an angle,
a, satisfying equation (29) for the phase speed of the wave at the peak
of the wavenumber spectrum.
The second mechanism of wave generation relies on the coupling of
the air and wave motions to induce the atmospheric pressure fluctuations
that force energy to flow to the water motion. Miles (1957) proposed a
model that coupled the equations for the flow of both media. The air
flow is assumed to be inviscid, incompressible, and to have a mean shear
flow specified by a logariticmic variation with height. The mean flow is
perturbed by small two dimensional disturbances induced by the shape of
the surface waves. The disturbances are assumed to be small enough to
allow linearization of the equations of motion. Turbulent fluctuations,
although necessary to maintain the specified velocity profile, are not
taken into account in the original problem. The water motion is assumed
to be inviscid, irrotational, incompressible, and to have no mean flow.
The wave motion itself is considered small amplitude.
From the average of the energy equation one is able to deduce the
rate of energy flow per unit area of the interface from the air to the
water due to the wave induced Reynolds stress. The flux rate is
dE U
 = 0p L  dz (210)
dt a z
where
0 = air density,
a
U = mean horizontal air flow velocity,
u = wave induced contribution to the horizontal air flow
velocity,
and
wt = wave induced contribution to the vertical air flow
velocity.
The Reynolds stress is evaluated as in instability theory. For the
inviscid parallel flow assumption, the value is approximated by the
expression
a 
p u d =  V'
a k U/dz I z = z
C
for the region z < z and
c
p UW = 0 (211)
for the region z > z
The expression above is evaluated at a height, zc, the distance
above the water surface where U(z ) equals the wave phase speed, c.
Substitution of (211) into (210) yields the value of the rate of
energy flow per unit area from the air to the water. The approximate
evaluation of the integral in (210) is
dE2 2 2
dE rp cW U/ 9z
, (212)
dt k @U/ z z
c
which needs to be evaluated at z only, and where
c = water wave phase speed,
k = water wave wavenumber,
and
2
W = intensity of the vertical velocity fluctuations.
Using the definition of energy for a two dimensional wave field and the
deep water gravity wave dispersion relation, one can formulate a norma
lized rate, i, of increase in the wave energy from equation (212) and
the definition
1 dE/dt
, (213)
a E
where a = wave radian frequency. Since the energy ratio is multiplied
by the time per radian, i/a, we see that yields the fractional in
crease in energy per radian of change in the wave. For this analysis
the energy of the wave system grows at an exponential rate with time.
To evaluate the growth rate the inviscid OrrSommerfeld equation, which
arrives by using a stream function to describe the perturbed flow in the
air, must be solved.
Based on the two proceeding mechanisms, Miles (1960) developed a
model for the wave response when both types of pressure fluctuations
are acting. In doing this he modified equation (25), which was formu
lated for the resonance model, to include the wave induced pressure
forcing function. Equation (25) becomes of the form of equation
(214),
I. k
A"(k,t) + oA(k,t) =  [P0(k,t) + Pl(k,t)] (214)
o
w
where P (kt) and Pl(k,t) are the turbulent and wave induced pressure
fluctuations, respectively. The pressures have been represented in
equation (2.4) through the use of the transforms
am
Po(x,t) = Po(k;t) ei(k x) dk
CO
and
p (x,t) = PI k; t) e x) dk
_O:
where time remains a parameter. Considering the wave induced pressure
field, which is assumed to be
p (x,t) = (a + ib) p cknri(,t) ,
it can l.e said that only the component in phase with the downward sur
face velocity will do work on the wave. The component of interest is
then the one proportional to that velocity, since it is wave induced.
This pressure may be represented as
P (k,t ) = ,' (k, t) (215)
The energy equation, when integrated over the water column, contains the
term, P (3r/L), which specifies the rate of energy input to the wave
11
motion due to the atmospheric pressure at the surface, P In spectral
terms the mean rate of energy input due to the wave induced pressure of
equation (215) can be written as
S2
P = aA'(k,t)A'*(k,t) = an (216)
St
where the use of the complex conjugate of the surface height is allowed
because the surface height is a real quantity.
The evaluation of a comes from directly equating the mean rate of
energy input of equation (216) to Miles' original result in equation
(212), where only wave induced pressure was considered. Solving the
relationship for a yields
Tp cW2 2U/3z2
a
k @U/@z
z
a c (217)
which can then be used to represent the pressure in terms of the wind
and wave parameters. In terms of the normalized energy growth rate,
a, a is given by
a = cp w (218)
where t is defined by equation (213). Then the equation of motion,
(214), can be written as
2 k
A"(k,t) + CoA'(k,t) + a A(k,t) =  P(k,t) (219)
. 0
Pw
where P0 is the only external force. From the solution of this problem
the energy spectrum of the water surface elevation can be developed.
The asymptotic solution for t much greater than the turbulence time
s.:al yields
2 i::t
k e 1
*(k;t) = 2F(k,a) (220)
4p a0 o
w
where '(k;t) is the wave height energy spectrum with time as a parameter.
It is evident that the energy spectrum, depending on the duration, can
grow linearly as with a purely resonant interaction, or exponentially as
with the shear flow mcdel. The power series expansion of the expo
nential term of equation (220) is
eot 1 ot2
= t +  + ... (221)
LO 2
which yields the two following time dependencies of
L. t
*.(k;t) t for  < 1
2
and
Lot
'(k;t) a e for 1
2
Neither of these two fundamental mechanisms are able to adequ:;'ely
explain the formation and growth of short gravity and capillary waves.
Turbulent pressure fluctuations cannot be expected to excite disturbances
of short wavelength because the turbulent fluctuations of correspond
ingly short wavelengths are not energetic enough, and are convicted
downstream too rapidly to account for the straight crested waves ob
served. The inviscid shear flow mechanism cannot supply enough energy
to capillary waves to overcome the laminar dissipation associated with
the waves (Miles, 1962). Miles (1962) investigated the importance of a
mechanism formulated by Benjamin (1959) with regard to short waves. In
this mechanism the energy transfer to the waves results from the presence
of a stress term which Miles called the viscouc Reynolds stress. This
stress enters the problem through Benjamin's formulation of the equation
governing small perturbations in the aerodynamic viscous .naar flow.
The aerodynamic viscous flow is assumed to be parallel and incompres
sible. The velocity profile is assumed to be linear within the viscous
sublayer and asymptotically logarithmic at larger elevations. The water
motion is considered to be inviscid, incompressible, and irrotational.
An eigenvalue problem is solved for the wave phase speed. The phase
speed turns out to be a complex value, (c + ic.), which leads to a
growth rate of kc. due to consideration of the wave amplitude
[kc.t + i k(x c t)]
n = ae (222)
where the complex value has been substituted. The results of the
analysis indicate growth rates increasing rapidly with increasing wind
speed and decreasing wavelength. This mechanism predicts rapid growth
of short waves having lengths 1 to 3 cm under the action of wind.
II.B. Interactions Among Components of a Uave Svstem
The complexiic of the wave development process arises from the fact
that the process is composed of several more or less equally important
components which are not well enough defined to be separable from one
another. The process of wave development contains generative and de
generative components corresponding to energy flow to and loss from the
organized motion of the waes, respectively. Analyses that attempt to
predict the growth rare of a wave component based on energy inflow from
the air streak. have been mentioned previously. Analysis of the decay
rate of a wave component is based on the estimation of the nonconserva
tive dissipative mechanism such as breaking, turbulent stress, and
viscous stress.
The aforementioned mechanisms deal with energy fluxes to a single
wave component of a discretized spectrum. Of course, waves rarely exist
in a solitary situation. Rather, m'ny wave motions of various charac
teristics combine to form a wave system. It is on the complete wave
syait that the net result of the process of wave developn.nt is observ
able. Because the wave .system and growth process are each made up of
so many parts and are imperfectly understood, we must resort to a spec
tral representation o[ the system tor its study. When the system is
broad banded, that is consisting of more than a sir.;!le spectral compo
nent, the process of wave development contains interactive mechanisms in
additionn to generative and degenerative mechanisms. These interactions
result in the conservative transfer of energy mo:ng two or more wave
components contained within the wave spectrum.
The interactive mechanisms and their effects are discussed below.
All of these mechanisms assume the motion of the wave system to be
eSentially linear with only small corrections required in the mathe
matical formulations. This approach seems justified by the generally
good results obtained through the application of the purely linear
theory.
II.B.1. Tick Second Order Perturbation of the Wave Spectrum
Tick (1959) attempted to remove a deficiency from the use of the
analytical firs: order spectrum to represent a natural wave field. The
deficiency is due to the representation of a nonlinear process with a
linear statistic. Tick's solution to this problem resulted in a theo
retical spectral calculation given by the linear combination of the
first order spectrum and a second order correction term.
The wave motion is assumed to be the two dimensional, irrotational
flow of an inviccid, incompressible, nininitely d;ep fluid. As such, he
applied the potential flow formulation retaining all tT rs up through
second order in wave height. The kiematic and dynamic freL surface
boundary conditions are expanded as Taylor series about the mean surface,
z 0. The equation defining the problem results from the combination
and perturbation of these two equations. The equation of the free
surface is z = n(x,t). A velocity potential, :(x,z,t), that satisfies
the placee equation is assumed to exist for the flow. The Laplace
equation is
2 2
d 0 d 0
+ = 0 (223)
3x Jz
where the z axis is the vertical, and originates at the mean surface
level. The bottom boundary condition restricts the vertical velocity
w = = 0 at z = m (224)
LIZ
where w is rhe vertical velocity. Surface stresses are not considered,
so pressure and surface tension are taken as zero. The dynamic free
surface boundary condition to second order may be written as
gz + + = 0 at z = n(x,t) (225)
it 2 rx ) z /
whe.e n(x,t) is the vertical surface position. The kinematic free
s:.:face boundary condition is
a4 an a4 an
=+ at n = n(x.t) (226)
;z at 3x bx
to second order. The surface boundary conditions are Taylor series
expanded about the mean surface elevation, z = 0. The velocity potential
and surface elevation are approximated as the sum of the first and
second order components,
S (1) (2)
>}, = >, + j, ,
and
n = n( + n(2 (227)
Combining the expanded and perturbed surface conditions yields the
second order equation governing the wave motion. The combined free
surface boundary condition is
(2) (2) (1) (1) (1) (1) (1) (1)
tt z t zz .x xt z zt
+ [4.(i ) + ( 1 ] at z = 0 (228)
zt tt t ttz
g
The two velocity potentials may be represented by their generalized
Fourier transforms, since the surface elevation is assumed to be a
homogeneous, stationary, random process. The first order problem is
solved after the transformation. The nonlinear problem is solved by
Fourier transformation. In the nonlinear product terms of the second
order problem Tick used two variables of integration, o' and 0", in the
Fourier transformation. In this way he was able to solve the nonlinear
problem by stepping through the integration required by the linear
solution mechcd. What arises from the problem is a second order correc
tion to the spectrum produced by the first order spectrum interacting
with itself. The correction to the frequency spectrum is given as
(2)() = K(o',o) (1(o o') ((o') do' (229)
g f
where
jo'(o' 2oo' + 2o') 0 < a' < o, o > 0
K(O''o) =<
(o 2o')" oo' o' < 0, o' > o, o > 0
for the given regions of the c, o' plane. Equation (229) indicates the
nonlinear effect quite clearly. Since the expression for 1 ()() is j
convolution operation, we see that for a continuous function, (1)(o),
(2)
e'ery contribution to .) in a band around a contains jn effect from
the first order energy at every other frequency in the spectrum. The
result on the spectral shape is the production of a small bump at a
frequency about twice that of the wind wave peak.
1I.b.2. Benjamin and Feir Side Band Interactions
benjamin and Feir (1967) discussed the existence of side band
interactions. This idea can account for the change in shape of a sinu
soid as it proceeds, unperturbed by external effects, from its point of
origin. This is accomplished by passage of energy from the basic fre
quency to its side band frequencies, which are present in infinitesimal
proportion from the point of origin of the basic wave. This interesting
result eliminates the possibility of the existence of a wave of perma
nent for..
The flow is considered to be two dimensional, irro: .ional motion
in an inviscid, incompressible fluid of infinite depth. The mean
surface level is at z = 0, and the equation of the free surface is
2 = n(x.t). The potential flow problem is formulated in the usual way.
A velocity potential, q(x,z,t), satisfies the Laplace equation. There
is assumed to be no motion at infinite depth. The dyramic free surface
boundary condition yields the condition for constant interfacial pres
sure with surface tension not included. The boundary value problem is
specified by the Laplace equation
7?.(x,z,t) = 0 (230)
the bottom boundary condition
7q.(x,z,c) = 0 at z  (231)
the kinematic free surface condition
S+ n 4 = 0 at z =n (232)
t X. z
and the dynamic free surface condition
Sg + + + 4.) = 0 at z = n (233)
t 2 x z
The Stokes solution to the nonlinear boundary value problem is perturbed
according to the form of small side band modes of oscillation and sub
stituted back into the boundary value problem specified by equations
(230) to (233). A new boundary value problem in the perturbation
variables results, the solution of which indicates the direction and
rate of energy flux to the side band components of the oscillating
system. The Stokes solution to the original problem to the order re
quired for this problem is given by
1 2
n = N = a cos X + ka cos 2X (234)
2
and
Skz
k
where X = kx ot is the phase function of the primary wave. The ex
pressions for velocity potential and surface level perturbed around the
Stokes solution in the ordering parameter, e, are given as
4 = ) + 4 (236)
n = N + en (237)
Substitution of equations (236) and (237) into the boundary value
problem equations, yields the new boundary value problem specified as
the Laplace equation
2
V2 (x,z,t) = 0 (238)
the bottom boundary condition
Vi = 0 at z (239)
the kinematic free surface condition
nI + n x + n(zz + N xz) + (4 + N ) = 0 at z = N
c xx zz xz z xx
. . . . . . (240)
the dynamic free surface condition
gn n( xXz + z + tz) + (4 + + 4 ) = 0 at z = N
. . . . . . (241)
where terms of order higher than e are not included, and the factors
represented in upper case letters are known.
The solution to the new perturbation problem is assumed to consist
of the sum of a pair of side band modes symmetrically located on each
side of the primary wave in phase and the results of the side band
interactions with the primary wave. Tne solution for wave height takes
the form
T = n1 + r (242)
where for each of the two side band modes we have
n. = b. cos X.
1 1 L
+ akb.[A. cos (X + X.) + B. cos (X X.)]
9 9
+ O[ak b.] (243)
where b. are the side band amplitudes, and the phases are
1
X1 = k(l + c) x o(] + 6) t y
and
X, k(l a) x o( ) t 
where a and 6 c:re much less than one. The results of the analysis indi
cate that under certain conditions the side band amplitudes, b., will
grow in time ir! an unbounded manner. Tf the perturbation, 6, about the
primary wave frequency is i:ithin the limits given in equation (244) the
Stokes wave will not maintain a permanent form, because its side band
modes will draw energy from it. The frequency spread allowing inter
action is
0 < 6 < 7ka .(244)
The value of t yielding maximum growth rate was found to be within these
limits, at 6 = ka. If there were minor oscillations present at a fre
quency in the neighborhood of the primary wave frequency, then the waves
satisfying the gravity wave dispersion relation and with frequencies,
oi = 0(1 + ka) and therefore wave numbers, k. = k(l + 2ka), would
project energy at the group velocity of the primary wave and be selec
tively amplified.
II.B.3. Phillips Resonant Interaction Among Gravity Waves
Phillips (1960) found an interaction of the third order in wave
amplitude among three wave components that results in an energy transfer
from them to a fourth new wave producing a growth in it that is linear
in time. Initially of third order in amplitude, the new wave could
reach a magnitude of the same order as the original three waves. In
this conception the spectrum is still discretized, has become somewhat
broad banded, yet by assumption is limited to the wave motions con
trolled by gravity. Phillips investigated the interaction of pairs of
infinite sinusoidal wave trains of small amplitude. These primary waves
are solutions to the linear equations of potential flow theory. The
nonlinear terms of the potential flow equations produce traveling sinu
soidal pressure and velocity fields with wave number and frequency equal
to the sum or difference of the wave numbers and frequencies of the
primary waves and with amplitudes proportional to the product of the
primary wave amplitudes. If the frequency of an infinitesimal free v'ave
happens to be tic same as that of the nonlinear fields of the same
wavenumber, reason irce will occur dnd the free wave will be forced.
Phillips assumed an irrorational motion in an inviscid, incompres
sible fluid of infinite depth. The wave numbers are allowed to be
vector quantities, so the wave propagation directions in the xy plane
are not restricted. The z axis is vertically oriented with z = 0 at the
mean water level. A velocity potential, ..(x,y,z,t), does then exist
that satisfies Laplace's equation. It defines a three dimensional
velocity vector, = E,. The kinematic and dynamic free surface bound
ary condition? are combined. The combined equation is assumed con
cinuouslyv valid and extended to the local surface, z = n(:.,y,t), through
:aylor expansion, from the mean water level. The resultant equation
before Taylor expansion, when the surface pressure is assumed constant,
is given as
.2
g  + + V 7 V = 0 at z = (245)
dat z t 2
which arises as the difference between the material derivative of the
dynamic condition and g times the kinematic condition. Note that the
cubic terms resulting from the material derivative are retained, due to
the ncccssiy of finding an effect at third order. Each variable is
perturbed with the first crder terms being the first order approximation
to the effective wave system created by two intersecting wave motions.
in Kinsman's (1965) notation, the perturbations are
=(c.'10 + Ol01 + ( 20 + 0811 02 (266)
S= (cV + ) + (a, + ( + a11 + BT, ) .... (247)
S(10 01) + 20 + 11 02
and
n = (anir + fn01) + (an20 + a.*iiri + n2 n,) + .... (248)
The coefficients, a and f, are small and proportional to the surface
slope of the wave they refer to. The solutions to the first appro::i
mation in the perturbation problem are the familiar linear wave solu
tions given as
n 10= al cos X1 (249)
alo1 klz
e0 = e in v (250)
k10
n01 = a2 cos X2 (251)
and
a202 k2z
01 = e sin X2 (252)
k2
where the phase function,
X. = k. x .t
i i i
and
2
i = glk k
There are three forms, depending on the combination of perturbation
variables, of the combined surface condition of the third order that
contain forcing functions. These functions are of third order, and some
will resonantly excite the third order problem. The third order form of
2
the combined surface condition that goes with the coefficient a 2 is
given as
S+g = +g =
2t 2z 2Bz t zt
2 2
+ 2V V + n 2V V
2t 10 10 10 O1
+V V(V V ) +V V V at z = 0
10 10 o01 o 01
.............. (253)
. . . . . . . (253)
after being considerably simplified by the reduction of terms allowed by
the second order problem. The first and second order problems yield the
expressions for all of the remaining forcing terms on the right hand
side of equation (253).
The behavior of 21' as the solution to equation (253), indicates
tch behavior of a wave of frequency, 2o 2, which can receive energy
.. 1 2'
from ihe combination of waves identified by wavenumbers, k k,, and k ,
where in this case k3 = k1. It turns out that on the right hand side of
equation (253) there are terms containing the phase function, 2 X
So resonant excitation of '21 does occur. The corresponding surface
elevation, ri grows linearly in time as
31'
Kt
n, (xy,t)  sin (2', X) (254)
2g
where i: is a constant depending on the amplitudes, wavenumbers, and
frequencies of the primary waves.
In general the resonant interaction excites a new wave at wave
number and frequency, k, and 0,, respectively, that grows linearly in
amplitude with time. For this to occur the wavenumbers and frequencies
must bear the following relationships to one another
k + k2 + + k = 0 (255)
1 3 3 
and
c + 0 + o + 0 = 0 (256)
1 3 H
where each wave obeys its own dispersion relationship,
oi = g'kil (257)
Analytically, this theory suffers difficulties because the wave
that begins at third order ana is assumed to be of third order, is able
to grow as large as the primary components. Phillips did not consider
the energy balance between the interacting components.
In separate experiments LorguetHiggins and Smith (1966) and
McGoldrick ec al. (1966) tested out the possibility of a resonant inter
action of the type suggested by Phillips. In both cases the environment
was reduced to that of two mutually perpendicular primary wave trains.
Both experiments cok place in rectangular wave tanks. McGoldrick's
apparatus was buil: with special attention given to the elimination of
external effects that would mask the measurement. For the interaction
geometry the production of a tertiary wave of frequency 2o1 a2 was
expected when the primary wave frequency ratio was, a1/02 = 1.7357.
Both tests found tertiary wave generation to occur and to closely follow
the theoretical growth rate when the resonance condition was met.
II.B.4. McGoldrick Resonant Interaction Among Capillary Gravity Waves
McGoldrick (1965) modified the resonance problem to include waves
from a broader band so that surface tension effects could be included.
He found an interaction at the second order in this case that is more
physically satisfying than Phillips' result because he considered the
balance of energy among the interacting components. His analysis
resulted in a group of three discrete waves exchanging energy among
themselves. The amplitudes of all three waves are of the same order,
and in its turn, each individual wave grows at the expense of the others.
The process repeats itself cyclicly.
McGoldrick assumed a three dimensional, irrotational motion in an
inviscid, incompressible fluid of infinite depth. The vertical axis, z,
equals zero at the mean water level. The actual surface is considered
to be composed of three waves. The equation of the surface is
3
z = n(x,y,t) = ni(x,y,t)
i=l
The combined free surface boundary condition defines the problem, as
usual. The equation below is the same as equation (245) used by
PhiliiDp with additional terms considered to allow for the surface
pressure variations due to surface tension. The combined condition is
ds _. 12)
2+ g +  + V 2
at 37 t 2
T :3 (V2r)
 + V ( r) =0 at z = (258)
P w L 1t
where the additional terms allow for the consideration of surface pres
sure variation due to surface tension. The equation is not perturbed in
this analysis, but it is extended by Taylor series to the surface from
z = 0, ard terms through second order are retained. A first order
potential function that satisfies Laplace's equation, equation (258),
and the bottom boundary condition
V(x,y,z,t) = 0 at z = " (259)
is wricten
3 k.z
(x,y,z,) a.(t) e sin (260)
1 1
i=l i
where
i = k.x c.t + c.
is the phase fu'.,ction.
The corresponding c:pression for the surface is
3
(x,y,t) = ai(t) cos (261)
i=l
Equations (260) and (261) are substituted into the combined and kine
matic surface conditions. The wave amplitudes are considered to be
slowly varying functions of time, so second order time derivative terms
are neglected. Three differential equations for the three amplitudes,
a.(t), result. Each equation is integrated directly in terms of Jacobian
elliptic functions with real parameters. The set of solved equations
represent a coupled system of three waves in which energy flows peri
odically into and out of each component from the others. The solutions
are bounded, satisfy conservation of energy, and have no analytical
difficulties concerning wave size or time range of solution.
Tbe conditions for the resonance interaction to occur require only
three waves and are
k + k = + k (262)
1 2 3
and
1 + 2 = 3 (263)
where each wave satisfies
T
2 3
o = gki + ki (264)
Pw
It is seen that if the k corresponding to k + k2 produces a o3,
calculated by equation (264), that equals o + a resonance will
occur.
II.B.5. Hasselmann Fifth Order Nonlinear Interchange
Hasselmann (1962) extended the perturbation type of analysis to a
truly continuous wave spectrum. For the first time the mathematics
represented the nonlinear energy transfer in an ocean wave spectrum.
The assumptions limit the wave motions in the theory to those of gravity
waves. The transfer of energy in a gravity wave spectrum was found to
be of third order in the wave energy. In the extension to a continuous
spectrum Hasselmann had to resort to a fifth order analysis in wave
amplitude because it contains the nonstaticnary quantity that leads to
the resonant interacrion. Hasselmann (1963) calculated that energy
flows from intemediate frequencies to lower and higher frequencies.
The problem is assumed to be one of irrotational motion of an
inviscid, incompressible, infinitely deep fluid. As such, a potential
function exists that satisfies the potential flow problem. The poten
tial function and corresponding surface elevation are used in the form
of the generalized Fourier transforms given by
0o
4(x,y,z,t) = A(k;x,y,z,t) e kzei x) dk (265)
oo
and
Co
Si(k x)
r(x,y,t) = B(k;.,y,t) e dk .(266)
The problem, which starts with the perturbation of the combined surface
condition, is not restricted to two dimensions. The linear motions are
assumed to be statistically independent, so that the first order spec
trum completely describes the surface. This property allows all energy
fluxes to br. represented in terms of the first order spectrum.
The resonant energy transfer could occur among four wave components
given the same set of conditions given by Phillips in equations (255)
through (257). Hasselmann (1962) found that an interaction does occur
when
k, +k = k + k (267)
a 2 3 4n
and
01 + 02 = 03 + C4
(268)
The energy transfer is the cubic function of the two dimensional
first order energy spectrum, $(k;x,t), given by
Snl T('1,k2,k3,=4) ( 14 12 3 + G3124 21i344
o
01 2 34) 6(04 + 03 2 a0) dkx ddk dkdk (269)
The subscripts on the spectra are a shorthand notation indicating at
which of the wavenumbers the spectral values are to be taken. The
transfer coefficient, T, is a lengthy function of only the wavenumbers.
The Dirac delta function only allows contributions to the integral from
the appropriate resonant conditions.
The effect of the interaction is to redistribute the energy toward
the formation of a more uniform spectrum. That is, peaks would tend to
be removed in favor of a white noise spectrum.
II.B.6. Valenzuela Capillary Gravity Wave Resonant Interaction
Valenzuela and Laing (1972) pursued a Hasselmann type of analysis
where capillary waves were allowed. The formulation of the problem is
exactly the same as Hasselmann's, except for the consideration of surface
tension. The interaction was found to be more pronounced with only a
third order analysis in wave amplitude needed to calculate the unsteady
term in the second order energy, that creates the energy flux.
The interactions are known as sum or difference resonances depend
ing upon which of the following conditions are met
k = k + k and 3 = 02 + (270)
or
k = k k and a = 0 ao
3 2 1 3 2 1
(271)
where each wave satisfies the dispersion relation. The resultant inter
action is expressed in terms of the products of two first order spectra
as
S11 T(k 2k3) (03 1 2 02 3 0123) (3 2 )dkxdk
+ 2 T((kk,k3) (3~~2 ~ 3 1+ oi ) 2 (o3 + ol)dkdkx y
. . . . . . . (272)
Ag before, the subscripts on the energy spectra indicate the appropriate
wavenuimber of evaluation, and the transfer coefficients are lengthy
relationships between k1,k2, and their associated frequencies.
The results of the analysis predict an energy flow from the region
of waves of minimum phase speed to both gravity and capillary waves.
II.C. Experimental Studies of High Frequency Waves
The experimental investigation of waves of short and intermediate
length did not begin in earnest until the 1950's. Short waves were
neglected prior to that time, not because they were not noticed or con
sidered irrelevant, but because the measurement and statistical tools
required to facilitate their study had not been developed. By the
middle 1950's the scientific environment was becoming conducive to the
many investigations which were to follow.
Using a photographic, optical reflection method, Cox and Munk
(1954) and Schooley (1954) studied the statistical distribution of wave
slopes in two dimensions as a function of wind velocity. When a portion
of the water surface attains an appropriate slope angle it will reflect
a glitter of light from some overhead light source into a receiver.
Knowing the true angle between the source and Lhe receiver, one is able
to infer the slope angle creating the reflection. Cox and Munk photo
graphed the glitter patterns from an airplane, using the sun as t:e
source of illumination, principally to measure mean square surface
slope. Schooley illuminated the surface with flashbulbs held 45 feet
above the water surface at night. Taking the glitter statistics as
representative of the slope distribution statistics, Cox and 'unk (1954)
found the slope distributions to be nearly Gaussian and the nrms value in
radians to be tan 160 at a 14 m/s wind speed (Munk, 1955). The cross
wind distribution was slightly more peaked than Gaussian. The upwind 
downwind distribution was slightly peaked and skewed 2.5 degrees upwind.
The skewness is thought to be due to the effect of wind stress on the
wave slope (Kinsman, 1965). The value of the distribution functions
fall to near zero by the slope values of + 250. The mean square slope
value was found to increase linearly with wind speed from 9.5 m/s to
13.8 m/s. Wentz (1976) showed that Cox and Munk overextended their data
to estimate the surface variance. He proposed that a more realistic
estimate would be a lower bound variance given by 0.8 times the Cox and
Munk value.
Subsequent to Munk's study, Schooley (1955) measured wave curvature
photographically in a small windwave tank to develop size distribution
information. He used the reflection method and considered glitter area
as well as angle. Approximating the glitter facets of the surface as
spherical reflectors, he found the average radius of curvature to be
greater across the wind than in line with the wind. He also found a
minimum mean wind required for wave generation in his tank to be 3.6 m/s.
Up to this point time series of short waves had not been investigated.
Cox (1958) performed a series of experiments in a model windwave
tank that were designed specifically to investigate short waves. Using
an optical refraction measurement technique, he measured wave slope time
series. The source of illumination was located beneath the water sur
face, and was of variable intensity along the axis of measurement. The
light receiver was located above the water surface and focused at a
point on the surface. The surface slope at the point of focus then
controlled the position of origin of the light beam, and therefore, the
intensity of the light beam received. Hence the light intensity was
related to the surface slope.
Of particular interest to the present experiment are the wave slope
spectra calculated and the slope time series displayed for the case when
a mechanically generated swell is present in combination with the wind
driven wave system. The wind wave spectra, as well as the time series,
identify the wave system as having two scales of motion separated in
frequency space by about an order of magnitude. In addition, he found
that when the wind is strong enough to generate regular, large scale
waves the small scale waves congregate on the front face of the longer
waves. Cox generated large free gravity waves mechanically and found
that capillary waves were present on the front face of the long waves
even when no wind was blowing. The addition of wind enhanced the capil
lary wave amplitudes, but they remained concentrated on the long wave
forward faces until the mean wind speed exceeded at least 9 m/s. Finally,
Cox exhibited the wind speed dependence of wave slope through the spec
tral representation of the wave system and its integral. The mean
square slope increases rapidly with increasing wind speed, corresponding
to a systematic increase in peak slope energy. However, at frequencies
higher than the peak region the spectral energy does not show a syste
matic dependence on wind speed.
Wu et al. (1969) devised an optical reflection instrument for use
in a windwave tank that measured discrete occurrences of a given surface
slope and curvature in a given time interval. Time series could not be
generated, so the wave analysis is of a probabilistic nature only. Wu
(1971) illustrated the slope distribution functions calculated from his
data. The function shapes are generally Gaussian for each wind speed.
However, at the lower wind speeds skewness occurs toward slope angles
favored by the orientation of the forward face of the long waves. As
the wind speed increases, the skewness goes to zero. The skewness
indicates a high concentration of very short waves on the forward face
of the long wave that becomes a more uniform concentration along the
profile with increasing wind speed. The maximum skewness occurred at a
mean wind speed of 5 m/s. At the higher mean wind speeds the distribu
tion function becomes peaked. Wu attributed this to wave breaking.
Wu (1975) proceeded to the problem of the measurement of short wave
slope distributions in the presence of a mechanically generated long
wave. He calculated the fractions of the total number of capillaries
that occurred at various positions on the long w'.i' profile. The maxi
aoum capillary activity concentrated on the forward face of the long
wave. Only four wind speeds from 4.2 m/s to 12.3 m/s were tested. The
front face dominance had a maximum at 7 m/s and decreased elsewhere.
His results for other segments of the long wave did not show clear
trends with wind speed.
At the University of Florida Shemdin et al. (1972) recorded wave
slope time series, using an optical refraction measurement systLm. Th
statistics were based on the large scale magnitude of time, over which
the',' iere assumed to be stationary. The wind wave slope spectra ex
hibited a .s.i c in the peak to lower frequencies and high..r values with
increasing wind speed, as the height spectra do. Spectral observations
showed t;t the intensity of high frequency waves, for example 20 Hz,
incre.as... lii,,arlyv .ith mean wind speed from 5 to 10 m/s. Short wave
inLensitie. a somewhat lower frequencies, for example 8 Hz, were be
ginning to saturate at a reference wind as low as 7 m/s. It was found
that the addition to the wave system of a larger scale, mechanically
generated wave produced a reduction in the wind wave peak value, which
,:as in the vicinity of 3 Hz, and somewhat of an increase irt the energy
levels above 5 Hz.
Long and Huang (1975) described an optical device they had con
structed for the detection of wave slopes in a windwave tank. The
instrument operates on the refraction principle. The position of a
laser beam after refraction at the surface is sensed in one dimension by
an array of parallel photodiode strips. Each strip responds to the
presence of the laser beam with a different output voltage. The re
ceiving array has 19 elements, allowing 19 angles of slope measurement.
The discretized electrical output of the array is then a step approxi
mation to the true wave slope. This approach introduces a large quantity
of high frequency energy into the spectral computation. The mean square
slope values obtained for mean winds of 3 to 5 m/s are comparable to the
values of Cox (1958) and Wu (1971), and increase with wind speed. In
the range 5 to 10 m/s the values obtained by Long and Huang are higher
than in the earlier studies and maintain a rather constant value over
the wind speed changes.
The nonphotographic, optical devices used for the measurement of
wave slopes all require a transducer to convert an optical signal to a
continuous electrical signal. Two devices have been employed in the
previously mentioned experiments. A direct discrete measurement can be
obtained with an array of photodiode devices. A measurement based on
the intensity value of the optical signal can be obtained with a photo
multiplier and optical attenuator. A third type of device, used in this
experimental investigation, is a continuous Schottky diode that measures
optical position directly. Each method has weaknesses. The diode array
provides only a discontinuous signal, the attenuator and photomultiplier
combination is quite nonlinear, and the Schottky diode is very expensive.
Scott (1974) proposed the use of a device called a "photentiometer" to
inexpensively measure the position of a laser beam in a refraction type
of slope measurement device. The device operates as an optically
activated slide wire resistor, giving perfect linearity.
Mitsuyasu and Honda (1974) took a nonoptical approach to the inves
tigation of the high frequency spectrum of wind waves in a windwave
tank. They measured wave height with very thin wire, resistance wave
gauges. Both single and double wire types were used with wire diameters
of 0.1 mm. The frequency response was found to be reasonably flat to
80 Hz through a dynamic calibration scheme that oscillated the gauge in
a tank of water. Electronic differentiation of the wave height signal
was used to derive the time derivative of wave height to emphasize the
high frequencies. At frequencies beyond 40 Hz this technique could not
be used because the results, when converted to height spectra, did not
agree with the spectra of the direct b:ight measurement. At a fetch of
8.25 meters they found the wave height energy to increase with wind
speed over tie mean speed range of 0 Eo 15 m/s. No approach to an
equilibrium range for the energy level was identifiable as wind speed
increased to the maximum, even though breaking was clearly visible
beyond a mean wind of 10.0 m/s. It is possible that the somewhat low
stability of the spectral estimate, Lhere being only 10 FFT calculations
averageid, obscured a trend toward saturation.
Ruskevich, Leykin, ari Rozenberg (1973) described a measurement
system that can measure both time and spatial series of wave height in a
windwave tank. The device uses an array in either one horizontal
dimension or two of resistance wave gauges. The wires are 0.2 mm in
diameter. The signal from each wire passes through a correcting ampli
fier with a gain function designed to compensate for the drop off in
frequency response of the wave gauge wires at high frequenciesE. The
device is designed to operate on waves in tl:e range of 3 to 40 Hz. A
discretized spatial series is produced by sampling the output of ea.h
wire in turn with an electronic switch at a rate of 10 kH.t. The output
of the switch then represents an almost synoptic look at the wave height
at each gauge. At any time the spatial series can then be constructed.
Sinitsyn, Leykin and Rozenberg (1973) used this device to investigate
the effect a long wave has on a short wave field that it is passing
through. The experiment was performed for both mechanically and wind
generated ripples. The frequency and wave number assigned to the rip
ples were those of the peak of the ripple frequency and wave number
spectra, respectively. On this basis long wave crest to trough short
wave frequency shifts were measured and found to agree with the ex
pression
w(k,x,t) = w (k,x,t) + k U(x,t) (273)
where o is the free wave frequency, modified to account for the change
in body force due to vertical acceleration over the long wave. k is the
short wave wavenumber, and U is the long wave horizontal particle veloc
ity. The mechanically generated ripples followed the linear expression,
(273), somewhat more closely than the wind ripple. Frequency maxima
occurred at long wave crests, while wavenumber changes along the long
wave profile were insignificant.
CHAPTER III
MEASUREMENT OF WAVE SLOPE
III.A. Measurement of Slope Versus Heicht
Munk (1955) pointed out that, because the wave slope statistics
emphasize the higher frequency components of the wave system, the study
of high frequency waves would be facilitated by the measurement of slope
rather than amplitude. For a fixed point, one dimensional measurement
of the sort used in the present study, the high frequency emphasis can
be demonstrated in the following way.
The periodic height and slope descriptions of the surface motion
can be expressed as (Kinsman, 1965)
n(x,t) = A(!k,o) ei(k O) dkd (31)
00
and
ddo k (32)
n(x,t) = i kA(k,) e dkdo (32)
00
where A(k,o) is the spectral representation of Lhe water surface, and is
assumed to be a stationary and homogeneous function. k is the scalar
wavenumber in the direction of the y.axis. At the fixed position, x,
which may be selected as zero for simplicity, we may write
n(x1,t) = f A(k,o) eictdkdo (33)
and
t) =i ff kA(k,o) etdkdo (34)
00
The temporal correlation functions for the wave height and slope
time series may then be constructed by application of the expressions
(33) and (34). The correlation funccicns are defined as (Lee, 1960)
n n(T) f n(t) n(t + r) dt (35)
00
and
4n (T) = f ri(t) r.(t + ) dt (36)
0
IntrcducLJon of the transforms of the time series from equations (33)
andLi (3.) fields s
l (r0) = f A(k,I) eiL~tdkdo
A(k,o) e( + dkdo dt (37)
and
1 (T) = F kA(!c,o) e Ptdkdo
x ;. I/ L i ff
S 1 kA(k,a) e i(t + T) dkdo dt (38)
Since the surface being described in equations (33) and (34) is real,
one can substitute the complex conjugates, n*(t) and r*(t), for the wave
height and slope, respectively, at time, t, with no change in physical
meaning (Kinsman, 1965). The result of the substitution for one group
of terms in each of equations (37) and (38) is
in(T) = j [A(k,a) A*(k,o) ei] e d dkdo ,
. . . . . . . (39)
and
ior
S(T) = i [kA(k,a)] [kA(k,o)] e
00
f ei(a o')t dt dkdo ,(310)
where the terms are regrouped to allow the time intgral to srand alone.
T'e L i:c integral of equations (39) and (310) can be reprsenr.ed by a
unit i[.pu]se function (.ee, 1960) where
1 F I(o o')t
6(a o')  J dt (311)
2n J
O
Ihen f:.r equations (39) and (310) we may write
OC,
4, n(T) = 2n f A(k,o) A*(L.) e kdo (312)
and
S(T) = 2n [kA(t! ,, )] [kA(k, )] e r dkdo (313)
00
The wave field being considered is two dimensional so there is only one
wavenumbcr associated with each wave frequency, and it is given by the
dispersion relation as f(o). Equations (312) and (313) are written as
nn( ) = 2n IA() 12 6[k f(a)] dk ei do (314)
S2 () = 2n ( kA(o)) 5[k f(,)] dk e do (315)
xx m
where the wavcnumber dependence of A(k,a) is expressed by A(O) and the
delta function representing the physically correct result of the disper
sion relation. It is then appropriate to write
2 io
() = 2 I dA()1 e do (316)
and
(1) = 2 k() 2 e do (317)
nxix 2
where k is restricted to those values given by the wave dispersion
relation.
By definition the spectral density functions, i (o) and Cn n (a),
xx
are the coefficients of the exponential term in the integrands of equa
tions (316) and (317), respectively (Lee, 1960). It is then clear
that
n n (a) knn (a)
xx
Wave slope intensity is, therefore, emphasized toward the higher fre
quency range by the value of the wavenumber squared.
The measurement of the slopes of high frequency waves, rather than
the corresponding heights, has another advantage. Up to the present
time devices used to measure short wave height have all required physi
cal contact with the flow. Obviously the smaller the waves being
43
measured, the more significant the distortion of the flow created by the
insertion of an obstruction. Slope measurements can be made without
resorting to physical contact with the surface, leaving the flow un
disturbed. So slope measurement seems to be the more attractive tech
nique for measurement of small waves.
III.B. The WindWave Facility
The measurements required for the present study were conducted in
the windwave facility at the University of Florida. Figure 1 illus
trates the general size and shape of the windwave tank. Waves may be
generated in the model with a hydraulically powered, mechanical wave
generator and through the action of the air flow created by a large
blower.
Shewdin (1969) described the windwave tank in detail, but the
major features will be mentioned here. The tank is a 36.6 meter long
channel, divided along its length into two equal bays of 0.86 meter
width and 1.9 meter height. At the upwind end of the tank a wind duct
system, shown in Figure 2, conveys the air flow produced by the blower
into one of the wave tank channels. The air inlet modifies the flow to
simulate rough turbulent air flow in the wind channel. Below the duct
work the hydraulically operated wave generator paddle produces large
scale waves as specified by a signal generator. The waves propagate
through the 36.6 meter long test section. A 5.8 meter long wave absorber,
composed of baskets filled with stainless steel turnings, is placed at
the downwind end of the tank.
FAN WiO DuCT rGAIL ,TP OF *IND TUNNEL GATE, ,BEfAC
bUL 1 .EAD
Side View
nh.riiicSIO5 ins rin lel
r. 4 _AS S I_ f
 * .   _ __   
I IaLO T 1 \ bASIN
,F' _
I .
_.WIND .._..__________..;_
____ NO NINAN
k VILNOLITv M*A{ ir CN 1*
[N'1 , 4 L..AIE. 
SE,,'hLi ,I', I .. I_ P n ~e
waTlh DAMNli 10 r:_ \n
S sRsoin Plan View
Figure 1. Plan and side views of the windwave tank model.
Meosuremen;
Section
I I
L 7.3m
F~L 2m
n
.Pitot Tube
L_
4
4
4
SFlow Conditioner
* Air Flow
0
Bulkhead
I p p
Figure 2. Wave generator section and test section of the windwave model.
4
4
4
n r 
111.C. LaserOptical S.stem for Measuring Slope
The objectives of this experiment required the accurate detection
of high frequency wave slopes. A device was designed and built at the
University of Florida (Palm, 1975) to accomplish this task. The instru
ment is a laseroptical system which operates on the principle of optical
refraction at the airwater interface. It is capable of obtaining
analog time series records of wave slope along two principle axes in the
interfacial plane. It features an Insensitivity to the local wave
height and to light source intensity variations.
The quantities required to discuss the principle of operation of
the instrument for one axis of measurement are shown in Figure 3. For
simplicity consider a system of plane waves tobe passing through the
fixed point of measurement. The laser beam pathway is stationary and
aligned with the vertical while in the water layer, before incidence on
the airwater interface. The angle formed between the submerged laser
pathway and the surface normal, the angle of incidence, is designated as
Si(t). The light beam is refracted at the interface and proceeds along
a pathway in the air forming an angle, 0 (t), the angle of refraction,
with the local surface normal. The relationship between the instantane
ous angles of incidence anid refraction is known as Snell's Law and is
specified by
n sin 0.(t) = n sin 0 (t)
1 r r
where n. is the index of refraction of the water, which contains the
1
incident beam, and nr is the index of refraction of the air, which con
tains the refracted beam. By virtue of the geometry shown in Figure 3,
it is possible to define a deflection angle, t(t), in terms of the
angles of incidence and refraction that specifies the deflection of the
Vertical
Path of laser beam
in air
/ Local Surface Normal
: .: .
/' r
f.
..* .. . .
"" ... .:'i... :. ::: :::.; I "O i .
" : : .' ".
 Laser beum path
. *. m water
Figure 3. Orientation diagram for laser beam refraction at the
air water interface.
Air
Water.
refracted beam away from the true vertical axis as
C(t) = C (t) O.(t) (318)
r 1
Using Snell's Law in equation (318), one can express the deflection
angle in terms of the angle of incidence
1
4((t) = sin1 [n sin C'.(t)] C.(t) (319)
where n = n./n is th.. relative irdex of reflection, and it is ascribed
i r
a constant value of 1.333. Since the laser beam is aligned with the
vertical axis, the angle, 0.(t), defines the time history of the local
wave slope in the x direction as shown in Figure 3. Solving equa
tion (319) for 0.(t) shows that measurement of the deflection angle
yields information about the local slope. The expression becomes
sin .(t)
0(t) = C.(t) = tan , (320)
n cos ,(t)
where 0(t) is understood to be the local value of slope. This relation
is plotted in Figure 4. The instrument receiver measures the deflection
angle through the means of electronic detection of the corresponding
deflection distance in a horizontal plane on which the beam is made to
impinge. The resultant electrical signal is related directly to deflec
tion angle through calibration, and then to wave slope through relation
(320).
Figure 5 schematically depicts the optical receiver. It consists
of an aluminum housing containing four essential components; the objec
tive lens, the diffusing screen, the imaging lens, and the photodiode
detector. The refracted laser beam enters the receiver through the
objective lens at the lower end of the receiver. The lens is an Aero
Ektar, f/2.5, 30.48 cm focal length lens. After passing through the
objective lens the laser beam is incident on the diffusing screen of
Figure 4. The conversion of deflection angle to wave slope.
45
40
30 eTcn s
3n ncos <
n = 1.333
< 20
a,
0. 0
U)
10
O I I I I I
0 5 10 !5 20 25 30
SDeflection Angle (..o)
imaging Lens 
Objective Lens
Detector Diode
"Oiffus;ng Screen
Short
. . .. . . .. '. .. %. ..
::.. : :: :.. .... .~i .... ...e
. ... "' .. .. .:.::::'" . : "" :.. .:. ..'.....' : :
:}:~i::::ii:ii iN ~iii~ i i % i i ii :!: : :: :: i ri : ii!j:!:.: :. i ::: i :::: : ii
. . . .. ... .. ... . ... ... : . :
.... ..... ...... ... ... ... y : .~ c~r
... ............... ......i
. . . . . . . . . . . . .
A3
Loser Pthwa... ... .............
Schematic view of the receiver of the optical wave
slope measurement system.
Figure 5.
frosted acrylic, placed in the lens tear .': plane. The objective
lens is set to focus at infinity, so that a ray of light entering the
lens at an angle, i, away from the instrument axis is focused at a fixed
distance from the axis on the diffusing screen, regardless of its point
of entry through the lens. Hence, at the screen the problem has been
converted from one of angle measurement to one of displacement measure
ment. The fact that the displacement in the plane ot the screen is
uniquely related to the deflection angle means that the beam can origi
nate at any distance from the lens and still have the same effect. So
the changing wave height will not affect the slope measurement. The
imaging lens is necessary to reduce the size of the diffusing screen to
rhat of rhe photodiode area. The lens is a 35 mm camera lens, the input
to which is bandpass filtered at 6328 Angstroms with 100 half poer
width. The lens is an f/l.4, 55 mm focal length, Super Takumar. The
image of the laser spot on the diffusing screen is thereby focused on
the surface of the detector, which is a United Detector Technology model
SC/50 Schottky barrier two dimensional photodiode with a 3.56 cm square
active area.
The maximum deflection angle that is detectable is a function of
the objective lens aperture and distance from the water surface. This
relation is
1 1 d
max = tan  (321)
where d is lens diameter and h is distance between the lens and the
water surface. Figure 6 shows the maximum measurable values of deflec
Lion and slope angle for a given instrument height above the water
surface.
0
o
lO
C
a 5
00
F.
Figure 6.
0
20 V
C
C
10
10 20 30
h(cm)
Maximum deflection and slope angles measurable as a
function of distance from the objective lens to the
local water level (Palm, 1975).
In reality, of course, the water surface is two dimensional and
the laser beam is deflected along the two orthogonal dimensions of any
horizontal plane it is incident on. In this situation the location of
the laser beam in the incident horizontal plane is described by the
deflection angle, 0(t), and the azimuthal angle, 0 (t). The electrical
a
outputs from the diode are combined as shown in Figure 7 to yield elec
trical signals relating to slope along two orthogonal directions, x and
y, that correspond to the physical deflections of the laser beam along
the same axes. The result is the analog measurement of wave slope in
two dimensions at an interfacial point. The normalization operation
performed by the analog divider shown in Figure 7 is very useful,
because it eliminates the effect of a variation in laser beam intensity
on the slope output.
The errors in the measurement result from several sources. These
are calibration arnd alignment, laser beam size and nonuniformity, opti
cal aberrations, and amplifier drift. Searching the system behavior for
local anomalies indicated that optical aberrations were negligibly
small.
When calibrations were performed on an optical bench in an optics
laboratory at a constant temperature, it was possible to duplicate them
to within 2 to 3%. Ho;:ever, when installed in the windwave tank,
changing instrument temperature produced sizeable amplifier drift. It
was found that the outpuL drift could be reduced by performing the
signal division pictured in the output circuitry of Figure 7 in the
computer analysis rather than in the analog circuit of the instrument.
To further reduce temperature drift, it was possible to adjust the
amplifier offset voltage without changing the calibration. The maximum
analog yaxis
divider output
 #4 amp sum amp
\2
3 photodiode
*3aiap * diff amp
analog xaxis
divider output
#1 amp sum amp
Figure 7. Analog conversion of the photodiode output signals to
orthogonal axes of deflection (Palm, 1975).
uncertainty in the calibration after moving the instrument to the field
was +7%.
Laser beam size and nonuniformity restrict the size of the smallest
resolvable water wave lengths. Cox (1958) performed an analysis of his
measurement, assuming a uniform light intensity across the beam diameter,
2r He concluded that waves of measurable length, A, satisfy the
relation, A > 6.8r Palm (1975) performed a more complex analysis for
the present instrument, assuming the laser beam to have a Gaussian
distribution of light intensity across its diameter. In this case the
2
distance, r is the radius at which the intensity reaches 1/e of its
maximum value. The calculated error in the slope measurement resulting
from the finite size and Gaussian intensity distribution of the laser
beam is shown in Figure 8, as a function of the actual maximum slope and
beam radius normalized by water wavelength. The beam radius to the I/e
points is 0.4 mm. Therefore, if a wave of the highest measureable pepk
slope of 350 is passing through the laser spot. Figure 8 would predict
a 10% measurement error if the length of the wave is 2.4 mm. Figure 8
shows that this error decreases as ) increases, increases rapidly as A
decreases, and decreases if th>o slope of a wave with a given r /\ ratio
o
is decreased.
The frequency response of the electronics was estimated by chopping
the laser beam input to the receiver. The response was fund to be flat
to above 400 Hz.
The calibration of the instrument was carried out in an optics
laboratory. The laser source and instrument housing were lined up at
right angles to each other, such that their axes lay in the same hori
zcntal plane. Light from the laser was reflected by a rotatable prism
into the receiver. Rotation of the prism caused the laser beam to sweep
3.6
0.05
0.10 0.15
ro/x
0.20
025
Error introduced into the slope measurement due to the
finite size of the laser beam (Palm, 1975).
Figure 8.
across the objective lens along the horizontal axis. The device output
was recorded in deflection angle increments of 20. To ascertain the
response of the device over its entire active area the instrument hous
ing was rotated and the deflection angle sweep was performed along the
new azimuth. The result is a calibration net, as shown in Figure 9
(Palm, 1975).
In this study the xaxis of the receiver is aligned with the longi
tudinal axis of the wave tank. To measure the xcomponent of surface
slope only a one dimensional calibration, in this case the xaxis cali
bration, is required. Combining the instrument calibration of deflection
angle versus voltage output with the conversion to slope angle of equa
tion (320) gives the calibration result of Figure 10. It was found
that the slope could be calculated with more precision and less concern
about thermal drift if the analog divider was bypassed and the computa
tion performed by the computer. Figure 10 applies to the situation when
computer computation of the quotient was done.
The measurement system is installed roughly at the center of the
windwave channel at a fetch of 7.3 m. Figure 11 schematically illus
trates the orientation of the measurement system components in the
cross section of the wave tank. The laser is a Coherent Radiation
Model 802S, 2 mw, 12 volt unit. It is submerged and held in place by a
pipe mounting so that its highest point above the bottom is 25 cm below
the still water level. The beam is aligned with the vertical by ad
justing the clamping bolts that penetrate the wall of the pipe. The
receiver is supported by a gimbal mount, rigidly suspended from the top
of the tank, such that the x measurement axis coincides with the longi
tudinal axis of the tank. The center cf the objective lens is placed
directly over the laser beam with the use of a template and lowered to a
Lines of Constont 80
Static response of the instrument receiver to deflection
and azimuth changes (Palm, 1975).
Figure 9.
a~150
40 T Surface Slope Angle (degrees)
30
20
Slope = 36.47Volts
Instrument Output Signal
(Volts)
X channel instrument callbraticn curve.
10
30
40
1.5
1.0
Figure 10.
Looking in the upwind direction
Figure 11.
Schematic cross section of the wave tank at the
instrument installation site.
height of 8 cm above the still water surface. Alignments with the verti
cal is achieved by rotating the receiver about the gimbal pivots until
the beam passing through the center of the objective lens strikes the
center of the photodiode, giving a zero signal on both channel outputs.
Figure 12 shows the slope detector receiver, counted and emplac..d
in the wave tank. Figure 13 shows the physical relationship betwec:i h.
submerged laser and the end of the receiver.
The optical receiver was placed close to the water surface to allow
high angle measurements. At reference wind speeds exceeding 11.9 m/s
wave slopes exceeding the 350 measurable maximum at the instrument
height of 8 cm were not uncommon. It is reasonable to assume that the
instrument produced some alteration of the air flow that was related to
the instantaneous height of the water surface directly below it. The
short waves remained in the region of most severe air flow modification
at the long wave crest (taken as 20 cm in length), due to the reduced
distance between receiver and water surface, for 0.1 to 0.2 s. It is
felt that changes in the short waves brought about by the altered wind
field in this period of time would be small. Visually no effects due to
the presence of the receivers could be seen.
Slope measurements were made, prior to the development of the
device described here, with a slope detector that had a limited slope
range, but was located 35 cm from the still water surface. Slope
intensity variations calculated from time series obtained from the
35 cm height had maxima located at the long wave crest and forward face.
The similar phase angles obtained from these measurements with two
different devices leads to the conclusion that the high slope intensities
at the long wave crest are not induced by the modification of the air
flow created by the presence of the instrument.
wm_ __ ___
gC~'~ ~~ CI:~_: ;T_~l~s3~~IftW~
Figure 12.
The slope measurement receiver in operating
position in the wave tank.
I **;P;xiai~~ l~;r~ I
N\
Figure 13. Side view of the installed relationship
between the system light source and receiver.
III.D. Supporting Measurements
The wind speed at the reference location was measured with a Pitot
static probe and a Pace differential pressure transducer. The flow
through the fan was set according to the velocity figure obtained at the
reference location.
A capacitance wire wave gauge was installed alongside the slope
detector, about 20 cm laterally from and 10 cm downwind from the point
of laser beam surface penetration. The dynamic response of the wave
gauge is such that it is not able to respond to waves of high frequency.
Its sole function was to monitor the mechanically generated long wave,
so that the phase of the slope measurement relative to the long wave
could be determined. The wave gauge static calibration curve obtained
during experiment three is shown in Figure 14. The figure shevw the
calibrations obtained before and after the experiment. The DC shift
between the two is a result of electrical drift and water level change.
The final calibration was selected for use, although either would have
been satisfactory because the DC level is removed in the analysis.
Three local wind velocity profiles were made at reference wind
speeds of 4.4, 8.3, and 11.9 m/s. The leading tip of the Pitotstatic
probe was located 71 cm upstream of the most forward portion of the
slope detector receiver, and traversed the wind section from about 1.0
to 54.0 cm above the still water level. A large diameter United Sensors
probe was used in conjunction with a type 1014A Datametrics electronic
manometer. The measurement transducer was a Barocel unit of 10 mm Hg
full scale range. The DC output of the manometer, representing the
local dynamic pressure, was electronically time averaged over a 20 s
interval to obtain a reliable estimate of its mean value. The wind
6 T Wove Height (cm)
.Final Calibration
First
Calibration
0.9 0.6 03 N
2
Final Calibration
Height (cm)= 6395X
Wave Gauge Output (Volts)
Wave Gouge Output (Volts)
The wave gauge calibration for Experiment 3.
Figure 14.
section was completely enclosed from the fan to 1.2 m upstream of the
local wind measurement. From that point and on downstream the tank top
was left open. Figure 15 shows the three velocity profiles obtained.
The friction velocity, U was calculated in the usual way from the
profile data by assuming that they have a logarithmic distribution with
height. This assumption seems reasonable based on data in the lower
levels sho.nm in Figure 15.
100.0
10.0
I
3

0
(>
U)
1.0
0 L
4.4 m/s
..**
8.3 m/s
11.9m/s
L_ I  I I I
.I I  i 
S2 3 4 5
Mean Wind Speed (m/s)
Figure 15. Three local wind velocity profiles. Reference wind
b) 8.3 m/s, c) 11.9 m/s.
speeds: a) 4.4 m/s,
Uref
ni
. .
II.E. ExperimenLal Conditions
Several experiments utilizing the laseroptical system were con
ducted in the windwave tank. The data to be discussed presently were
all obtained in the nine tests of experiment number three (E3). Each
test corresponded to a different set of conditions as detailed in
Table I, the experimental test conditions. The tests are numbered T13
through T21 and are completely identified as E3T!3 through E3T21. The
basic long wave parameters of height and period were the same for each
test. To aid in comparison with results of other investigations, the
wind speeds are reported as reference value. local value at the measure
ment site, and friction velocity value at the measurement site. For the
tests with reference wind speeds other than 4.4, 8.3, and 11.9 m/s the
local velocities were not measured, but were projected from those values
that were actually measured.
Prior to the start of the experiment, the wave tank walls and
bottom were cleaned. For five hours immediately before the experiment
the surface layer in the water was skimmed by an overflow weir. These
measures were designed to minimize surface contamination. During the
experiment, a light, oily film appeared on the water surface. The oil
source is unknown, but it must have been either leached from an accumu
lation on the wave absorbing beach material or washed from the hydraulic
piston of the wavemaker. Since the formation of a surface slick was
anticipated, surface water was sampled during the experiment and the
surface tension measured with a ring tensiometer. Before the experiment
began the surface tension value was 68.8 dynes/cm. The water sample
obtained during the experiment indicated a surface tension value of
70.6 dynes/cm.
TABLE I
EXPERIMENTAL TEST CONDITIONS
Experiment Reference Local Wind U, at Wave
and Test Wind at 7.3 m Fetch 7.3 m Fetch Period Height
Designation (m/s) (m/s) (m/s) (s) (cm)
E3T13 3.8 2.7 0.099 2.0 10.0
E3T14 4.4 3.4 C.110 2.0 10.0
E3T15 4.9 3.5 0.127 2.0 10.0
E3T16 5.6 4.0 0.145 2.0 10.0
E3T17 5.9 4.2 0.153 2.0 10.0
E3T18 7.2 5.1 0.187 2.0 10.0
E3T19 8.3 5.7 0.223 2.0 10.0
E3T20 9.2 6.5 0.239 2.0 10.0
E3T21 10.1 7.2 0.262 2.0 10.0
DATE: Sept. 18, 1975
TLIE: 12:00 p.m.
Before testing began the laseroptical system and wave gauge were
installed in the windwave channel at a fetch of 7.3 m. The laser beam
was aligned with the vertical and the optical detector was aligned with
the laser beam, allowing the use of the calibration data obtained in the
optics latoratorv. The wave gauge was calibrated by changing its posi
tion along the vertical axis and by physically and electrically measur
ing the changes.
After alignment, warm up, and calibration the test was begun by
setting the long wave parameters. The mechanical, wave generator was
set in motion during the no wind condition in the windwave tank. The
wave form produced by the wave generator was specified at the input of
the mechanicalhydraulic system by a function generator. The input
function .:as a sinusoid. Thus, the wave period was set by control of
the period of the input sinusoid. The wave amplitude was varied by
control of the amplitude of the input sinusoid, and set according to the
direct measurement of its height in the windwave tank. The long wave
profile for all of the tests was a slightly irregular sinusoid of 2.0 s
period and 10.0 cm height.
The fan was then turned on to provide a wind field for the develop
ment of the wind driven portion of the wave spectrum. The wind condition
was started at the lowest value and increased for each succeeding test
to the maximum. Each individual test required a minimum of 15 minutes
to complete. Five minutes were used for the recording of the data sig
nals. The remainder of the time was allowed to permit the water motion
to reach a steady state at each wind speed.
The testing procedure was halted after approximately every hour to
check water level, surface condition, and instrument drift. In each
73
case the laser beam was turned off and the instrument amplifier outputs
rezeroed for the no light condition. The DC drifts generally experi
enced were within 40 mv, representing somewhat less than 5% of the full
scale slope.
III.F. Digital Data Acquisition and Reduction
The data signals were recorded in the FM mode on a Hewlett Packard
model 3960A instrumentation tape recorder. According to Hewlett Packard,
the tape deck and cape combination had a frequency response of 0 to
312 Hz with a variation of less than +1.0 db within that range. The
signal to noise ratio was 45 db and the peak to peak flutter was 0.70%
of full scale.
The recorded data signals were sampled at a rate of 400 Hz without
prior filtering, and rerecorded in digital format on magnetic tape. The
three signals recorded for digital processing were the water surface
displacement, and the xchannel sum and difference outputs of the slope
detector. The three channels of data were sampled essentially at the
same instant to maintain their parallel relationship to one another in
time.
The digitized data were used in two different analysis schemes.
The first, to be discussed below, yields the experimental short wave
modulation figures. The second, to be discussed in Chapter VI, yields
the first order spectral parameters required to calculate modulation by
way of the Hamiltonian formulation of the problem.
The purpose of this analysis is to produce a stable estimate of
short wave slope intensity at selected regions of the long wave profile.
In doing this the technique of ensemble averaging is applied co specific
subsets of the set of data epics available.
For every test condition data from 133 cycles of the long wave are
available for data processing. For each of the long wave cycles the
positive going wave height zero crossing is located. Starting at each
zero crossing, the slope time series is divided into eight overlapping
segments, as shown in Figure 16. Each segment spans 200 data points and
overlaps 100 points of the previous segment. Each segment may be thought
of as an individual member of several different ensembles of data epics.
For example, two of the ensembles, those of every epic and of every
other epic, can be operated on to yield good estimates of the first
order spectrum of the short wave slope record. If the ensemble is taken
as being composed of all the segments appearing at the same phase loca
tion of a long wave profile (for example, all segr.ents numbered one in
Figure 16) it can be operated on to yield an estimate of the phase
averaged spectrum of the slope record. The phase averaged spectral
estimate calculated for each segment is assumed to be representative of
the "typical" spectrum associated with a point central to the segment on
any given long wave cycle.
The slope intensity spectrum for each data segment results through
the application of a Fast Fourier Transform (Robinson, 1967) routine.
Before Fourier transformation a cosine bell is applied to the first and
last 10% of che data points in each segment and 56 zero value points are
added to each segment. The FFT is then performed on the eight segments
in the long wave cycle. Without further smoothing the slope intensity
spectra are calculated for each of the transformed segments and retained
in memory for subsequent averaging with the 132 additional spectra to be
calculated that have the same phase relationship co the long wave height
signal.
The slope time series measured at the low wind speeds of 3.8 m/s to
5.9 m/s contain a slope component due to the long wave that appears to
be significant relative to the wind driven wave slopes. The low fre
quency component appears as a trend in the segmented data. The low
resolution of the spectral calculation and the large amount of energy
Wind and Wove
SDirection
IN~y
Figure 16. Wave record segmenting scheme used to produce short wave slope epics for phase
averaging.
r
placed at low frequency by the trend create concern that the low fre
quency energy may spread to the higher frequency portion of the spectrum.
To alleviate this difficulty the trend is removed by fitting a third
degree polynomial to the data of each time series segment and subtracting
it point by point from the original series. Figure 17 illustrates the
effect on the time series of the trend removal operation for a typical
segment of data. For low wind speeds the constructed polynomials fit
the observed trends very well. At higher wind speeds the polynomial fie
becomes erratic because it tends to follow the larger and longer period
excursions in slope related to the wind driven wave system. Figure 18
illustrates the effect that trend removal has on the spectrum. For the
conditions considered the figure shows that the long wave energy did not
spread significantly into the high frequency region of most interest
(above 9 Hz). However, good practice and the cases where the high
frequency regions were affected to a greater extent dictated that the
trend removal routine be applied to the low wind speed data. At higher
wind speeds the low frequency trend assumes much less importance rela
tive to the short wave slope values, so trend removal was not attempted
for reference wind speeds above 5.9 m/s.
Simply stated, the results of the analysis for each test condition
yield eight phase averaged slope spectral estimates, each attributed to
a phase location along the wave profile as in Figure 16. The 256 point
spectra have a frequency axis resolution of 1.56 Hz. The window func
tion, W(j), has the value 1.0 for the central 160 points of actual slope
data in each segment and assumes the slope of half of a cosine bell for
the data points 1 to 20 and 181 to 200. The spectral intensity esti
mates, I (n), for the k time series segments, X (j), of length, L, are
corrected for the reduction in energy due to the window by a formulation
Figure 17.
Visualization of a typical trend removal sequence. Reference
wind speed: 4.9 m/s. Long wave: T = 2.0 s, H = 10.0 cm.
a) Appearance of the front face total slope time series
segment.
b) The corresponding trend by third degree polynomial fit.
c) Appearance of the high frequency slope time series after
trend removal.
I%,,, i /" i ., \ ) \ I
0.30
I _
0.30
0.30
Time (s)
0.00
0.25
S.10
0.IO
0.20
(a )
0.40
0.50
0.25 T
0.00
0.25
0.10
0.20
0.40
i t(b)
0.50
0.25r
0.00
V \J
0.10
0.25 1
0.20
" i (c)
0.50
0.40
 I r __

I
 , I~ I l
i '
0.25 T
, v
I v
Figure 18.
Phase averaged slope energy spectral estimate at the front
face of the long wave crest. Reference wind speed: 4.9 m/s.
Long wave: T = 2.0 s, H = 10.0 cm.
a) Without trend removal  .
b) With trend removal 
0
C
CI
O
d)
C
0
CF
05
I 0 I
10 10 100
Frequency (Hz)
suggested by Welch (1967) in his important paper on the FTT. The finite
Fourier transform of each of the k segments is given by
1 L1
LI 2i (nkj/L)
Ak(n) = ,(j) W(j) e (322)
L
j=1
where L equals the 200 points of real data and i = r1. The estimate of
the energy spectrum for the kth segment is then given by
Ak (n)2
Ik k (= (323)
U
where U is given as
Ll
U = (j) (324)
L
j=l
Hence, the effect of the cosine tapered window to reduce the spectral
intensity values calculated by the FT, as in equation (322), is
normalized out by dividing each spectral point by the mean square value
of the window, equation (324). For the computations here two windows
were utilized. The 200 point window described above has a mean square
value, U = 0.880. The 4064 point window to be described in Chapter VI
has a mean square value, U = 0.938.
The stability of the estimate based on the ensemble of 133 inde
pendent sampled epics from a Gaussian process is specified in terms of
the number of equivalent degrees of freedom (EDF) of the approximating
chisquare distribution (Welch, 1967). For the short wave slope in
tensity estimates described above we have
EDF = 266 .(325)
This value holds for all but the end points of the spectrum, and implies
a very stable spectral prediction.
CHAPTER IV
EXPERIMENTAL RESULTS
IV.A. Time Series Data
The time series data displayed in Figures 19 through 25 illustrate
in a literal way the effects of the long wave motion on the short wave
field at the nine different reference wind speeds from 3.8 m/s to
10.1 m/s. For the four lowest wind speeds it is seen from Figures 19
and 20 that the slope of the long wave represents a large portion of the
total slope value measured at any instant. At a reference wind speed
slightly larger than 5.0 m/s the intensity of the wind waves increases
significantly. This is indicated in Figure 20, when the wind speed
increases from 4.9 m/s to 5.6 m/s, by the increase in peakedness of the
slope time series from parts a) to b).
At the reference wind speed of 5.6 m/s the effect of the presence
of the long wave on the short wave field becomes manifest in the short
wave time series. The resultant changes in the short wave field are
most clearly visible in Figures 20b through 23. The long wave short
wave field interdependence results in two distinguishable effects on the
short wave field characteristics. Firstly, if one thinks of a curve
lying, as an envelope along the outer edges of the slope time series,
containing the slope maxima and minima; it is clearly seen from Fig
ures 20b through 23 that the envelope width is not constant and is in
fact related to the long wave phase. Secondly, the frequency range
Figure 19. Wave slope and height time records.
a) Wave slope for reference wind speed = 3.8 m/s.
b) Wave slope for reference wind speed = 4.4 m/s.
c) Wave height: T = 2.0 s, H = 10.0 cm.
4o
211 25 1.2 5.00e 
WV S
7
)
^\ >
r^
;'
Ja 
<. LL '$ L
1
4;.
2 )
3~ i.OO
WAVE SLOPE (RAO)
r
I
*
V
' If
o
I.
K
I ,=
I
WAVE HEIGHT (CM)
Figure 20. Wave slope and height time records.
a) Wave slope for reference wind speed = 4.9 m/s.
b) Wave slope for reference wind speed = 5.6 m/s.
c) Wave height: T = 2.0 s, H = 10.0 cm.
S
Y
.o
I z
WAvE SLOP:E ( AO)
< "
, _
f
,,
?'
r L *3 u^ *1.i5 ci 
WAVE SLOPE (rA)~0
* J
I

w
u
z
0
IL
I
',: 25 '5.'. '
a JD 3.00
WAVE HEIGHT ICM)
Figure 21. Wave slope and height time records.
a) Wave slope for reference wind speed = 5.9 m/s.
b) Wave height: T = 2.0 s, H = 10.0 cm.
zzczL
. .
0.50 0.25 0.' .2 . i' 0 .
WAVE SLOPE (1R40) WAVE HEIGHT (CM)
Oil
WAVE SLOPE (RO AEHEGT(
