Title Page
 Table of Contents
 List of Figures
 Investigations of waves and associated...
 Measurement of wave slope
 Experimental results
 Theories of interaction between...
 Comparison between experimental...
 Appendix I: The Boltzmann transport...
 Appendix II: The wave-wave interaction...
 Biographical sketch

Title: Modulation of wind generated waves by long gravity waves
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00097513/00001
 Material Information
Title: Modulation of wind generated waves by long gravity waves
Physical Description: xi, 196 leaves : ill., diagrs., graphs ; 28 cm.
Language: English
Creator: Reece, Allan MacDonald, 1947-
Publication Date: 1976
Copyright Date: 1976
Subject: Wave mechanics   ( lcsh )
Ocean waves   ( lcsh )
Gravity waves   ( lcsh )
Ocean-atmosphere interaction   ( lcsh )
Civil Engineering thesis Ph. D
Dissertations, Academic -- Civil Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Statement of Responsibility: by Allan M. Reece, Jr.
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 192-195.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097513
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000187963
oclc - 03418215
notis - AAV4567


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Table of Contents
    Title Page
        Page i
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Figures
        Page v
        Page vi
        Page vii
        Page viii
        Page ix
        Page x
        Page xi
        Page 1
        Page 2
        Page 3
        Page 4
    Investigations of waves and associated energy transfers
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
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        Page 18
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        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
    Measurement of wave slope
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
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        Page 49
        Page 50
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        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
    Experimental results
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
    Theories of interaction between short waves and long waves
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
    Comparison between experimental and theoretical results
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
        Page 144
        Page 145
        Page 146
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        Page 148
        Page 149
        Page 150
        Page 151
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        Page 156
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        Page 176
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        Page 178
        Page 179
        Page 180
        Page 181
        Page 182
    Appendix I: The Boltzmann transport equation
        Page 183
        Page 184
        Page 185
        Page 186
        Page 187
    Appendix II: The wave-wave interaction integral
        Page 188
        Page 189
        Page 190
        Page 191
        Page 192
        Page 193
        Page 194
        Page 195
    Biographical sketch
        Page 196
        Page 197
        Page 198
        Page 199
Full Text

MODULMiLc!:" OF '[.-T. C;..:7...-\TErD WAVE.
.."'! C--L''VITY ,.V'ES


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 .


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 Oce-o.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 profess-onal 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..sc-rrt ticn 'as written.

This work was sponsored by NOAA Space Oceanography Prograrn I: i'..r

Crant NG-29-72 anu by the Jet Propulsion laboratoryy under Contract




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
Miles-Phillips 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 Capillary-Gra.vicy
Wa.ves . . . . . . . . ... . . . 26
5. Hasselmann Fifth Order Nonlinear Inrtrchange . . .. 28
6. Valenzuela Capillary-Gravity 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 Wind-Wave Facility . . . . . . . . .. 44
C. Laser-Optical 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)


Chapter V. Theories of Interaction Between Short Waves and
Long Waves . . . . . . . . . . . 117

A. The Two-Scale Model of Longuet-Higgins and Stewart . . 118
B. The Solution of the Two-Scale Wave Model by Direct
Integration of the Wave Energy Equation. . . . . .122
C. The Solution of the Two-Scale 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 Wave-Wave Interaction Integral . . . . .

List of References . . . . . . . . ... .. . . 192

Biographical Sketch . . . . . . . . ... . .. 196


Figure Page

1 Plan and side views of the wind-wave tank model . . .. 45

2 Wave generator section and test section of the wind-wave
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)




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
tc-r 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 . . . . . . . .


















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





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







. 174

. 176


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



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

re-ion 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 wave-wave 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


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 wave-wave 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) Wave-wave 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.



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 s-Iort. ;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 gre-t 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 developz-d 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 a-sociaLte '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


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. Lonquet-Higgins (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]o-Christensen

(].972) made measurements of wave response to wind gusts, indicating that

ther-e '.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.



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 inte-face

are capable of passing on energy in such a manner that irrotational

motioni results. Irrotational analyses have been sho.'n to yield consid-

erablt s-irplifications 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


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 (2-1)
at Pw Pw


S= velocity potential of water motion,

p = water surface displacement,

p = surface pressure,

T = surface tension,


p = water density.

Fourier transformation nicely converts the equation, (2-1), to a differ-

ential equation in terms of the transformed variables. The variables

are written in equations (2-2), (2-3), and (2-4) in terms of their

generalized Fourier transforms in space. The equations are

n(x,t) = A(k,t) ei ( k xdk (2-2)

p(x,c) = j P(kt) ei(k )dk (2-3)


*(x,z,t) =- e 9- dk (2-4)

where a prime indicates a time derivative. The resultant differential

equation, is in a fomn describing a system in forced oscillation

A"(k,t) + -A(k,t) P(k,t) (2-5)

where o is given by the familiar dispersion relation for free surface

waves with surface tension included,

o = gk + .(2-6)

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 (2-7)
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 (2-7), 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

(k;t) = F(k,o) ,(2-8)
4p a

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 (2-9)

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 (2-9) 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 (2-10)
dt a z


0 = air density,

U = mean horizontal air flow velocity,

u = wave induced contribution to the horizontal air flow


wt = wave induced contribution to the vertical air flow

The Reynolds stress is evaluated as in instability theory. For the

inviscid parallel flow assumption, the value is approximated by the


a -

p u d = -- V'
a k U/dz I z = z

for the region z < z and

p UW = 0 (2-11)

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 (2-11) into (2-10) 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 (2-10) is

dE2 2 2
dE rp cW U/ 9z
-, (2-12)
dt k @U/ z z

which needs to be evaluated at z only, and where

c = water wave phase speed,

k = water wave wavenumber,


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 (2-12) and

the definition

1 dE/dt
-, (2-13)
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 Orr-Sommerfeld 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 (2-5), which was formu-

lated for the resonance model, to include the wave induced pressure

forcing function. Equation (2-5) becomes of the form of equation


I. k
A"(k,t) + o-A(k,t) = -- [P0(k,t) + Pl(k,t)] (2-14)

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

Po(x,t) = Po(k;t) ei(k x) dk


p (x,t) = PI k; t) e x) dk

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) (2-15)

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

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 (2-15) can be written as

-P = aA'(k,t)A'*(k,t) = an (2-16)

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 (2-16) to Miles' original result in equation

(2-12), where only wave induced pressure was considered. Solving the

relationship for a yields

Tp cW2 2U/3z2
k @U/@z
a -c (2-17)

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 (2-18)

where t is defined by equation (2-13). Then the equation of motion,

(2-14), can be written as

2 k
A"(k,t) + CoA'(k,t) + a A(k,t) = - P(k,t) (2-19)
.- 0

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) (2-20)
4p a0 o

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 (2-20) is

eot- 1 ot2
= t + --- + ... (2-21)
LO 2

which yields the two following time dependencies of

L. t
*.(k;t) t for -- < 1

'(k;t) a e for 1

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 (2-22)

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

e-Sentially 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


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 t-T rs up through

second order in wave height. The ki--ematic 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 (2-23)
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 (2-24)

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) (2-25)
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) (2-26)
;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, ,


n = n( + n(2 (2-27)

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 (2-28)
zt tt t ttz

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' (2-29)
g f


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 (2-29) indicates the

nonlinear effect quite clearly. Since the expression for 1 ()() is j

convolution operation, we see that for a continuous function, (1)(o),
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 (2-30)

the bottom boundary condition

7q.(x,z,c) = 0 at z - (2-31)

the kinematic free surface condition

S+ n 4 = 0 at z =n (2-32)
t X. z

and the dynamic free surface condition

Sg + + + 4.) = 0 at z = n (2-33)
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

(2-30) to (2-33). 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 (2-34)




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 (2-36)

n = N + en (2-37)

Substitution of equations (2-36) and (2-37) into the boundary value

problem equations, yields the new boundary value problem specified as

the Laplace equation

V2 (x,z,t) = 0 (2-38)

the bottom boundary condition

Vi = 0 at z (2-39)

the kinematic free surface condition

nI + n x + n(-zz + N xz) + (-4 + N ) = 0 at z = N
c xx zz xz z xx

. . . . . . (2-40)

the dynamic free surface condition

gn n( -xXz + z + tz) + (4 + + 4 ) = 0 at z = N

. . . . . . (2-41)

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 (2-42)

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.] (2-43)

where b. are the side band amplitudes, and the phases are

X1 = k(l + c) x o(] + 6) t y


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 (2-44) 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 .(2-44)

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 x-y 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


g -- + + V 7 V- = 0 at z = (2-45)
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 nccc-ssiy 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 (2-66)

S= (cV + ) + (a, + ( + a11 + BT, ) .... (2-47)
S(10 01) + 20 + 11 02


n = (anir + fn01) + (an20 + a.*iiri + n2 n,) + .... (2-48)

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 (2-49)

alo1 klz
e0 = e in v (2-50)

n01 = a2 cos X2 (2-51)


a202 k2z
01 =--- e sin X2 (2-52)

where the phase function,

X. = k. x .t
i i i

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
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
.............. (2-53)
. . . . . . . (2-53)

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 (2-53).

The behavior of 21' as the solution to equation (2-53), indicates

tch behavior of a wave of frequency, 2o 2, which can receive energy
.. 1 2'
from i-he 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 (2-53) 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
n, (xy,t) -- sin (2', X) (2-54)

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 (2-55)
--1 3 3- -


c + 0 + o + 0 = 0 (2-56)
1- 3 H

where each wave obeys its own dispersion relationship,

oi = g'kil (2-57)

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 Lorguet-Higgins 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 c-ok 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

z = n(x,y,t) = ni(x,y,t)

The combined free surface boundary condition defines the problem, as

usual. The equation below is the same as equation (2-45) 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 = (2-58)
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 (2-58),

and the bottom boundary condition

V(x,y,z,t) = 0 at z = -" (2-59)

is wricten

3 k.z
(x,y,z,) a.(t) e sin (2-60)
1 1
i=l i


i = k.x c.t + c.

is the phase fu'.,ction.

The corresponding c:-pression for the surface is


(x,y,t) = ai(t) cos (2-61)

Equations (2-60) and (2-61) 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 (2-62)
-1 2 -3


1 + 2 = 3 (2-63)

where each wave satisfies

2 3
o = gki + ki (2-64)

It is seen that if the k corresponding to k + k2 produces a o3,

calculated by equation (2-64), that equals o + a resonance will


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 inte-mediate 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


4(x,y,z,t) = A(k;x,y,z,t) e kzei x) dk (2-65)


Si(k x)
r(x,y,t) = B(k;.,y,t) e dk .(2-66)

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 (2-55)

through (2-57). Hasselmann (1962) found that an interaction does occur


k, +k = k + k (2-67)
-a 2 3 -4n


01 + 02 = 03 + C4


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

01 2 34) 6(04 + 03 2 a0) dkx ddk dkdk (2-69)

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 + (2-70)


k = k k and a = 0 ao
-3 -2 -1 3 2 1


where each wave satisfies the dispersion relation. The resultant inter-

action is expressed in terms of the products of two first order spectra


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

. . . . . . . (2-72)

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 wind-wave 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 wind-wave

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 wind-wave 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 wind-wave 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 wind-wave

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

ave-rageid, 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

wind-wave 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-


w(k,x,t) = w (k,x,t) + k U(x,t) (2-73)

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,

(2-73), somewhat more closely than the wind ripple. Frequency maxima

occurred at long wave crests, while wavenumber changes along the long

wave profile were insignificant.



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) e-i(k O) dkd (3-1)


d-do k (3-2)
n(x,t) = -i kA(k,) e dkdo (3-2)

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 (3-3)


t) =-i ff kA(k,o) etdkdo (3-4)

The temporal correlation functions for the wave height and slope

time series may then be constructed by application of the expressions

(3-3) and (3-4). The correlation funccicns are defined as (Lee, 1960)

n n(T) f n(t) n(t + r) dt (3-5)


4n (T) = f ri(t) r.(t + ) dt (3-6)

Intrc-ducLJon of the transforms of the time series from equations (3-3)

andLi (3.-) fields s

l (r0) = f A(k,I) eiL~tdkdo

A(k,o) e( + dkdo dt (3-7)


1 (T) = F kA(!c,o) e Ptdkdo
x ;. I/ L -i ff

S -1 kA(k,a) e i(t + T) dkdo dt (3-8)

Since the surface being described in equations (3-3) and (3-4) 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 (3-7) and (3-8) is

in(T) = j [A(k,a) A*(k,o) ei] e d dkdo ,

. . . . . . . (3-9)


S(T) = i [kA(k,a)] [kA(k,o)] e

f ei(a o')t dt dkdo ,(3-10)

where the terms are regrouped to allow the time int--gral to srand alone.

T'e L i-:c integral of equations (3-9) and (3-10) can be repr--senr.ed by a

unit i[.pu]se function (.ee, 1960) where

1 F I(o o')t
6(a o') -- J dt (3-11)
2n J

Ihen f:.r equations (3-9) and (3-10) we may write


4, n(T) = 2n f A(k,o) A*(L-.) e kdo (3-12)


S(T) = 2n [kA(t! ,, )] [kA(k, )] e r dkdo (3-13)

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 (3-12) and (3-13) are written as

nn( ) = 2n I|A() 12 6[k f(a)] dk ei do (3-14)

S2 () = 2n ( kA(o)) 5[k f(,)] dk e do (3-15)
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 (3-16)


(1) = 2 k() 2 e do (3-17)
nxix 2

where k is restricted to those values given by the wave dispersion


By definition the spectral density functions, i- (o) and Cn n (a),
are the coefficients of the exponential term in the integrands of equa-

tions (3-16) and (3-17), respectively (Lee, 1960). It is then clear


n n (a) knn (a)

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


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 Wind-Wave Facility

The measurements required for the present study were conducted in

the wind-wave facility at the University of Florida. Figure 1 illus-

trates the general size and shape of the wind-wave 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


Shewdin (1969) described the wind-wave 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.


bUL 1 .EAD
Side View
nh.riiicSIO5 ins rin lel

r. 4 _AS S I_ f
-- * .-- - - _-- __ --- --- -

I IaLO T 1 \ bASIN
,F' _

I .
_.WIND .._..__________..;_
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 wind-wave tank model.



L 7.3m

F~L 2m-


.-Pitot Tube




S-Flow Conditioner

*- Air Flow



I p p

Figure 2. Wave generator section and test section of the wind-wave model.


n r- -

111.C. Laser-Optical 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 laser-optical system which operates on the principle of optical

refraction at the air-water 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 to-be 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 air-water 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
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


Path of laser beam
in air

/ Local Surface Normal

: .: .

/' r
..* .. . .
"" ... .:'i... :. ::: :::.; I "O i .

" : : .' ".

---- -Laser beum path
. *. m water

Figure 3. Orientation diagram for laser beam refraction at the
air water interface.



refracted beam away from the true vertical axis as

C(t) = C (t) O.(t) (3-18)
r 1

Using Snell's Law in equation (3-18), one can express the deflection

angle in terms of the angle of incidence

4((t) = sin1 [n sin C'.(t)] C.(t) (3-19)

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 (3-19) 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 -, (3-20)
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


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.



30-- eTcn s
3n n-cos <

n = 1.333

< 20-
0. 0



0 5 10 !5 20 25 30
SDeflection Angle (..o)

imaging Lens -

Objective Lens

---Detector Diode

"---Oiffus;ng Screen


. . .. . . .. '. .. %. ..
::.. : :: :.. .... .~i .... ...e
. ... "' .. .. .:.::::'" . : "" :.. .:. ..'.....' : :

:}:~i::::ii:ii iN ~iii~ i i % i i ii :!: : :: :: i ri : ii!j:!:.: :. i ::: i :::: : ii
. . . .. ... .. ... . ... ... : . :
.... ..... ...... ... ... ... y : .~ c~r
... ............... ......i
. . . . . . . . . . . . .

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 po-er

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 - (3-21)

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






a 5--


Figure 6.


20 V



10 20 30


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
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


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 wind-wave 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 y-axis
divider output

--- -#4 amp sum amp


3 photodiode

*-3aiap *--- diff amp

analog -x-axis
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

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

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



0.10 0.15



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 x-axis of the receiver is aligned with the longi-

tudinal axis of the wave tank. To measure the x-component of surface

slope only a one dimensional calibration, in this case the x-axis cali-

bration, is required. Combining the instrument calibration of deflection

angle versus voltage output with the conversion to slope angle of equa-

tion (3-20) 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

wind-wave 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 80-2S, 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.


40 T Surface Slope Angle (degrees)



Slope = 36.47-Volts

Instrument Output Signal

X channel instrument callbraticn curve.






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


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 Pitot-static

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


-0.9 -0.6 -03 N


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.






0- L

4.4 m/s


8.3 m/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,



. .

II.E. ExperimenLal Conditions

Several experiments utilizing the laser-optical system were con-

ducted in the wind-wave 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.



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 laser-optical system and wave gauge were

installed in the wind-wave 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 wind-wave tank. The

wave form produced by the wave generator was specified at the input of

the mechanical-hydraulic 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 wind-wave 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


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 x-channel 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


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


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


Figure 16. Wave record segmenting scheme used to produce short wave slope epics for phase


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
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


I -_



Time (s)





(a )



0.25 T






-i t(b)




V \J


-0.25 1


"- i (c)



- I r __



- ,- 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 -







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 L-1
L-I -2i (nkj/L)
Ak(n) = ,(j) W(j) e (3-22)

where L equals the 200 points of real data and i = r-1. The estimate of

the energy spectrum for the kth segment is then given by

Ak (n)|2
Ik k (= (3-23)

where U is given as

U = (j) (3-24)

Hence, the effect of the cosine tapered window to reduce the spectral

intensity values calculated by the FT, as in equation (3-22), is

normalized out by dividing each spectral point by the mean square value

of the window, equation (3-24). 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

chi-square distribution (Welch, 1967). For the short wave slope in-

tensity estimates described above we have

EDF = 266 .(3-25)

This value holds for all but the end points of the spectrum, and implies

a very stable spectral prediction.



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.


211 25 1.2 5.00e -


^\ >

Ja --
<. LL- '$ -L



-2 )

3~ i.OO





-'- -If


I ,=


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.



I z

<- -"

, _




r L- *3 u^ -*1.i5 ci --


* J






-',: 25 '5.'. '

a JD -3.00


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.


.- .

0.50 0.25 0.' .2 -. i' 0 -.


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