Title: Field investigations of the short wave modulation by long waves
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Title: Field investigations of the short wave modulation by long waves
Physical Description: x, 103 leaves : ill., map ; 28 cm.
Language: English
Creator: Tang, Shih-Tsan, 1949-
Publication Date: 1981
Copyright Date: 1981
 Subjects
Subject: Surface waves   ( lcsh )
Wind waves   ( lcsh )
Engineering Sciences thesis Ph. D
Dissertations, Academic -- Engineering Sciences -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1981.
Bibliography: Bibliography: leaves 99-102.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Shih-Tsan Tang.
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Bibliographic ID: UF00099373
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 - 000297665
oclc - 08480075
notis - ABS4040

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FIELD INVESTIGATIONS OF THE SHORT WAVE
MODULATION BY LONG WAVES










by



Shih-Tsan Tang


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY




UNIVERSITY OF FLORIDA

1981







































This dissertation is dedicated to Chi-Mei.













ACKNOWLEDGEMEN(TS


I would like to express my gratitude to Professor Omar H. Shemdin

for his patient guidance, sincere encouragement and kind support

throughout the course of this study.

Thanks are due to those who participated in the field measurement.

I also wish to thank Drs. M. L. Banner, W. T. Liu and S. V. Hsiao for

their helpful discussions and encouragement during my residence at

Jet Propulsion Laboratory, California Institute of Technology.

I extend my thanks to Rena Herb and Cynthia Vey for their

excellent effort in typing the manuscript, and to Lillean Pieter for

her help in the drafting of the figures.

The work was sponsored by the Office of N~aval Research under

contract number N~00014-81-F0069 and National Aeronautics and Space

Administration under contract NAS7-100.

















TABLE OF CONTENTS


vi

ix

x


ACKNOWLEDGEMENTS . . . . . . . . . . .

LIST OF FIGURES . . . . . . . . . . . .

LIST OF TABLES . . . . . . . . . . . .

ABSTRACT . . . . . . . . . . . .

CHAPTER

I INTRODUCTION.

1.1 Prologue. ....
1.2 Scope of the Present Study

II THEORETICAL BACKGROUND.. ....... ....

2.1 Momentum Transfer.
2.2 Properties of High Frequency Waves.
2.3 Two-Scale Model. ..... .....
2.4 Relaxation Model

III FIELD EXPERIMENT .........

3.1 Field Set-Up
3.2 Surface Wave Slope Measurement
3.3 Sea Surface Displacement Measurement .....
3.4 Current Measurements
3.5 Other Measurements ..... ......
3.6 Data Acquisition and Digi~qtization ....

IV DATA ANALYSIS .. .......

4.1 Spectral Analysis .....
4.2 Directional Spectrum Estimate, ........
4.3 Determination of Surface Wave Slope .
4.4 Demodulation Analysis.

V EXPERIMENTAL RESULTS. ......

5.1 Characteristics of Wave Slope Spectra.
5.2 Modulation of Mean Square Slopes by Long Waves.
5.3 Probability Density Function of Wave Slopes
5.4 Modulation of Short Waves by Long Waves...












CHAPTER Page

VI DISCUSSION OF RESULTS . .. ... .. .. 78

6.1 Wave Slope Spectra ..... 78
6.2 Modulation of Mean Square Slopes 79
6.3 Probability Density Function .. .. . . 80
6.4 Hydrodynamic Modulation Level .. .. . ... 80

VII CONCLUSIONS AND RECOMMENDATIONS . .. .. . 86

7.1 Conclusions . . ... .. . .. .. 86
7.2 Recommendations ... .. .. .. ... .. 87

APPENDICE~S

A Algorithm for Estimating the Directional Wave Height
Spectrum .. . .. .. . .. . . . 89

B Demodulation Procedure . .. ... .. ... .. 96

REFERENCES .. . .. . .. .. .. . .. .. .. 99

BIOGRAPHICAL SKETCH . . .. - - . . . 103















LIST OF FIGURES

Figure Page

1 Schematic energy balance for the case of negligible
dissipation in the main part of the spectrum. Qin =
atmospheric input, Qng = non-linear wave-wave transfer,
Qd = dissipation, and Q = net transfer (after Hasselmann
et al., 1973). . . . . . . . . 8

2 Hydrodynamic modulation transfer function for short
gravity waves from relaxation model . ... .. . 18

3 Geographic location of the Noordwijk Tower. The contour
lines are specified in fathoms . .. .. .. .. ... 21

4 The wave follower in the wave following mode . ... .. 22

5 Close view of the optical sensor mounted on the wave
follower . .. . .. ... .. .. . . 23

6 Schematic diagram of the wave follower . ... .. .. 24

7 Plan view of the relative location of sensors used in the
wave follower experiment .... . ... .. .. .. 25

8 The definition diagram of the refracted laser beam. The
unit normal vector 6 = (siny cost, siny sing, cosv) defines
the slope vector in xl and x2 directions as
-tany (cost, sing) .. 27

9 Definition sketch of true direction. The true direction e
is the angle of Va referred to the east counterclockwise .a31

10 Sample time series measured by the electromagnetic current
meter for Run 108 . ... . ... .. . . 35

11 Sample time series for sea surface displacement and
slope for Run 108 .. .. .. .. .. .. .. 36

12 Frequency spectra for n and u for Run 108. The solid line
gives the surface displacement spectrum. The dashed line
gives the u-component of the orbital velocity spectrum , 37

13 Attenuation of the orbital velocity. u/uo is the ratio of
u-component of orbital velocity at 4.75 m below the mean
sea level to that at the sea surface .. ... .. .. 38









Figure.Pg
14 Calibration results for the deflection angle; v is the
calculated value .42

15 Calibration results for the azimuth angle; 5 is the
calculated value ..43

16 The down-wind slope spectrum for Run 108; EDF = 100 .. 44

17 Probability density function (PDF) of the sea surface slope
summarized from 40,000 points of Run 108. The open circles
and solid points denote the PDF for down-wind slope and the
cross-wind slope, respectively. The normalized slope is the
ratio of the slope to the standard deviation. The solid
curve is a Gaussian distribution with the same variance.
The variance of down-wind slope is 0.0342 and that of
cross-wind slope is 0.0310 ... 45

18 An example of the sea surface displacement spectrum for
"mixed sea" case. Run ID = 102, U, = 6.3 m/s .50

19 Sea surface displacement spectrum of Run 328, a "well-
defined peak" case. Ua = 3.5 m/s .. .... . .. 51

20 Sea surface displacement spectrum of Run 325, a "well-
defined peak" case. Ua = 7.7 m/s .. .. .. ... 52

21 The down-wind wave slope spectra .. .. .. ... 54

22 The frequency-weighted down-wind wave slope spectra .. 56

23 The normalized down-wind wave slope spectra .. .. .. 57

24 Best fitted F(U,) from the cases of "well-defined peak". 58

25 Mean square wave slope as a function of the wind speed . 60

26 The cross-correlation function of mean square down-wind
wave slope and sea surface displacement for Run 325.
The lag of the first peak in the cross-correlation
defines T . .... 62
max
27 The phase lead emax as a function of wind speed Ua '. 63
28 Peak value of the normalized cross-correlation function as
a function of wind speed. The open circles and solid points
denote pmax Values Of "well-defined peak" cases and "mixed-
sea" cases, respectively ... .... 64

29 Probability density function for Run 328. The notation used
is the same as that in figure 17; Ua = 3.5 m/s, the
variances are 0.0094 for down-wind wave slopes and 0.0101
for up-wind wave slopes .. .. .. .. . .. 65










30 Probability density function for Run 325. The notation
used is the same as that in figure 17; U, = 7.7 m/s;
the variances are 0.0159 for down-wind wave slopes and
0.0314 for cross-wind wave slopes . .. .. .. ... 66

31 Probability density function for Run 102. The notation
used is the same as that in figure 17; U, = 6.3 m/s;
the variances are 0.0098 for down-wind slope and 0.0117
for cross-wind slopes ........ 67

32 Hydrodynamic modulation level for Run 328. U, = 3.5 m/s 71

33 Hydrodynamic modulation level for Run 325. U, = 7.7 m/s 72

34 Hydrodynamic modulation level for Run 108. U, = 11.3 m/s 73

35 Hydrodynamic modulation level for 23 cm waves .. .. 74

36 Hydrodynamic modulation level for 8 cm waves .. .. 75

37 Hydrodynamic modulation level for 3 cm waves .. .. 76

38 Hydrodynamic modulation level for a "mixed-sea" case.
Run ID = 102, Ua = 6.3 m/s . .. . .. .. 77
39 Comparison of hydrodynamic modulation level and modulus of
radar modulation transfer function for Ua = 3.5 m/s .. 82
40 Comparison of hydrodynamic modulation level and modulus of
radar modulation transfer function for Ua = 7.7 m/s .. 83
41 Comparison of hydrodynamic modulation level and modulus of
radar modulation transfer function for Ua = 11.3 m/s . 84















LIST OF TABLES


Table Page

1 Summary of the Cases Selected for Detailed Data Analysis. 30

2 Observed Values of the Significant Wave Height,
Dominant Wind Direction and Tidal Current .. .. .. 46

3 Mean Square Wave Slopes and Sea Conditions .. .. .. 53

4 The Wavelength Bands for Modulation Study . ... .. 69
















Abstract of Dissertation Presented to the Graduate Council
of the Uniiversity of Florida in Partial Fulfillment
of the Requirements for the Degree
of Doctor of Philosophy

FIELD INVESTIGATIONS OF THE SHORT WAVE MODULATION
BY LONG WAVES

Shih-Tsan Tang

December 1981

Chairman: Omar H. Shemdin
Major Department: Engineering Sciences


The study of short wave modulation by long waves has been

carried out in-the field. A laser-optical sensor is used to detect

short wave slopes; a water surface displacement sensor and an

electromagnetic current-meter are used to obtain sea surface wave

directional spectra. Measured down-wind slope spectra are shown

to be wind speed-dependent; the mean square wave slopes are

generally larger than those measured by the sun glitter method.

A technique has been developed to account for the orbital

motion due to two-dimensional waves. Hydrodynamic modulation levels

are calculated for the wavelengths 3, 8 and 23 cm. A decrease in

the modulation level is found with increasing wind speed. The results

indicate that the hydrodynamic modulation is sufficiently strong at

low wind speed to constitute a major radar imaging mechanism for

ocean waves.
















CHAPTER I
INTRODUCTION



1.1 Prologue

The studies of the high frequency structure of surface waves have

been receiving increased attention in recent years. This interest

comes on the one hand from oceanographers who are interested in

investigations of the momentum transfer at the air-sea interface and

on the other hand from remote sensing scientists involved in interpre-

tation of radar backscatter information from the sea surface.

When air blows over the water surface, momentum and energy are

transferred to waves and currents. Wind-generated waves are described

in a well-known survey by Ursell (1956) which concludes: "wind blowing

over a water surface generates waves in the water by physical processes

which cannot be regarded as known." Since then, many investigations

followed to understand this complex phenomenon. Much progress towards

understanding wave generation has been achieved since 1956, although

our understanding of momentum transfer to current and to high frequency

waves remains deficient.

It is well known that the momentum transfer plays a central role

in the wave generation process by wind. From visual observation of

surface waves we may consider the wind-generated wave system to consist

of a spectrum of waves with a dominant wave frequency which shifts

slowly with increasing wind speed and/or fetch, and a small-scale high

frequency structure which is acted upon by wind and the orbital










velocities of long waves. Dissipation forces and the interactions

among the dominant wave components also play important roles in the

development of the whole wave system. If the small wave amplitudes

are modulated by long waves, so is the wind stress (Keller and Wright,

1975); a modulated stress will contribute to the growth of the long

waves. Hence, a detailed investigation of the short wave structure

in the presence of long waves is essential to our understanding of

the interacting processes at the air-sea interface.

From microwave backscatter point of view, the high frequency waves,

which are of the order of centimeters in wavelength, scatter and

reflect electromagnetic and acoustic waves impinging on the sea surface.

These centimetric waves, characterized by their wavelengths, are

associated with radar backscatter and formation of radar images of the

ocean surface. The measurement of the imprint of ocean processes

on the sea surface is attributed to the dynamics of waves with

wavelengths of the same order as that of the radar. Because radar

has limitations in observing long ocean waves (McLeish et al., 1980,

Pawka et al., 1980), it is essential that further investigation of

short wave dynamics be pursued.

By using sinusoidal long waves, it has been found, under

laboratory conditions, that short waves undergo cyclic modulation

in their wave height levels (Keller and Wright, 1975; Reece, 1978).

In the field where the long waves are not monochromatic, the modu-

lation of short waves has also been shown (Evans and Shemdin, 1980;

Wright et al., 1980).

The radar observations by Wright et al. (1980) indicated that

the modulation levels under field conditions cannot be explained by









straining of short waves by long waves alone and that additional forcing

functions in addition to the straining must be considered. There is

considerable evidence suggesting that radar backscatter can provide

a direct measure of the hydrodynamic modulation and other

characteristics of the short waves. There remains a need, however, for

conclusive verification of such measurements. This study is motivated

partially by the need to interpret radar backscatter from the sea

surface and further in understanding the mechanisms responsible for

imaging of ocean surface signatures with real and synthetic aperture

radars.



1.2 Scope of the Present Study

The work reported in this dissertation is in response to the

need to measure wave slopes in the field to obtain insight on the

general characteristics of high frequency waves. Particular care is

taken to obtain the hydrodynamic modulation levels.

Because the height of short waves is only a fraction of that of

the long waves, a conventional wave gauge does not have sufficient

resolution and satisfactory frequency response to provide a quanti-

tative measurement of short waves. However, the sea surface wave

slope, where the wave amplitude is tuned by the wavenumber, provides

a measure that is of the same order of magnitude for both long and

short waves. In the analysis phase of this investigation, the long

wave directional properties are considered and the hydrodynamic

modulation levels of short waves are estimated for each long wave

frequency component.










A theoretical survey is given in chapter 2. The experimental

set-up and field conditions are described in chapter 3. Chapter 4

shows the general scheme for data analysis and provides sample results.

The experimental results are shown in chapter 5. A discussion of results

is given in chapter 6. Finally, conclusions and recommendations are

given in chapter 7.
















CHAPTER II
THEORETICAL BACKGROUND



2.1 Momentum Transfer

The generation of waves by wind is a significant geophysical

phenomenon that has received considerable interest in recent years.

The wind-wave generation mechanisms proposed by Phillips (1957) and

Miles (1957) constitute significant contributions towards understanding

the processes by which wind generates waves. The initial phase of

wave generation is associated with the turbulent aspects of wind over

the air-sea interface. Phillips (1957) postulates that waves are

generated by the normal stress fluctuations associated with a turbulent

wind. He suggests that the pressure field "resonates" with the wave

field as long as both the speed and length of the pressure field

match those of surface waves. His resonance mechanism predicts a

linear energy growth rate which is consistent with field observations

in the initial phase of the wind generation process. The mechanism

proposed by Miles (1957) suggests that the energy transfer from wind

to waves is primarily due to the wind-induced pressure perturbation

in the atmospheric shear flow. Miles' shear flow instability

mechanism yields an exponential growth rate of surface wave energy.

Later, Miles (1960) combined the two mechanisms discussed above to

account for both the initial and subsequent growth rates of surface

waves.










The field investigations of Snyder and Cox (1966) and Barnett

and Wilkerson (1967) concluded that Phillips' theory could account

for wave growth only if the atmospheric pressure fluctuation was

increased by a factor of 50. They also found the principal stage of

wave spectral development to occur at the rate with an order of

magnitude greater than that predicted by Miles' (1960) theory. These

discrepancies motivated new investigations to re-examine the mechanism

involved in atmospheric transfer from wind to waves.

Townsend (1972) re-evaluated the momentum transfer by solving the

linearized equation for turbulent flow over a progressive wave

numerically. He found the momentum transfer from wind to waves to be

of the same order as that predicted by Mliles' theory. Gent and Taylor

(1976) extended Townsend's work by including the nonlinear terms in

the governing equation. They showed the air flow pattern and shear

stress vary along the wave profile. The shear stress was found to

have a strong peak at the crest; they also suggested that the

momentum transfer rate is significantly enhanced if the surface

roughness is to vary along the long wave profile. An enhanced

momentum transfer rate associated with the flow separation over

breaking waves was suggested by Banner and Melville (1976). The

possibility of patches of concentrated momentum transfer from air to

water was first introduced by Dorman and Mollo-Christensen (1973)

following acquisition of an extensive set of field measurements.

A nonlinear wave-wave interaction mechanism has been invoked

that indicates transfer of energy from the high frequency part of a

wave spectrum to the peak and forward face of the spectrum. This

mechanism offers a satisfactory explanation of the rapid growth rate









of waves observed in the field. Figure 1 shows the schematic

representation of the energy balance proposed by Hasselmann et al.

(1973). The nonlinear wave-wave interaction process redistributes

energy from frequencies higher than the peak frequency to those lower

than the peak of the spectrum. The nonlinear energy input (Qin) to
the high frequency part is cancelled by the dissipative process

(Qd), and the main role of the wave-wave interactions (Q ~) appears
to be energy transfer toward the low frequency forward face of the

energy spectrum. The nonlinear wave-wave interaction can explain the

rapid growth rates observed by Snyder and Cox (1966) and Barnett and

Wilkerson (1967). A simplified mechanism for energy transfer from

short waves to long waves was explored by Languet-Higgins and

Stewart (1960, 1964). This will be discussed in section 2.3.



2.2 Properties of High Frequency Waves

The energy of waves in the capillary and short gravity wave

range is better characterized by the slope spectrum rather than by

the surface displacement spectrum. The high frequency waves are

small in amplitude and large in wavenumber; the wave slope, which is

proportional to the product of these two wave parameters, is a

significant quantity throughout the spectral range. Hence, the slope

is more easily measured in the high frequency range of waves compared

to wave amplitude.

The shape of the high frequency portion of the wave height

spectrum was proposed by Phillips (1958) based on dimensional

analysis and similarity considerations as follows


Elf) = (2*)- B g2 f-5


(2.1)
















































Fig. 1 Schematic energy balance for the case of negligible di5sipation~
in the main part of the spectrum. Qin = atmospheric input,
Qnt = non-linear wave-wave transfer, Qd -dsiain n
q ~= net transfer (after Hasselmann et al., 1973).









where e is the equilibrium constant, f is the circular frequency and

g is the gravitational acceleration. The constancy of B has been

questioned by many investigators (Liu, 1971; Hasselmann et al., 1973;

Mitsuyasu, 1977). Various other spectral forms for the high

frequency range of waves have been proposed; Pierson (1976), for

example, summarized Mitsuyasu and Honda's (1974) laboratory results

by proposing a displacement spectrum for the high-frequency range of
the form


Elf) = 0.1393 f-b (2.2)

where


b = 5 10gl0(U,) (2.3)

and U, is the friction velocity of the wind. However, the spectral

form shown in equation (2.2) does not follow Phillips' inverse fifth

power law shown in equation (2.1).

In the capillary wave range, if the viscous dissipation is

not important the equilibrium shape can be derived on dimensional

grounds as


Elf) = A ( )2/ -7/3 ,(2.4)

where A is a constant and r is surface tension.

The equilibrium range in the wave slope spectrum for both

gravity and capillary waves was derived by Phillips (1977, p. 152)
from dimensional considerations


5(f) = Df


(2.5)










where D is a constant but may assume different values for gravity

and capillary spectral ranges.

Cox (1958) used an optical-refraction technique to measure

capillary wave slopes. He found the mean square slopes increase

rapidly with wind speed. His results, however, do not show clearly

the wave spectral dependence on wind speed. Wu (1971) provided

useful information on the wave slope distribution function by

using an optical reflection technique in a laboratory facility.

He found the shapes of the distribution function to be Gaussian in

general. Long and Huang (1976) measured the wave slopes using a

laser device. They concluded that equation (2.5) is valid in the

capillary-gravity range; the spectra showed similar shapes and no

obvious dependence on wind speed. A recent experiment by Lleonart

and Blackman (1980) clearly shows the wind dependence in spectra of

capillary waves. They proposed the wind dependence of the down-

wind slope spectrum to have the form


S(f) = D2 1/2 f- (2.6)


where D2 = 2.95 x 10' and v is kinematic viscosity of water.
The experiments discussed above were all conducted in the

laboratory. It is noted that only few wave slope measurements are

available under field conditions. Cox and Munk (1954) used a

photographic and optical reflection technique to infer the charac-

ter of wave slopes from sun glitter patterns. They showed that the

mean square slope is wYind-dependent. The cross-wind slope was

found to be slightly more peaked than the Gaussian distribution;









the down-wind slope distribution peak was found to be shifted by

2.5 degrees in the down-wind direction.



2.3 Two-Scale Model

In a first examination of the interaction of short waves and

long waves, Languet-Higgins and Stewart (1960) considered the hydro-

dynamic interaction of two gravity waves noncontiguous in the

frequency domain with no energy input nor dissipation. They pursued

a perturbation analysis to find that, correct to the second order,

the free surface in deep water can be described by


n = as sinXs(1 + a~k sinxg) as cosys(a ks cosx ) (2.7)

where x is the phase of the progressive wave, a is the wave amplitude,

k is the wavenumber, the subscripts s and e denote the properties of

short waves and long waves, respectively, and ks >> kt is assumed.

Equation (2.7) represents a modulated amplitude


as = as(1 + a ki sinx ) ,(2.8)

and a modulated wavenumber


ks = ks(1 + a~k sinxq) .(2.9)

This means that both the amplitude and the wavenumber vary in phase

with the long wave profile. The results shown in equations (2.8)

and (2,9) represent the effect of the compression of the orbital

velocity of the long wave and the work done by the long waves against

the radiation stress of the short waves.










Phillips (1963) indicated that the energy dissipated by the

short waves is acquired from the long waves so that the interaction

of short waves with long waves would dampen the long waves. On the

other hand, Longuet-Hliggins (1969) pointed out that as the short

waves dissipate their energy, they impart their momentum to the

long waves and thus exert a stress which is in phase with the

orbital velocity of the long waves; the latter should lead to the

growth of the long waves. He also showed that if the short waves

were continuously regenerated by the wind, the input of energy due

to this "maser-type" mechanism should be more significant than other

competing mechanisms.

Hasselmann (1971) showed that the energy gain of long waves

predicted by Longuet-Higgins (1969) is cancelled by the potential

energy transfer, hence resulting in a weak decay of the long waves.

Garrett and Smith (1976) reported that the long wave growth can

result from the interaction of short waves with long waves provided

the short wave generation is correlated with the orbital velocity

of the long waves.

All of the above analyses have been with two-scale motions

where each scale is represented by a monochromatic wave. In a wind-

wave situation we have a spectrum of short waves modulated by a

spectrum of long waves. A simplified version, the modulation of a

spectrum of short waves by a monochromatic long wave, can be sim-

ulated under laboratory conditions (Mitsuyasu, 1966; Reece, 1978).

A refined model, originated from the radar measurements by Keller

and Wright (1975), will be discussed more fully in the next

section.









2.4 Relaxation Model

Keller and Wright (1975) first proposed a relaxation model to

interpret their microwave measurement of the modulation of wind-

generated short waves by longer plunger-generated waves. The basis

of their relaxation model is that in an equilibrium situation, with

a known distribution function, any departure of the system from the

equilibrium induced by external forces, results in a return to

equilibrium at an exponential rate. Their measurements are based on

radar backscattering, in which the short waves represent the Bragg-

scattering waves, through the relation


kS = 2kr cos6d (2.10)


where ks is the wavenumber-of the short waves, kr is the radar

wavenumber, and dd is the depression angle. They specify the energy

spectrum in the wavenumber domain and treat the hydrodynamic inter-

action between the short wave and long wave through the use of the

radiative transport equation in which the energy spectrum of the

short waves is modulated by a long monochromatic wave. The transport

equation, which includes the effect of the horizontal orbital current

component, for the short wave-long wave interaction can be written as


dt (ks. x, t) + ylF -x- = Q (2.11)


where F is the short wave energy spectrum in wavenum~ber space,

'l is the Longuet-Higgins' strain factor (Longuet-Higgins and Stewart,
1964) given by










1 + (3r k2/pg)
1(ks 2(1 (r /pg) ,(2.12)


U, is the horizontal component of the long wave orbital velocity

given by


U (x, t) = U2 eikx-t (2.13)

The source term Q on the right hand side of equation (2.11)

represents the energy input from the wind, nonlinear dissipation and

wave-wave interaction. The energy input from the wind is assumed to

be BF, where B is the exponential growth rate. Equation (2.11) can
be written as

aU
+F +F xF + B (2.14)
at ak s ax 1l ax =8


where the dot operator represents derivative with respect to time.

The functional H(F, ks) accounts for the nonlinear energy dissipation
and the energy transfer due to wave-wave interactions. The con-

servation of waves gives

k; s (2.15)
s ax

The dispersion relation is modified by the underlying current such that


0s = 5s(ks) = ks(U (x't) + Um) ,(2.16)

where as is the intrinsic frequency, Um is the medium speed contributed

by the tidal current and the wind drift. Also, the group velocity

Cgs is given by











x = Cgs. (2.17)

Equation (2.11) becomes


aF + F k o aF + F o F+H. (2.18)
:t gs ax s ax ak 1ax


In the frame moving with the long wave phase speed, C the
following transformation can be made


a (2.19)
ax ax

C 3 (2.20)
at at ax


U (x,t) + U (x) = Usek x (2.21)

For the stationary case in the moving frame, equation (2.18) becomes
aU
-(C~ C s) aF+ (y F k aF _o= OF + H (.2
R sax 1 s ;k ax

If F and H are expanded in terms of E, where


F = U /C~ (2.23)

which is of the same order as the long wave slope,


F = (0)(s) + cF( )(ksx) + ...(2.24)

H = H(0) + ,H(1) + ... (2.25)

equation (2.22) gives the zeroth-order equation

BF(0) = H(0) (2.26)

which can be interpreted as the energy input from the wind is being

balanced by dissipation due to nonlinear wave-wave transfer.









The terms proportional to a give to first order equation

C ,(1) a(0) ik x FO
-(1 gs) -- -ik k - e + t ylk F(0


e F(1) H(1)
C, C,


(2.27)


In the relaxation model, the relaxation rate p is introduced such
that


F(1) = (1) eik x


(2.28)


(2.29)


Letting


f(1) can be solved by substituting equation (2.28) into equation (2.27)


k (0)
(1 at 1
MEFp 92 e,(o) -ks C 2 (kCs
tk 1 gs

where oQ is the radian frequency of the long wave.
(2.25), (2.29) and (2.30) it follows that

F = F[O (1 + C~ Me x)


correct to the first order, where re is the long wave phase angle.
Since M is a complex number in general, its modulus |M1 is called

the hydrodynamic modulation level, its argument emod gives the phase
information of F(1 relative to the long wave crest.

In fact, the long wave phase speed is much greater than Cgs, 1.e.


o /k~ >> Cgs ,
and therefore, equation (2.30) can be written as


(2.32)


-a) + ip] (2.301)

Combining equations



(2.31)










k (0)i k 1

N = (-aa + iu)
i92 + 02

ks aF(0)

F s (2.33)



Assuming a Phillips' spectrum of the for-m F(ks)~ks or (Phillips,
1977, p. 145) the hydrodynamic modulation for the short gravity wave

where rl = 0.5 is

M = 4.5 a a/ )(2.34)
1 l+(o /p1)2
The variation in hydrodynamic modulation ranges from 0 at low qe/v

and asymptotically approaches 4.5 at high ot/. Figure 2 displays

M given in equation (2.34). Alpers and Hasselmann (1978) used the
ratio of the group velocity of the short wave to the group velocity

of the long wave and the long wave slope as the expansion parameters

in the action balance equation. Their solution is equivalent to

equation (2.33) where it is further assumed the long wave phase

speed is much greater than the short wave group speed.
In investigations of radar backscatter from the sea surface

Keller and Wright (1975), Wrright et al. (1980), and Plant (1980)

proposed the modulation of microwave power backscattering from the

Bragg-waves, P(t), to have the form

U ix,
P(t) = P (1 + Mra ( a) e ),(2.35)

where p is the mean backscattered power. For the natural environment

the radar modulation transfer function was generalized to the form


































2 3


OO -




$60







O 1 2 3




Fig. 2 Hydrodynamic modulation transfer function for short gravity
waves from relaxation model..







19


C E
M~ (2.36)
radar t--
p vv

where E is the cross-spectrum of the instantaneous received power and

instantaneous line of sight speed v and E~~ is the auto-spectrum of the
vv
line of sign speed. V(t) can be detected directly from the instan-

taneous Doppler shift of the backscattered signal.















CHAPTER III
FIELD EXPERIMENT


3.1 Field Set-Up

During September to November of 1979, the wave follower was

operated from the Noordwijk research tower located approximately

10 km offshore of the coast of Holland in 18 m water depth as an

element of the Maritime Remote Sensing Experiment (MARSEN). The

field operations were described by Shemdin (1981). Figure 3 shows

the geographic location of the Noordwijk tower. Figures 4 and 5

show the wave follower deployed from the Noordwijk tower in a

data acquisition mode.

The wave follower is a servo-driven electric motor that powers

an endless cable on which instruments are mounted. The general

features of the wave follower were described by Shemdin (1981).

The mechanism is shown schematically in figure 6. The instrument

frame was designed to follow a wave profile 6 m in height and 3 sec

in period. The capacitance wave height gauge attached to the wave

follower frame provides the drive signal to the servo motor which

then vertically translates the instrument frame to maintain the

capacitance gauge at a constant submergence depth. In the field,

the latter was accomplished to within an error of j- 5.0 cm.

The measurements obtained by mounting instruments on the wave

follower are discussed in the following. A plan view showing

relative positions of instruments is given in figure 7.


















































Fig. 3 Geographic location of the Noordwijk Tower. The contour
lines are specified in fathoms.
























































J~ii~z









--.ii~-
~
=5~r~ =C~iPIC-
~-LL ---
--c --
-.--
-~c~s~I~ch~i~Lc
=--`-*'
------ - 'ci-~
I--)T---2
--r- L- ----~-


Fig. 4 The wave follower in the wave following mode.


Elm

















































Fig. 5 Close view of the optical sensor mounted on the wave follower.










Support
Cable


Servo Drive
Electric Motor


1000#
Cable


..u


coEM-Current Meter
End ess Cable`
Assembly


Fig. 6 Schematic diagram of the wave follower.






















Wave Follower
- Stationary
Truss Frame



Current~ee


Error Gauge (also location of Wave
Height Measurement )


3. 66 m


Fig. 7 Plan view of the relative location of sensors used in the
wave fo1 Tower experiment.









3.2 Surface Wave Slope Measurement

A high-response laser-optical sensor developed at the University
of Florida (Tober, Anderson and Shemdin, 1973; Palm, 1975) was used

to obtain time series of the sea surface slope at a point location.

A vertically oriented laser beam eminates from a laser mounted on

the wave follower instrument frame below the water surface.

The refracted laser beam in air is then related to the water surface

normal which is defined by an azimuth angle (4) and a deflection angle

(y), as shown in figure 8. If the unit surface normal vector is

denoted by n, its components are specified by


n =- (sinv coss, sinv sine, cosy) (3.1)

The sea surface at a fixed time can be specified by


+(xl' x2, x3) = x3 n(x1, x2) = 0 (3.2)

where xi (i=1, 2, 3) are the three spatial coordinates and n is the

sea surface displacement. The unit normal vector to the surface 9

is given by


n = ve/|%~ = (-an/ax1, -an/ax2, 1)/ vt| (3.3)


Equations (3.1) and (3.3) give the slope components along two

major axes,


(arn/ax1, an/ax2) = -tanv (cost, sing) (3.4)

The down-wind slope is given by the scalar product of the unit

vector pointing in the down-wind direction and the slope vector given

in equation (3.4). For example, if a denotes the wind direction































X2














Fig. 8 The definition diagram of the refracted laser beam.. The unit
normal vector n = (siny case, siny sing, cosy) defines the
slope vector in xl and x2 directions as -tany (cost, sing).






28

referred to the xl axis then the down-wind slope is given by

s = -tanv cos(4-a) .(3.5)



3.3 Sea Surface Dispalcement Mleasurement

A linear displacement sensor which measures the vertical position

of the wave follower instrument frame was used to detect the sea

surf ace displacement. Let the displacement signal be V1 and the

error signal be V2, the sea surface displacement is given simply by


nljt) = a1 V1(t) + a2 V2(t) ,(3.6)


where t is time, a1 and a2 are calibration constants.


3.4 Current Measurements

Two horizontal components of current were measured at an Eulerian

level of 4.75 m below the mean sea level and at a position located

3.66 m from the wave slope sensor (see figure 7). The electromagnetic

current meter was mounted on the lower extremity of the stationary

structural frame of the wave follower, shown in figure 6. Hence,

the current was measured only in an Eulerian mode. It is noted that

the electromagnetic current meter measured the tidal component

of the current as well as the wave orbital velocity (01son, 1972).



3.5 Other Measurements

In addi ti on to the above, wi nd speed and di recti on were obtai ned

from an anemometer located at 10 m above the mean sea level.








3.6 Data Acquisition and Digitization

Eight channels of data were recorded on the AMPEX PR-2000

Instrumentation Recorder. These included four slope signals which give

the position of the refracted laser beam, two current signals, one

displacement signal and one error signal. The author has been responsi-

ble for the work of the analog-to-digital conversion and thereafter.

The analog signal tapes were digitized at 400 sps (samples per

second) to account for the high frequency encounter of small scale

waves. The sampling rates for the current and sea surface displace-

ment measurements were reduced to increase computational efficiency.

The method used was to take the arithmetic mean of the values of

neighboring data points.

After careful examination of the strip chart records generated

during the data digitization process, twelve cases were selected for

detailed analysis. They are summarized in Table 1. In this table

wind direction is given with respect to a reference frame specified

in figure 9. Hence, in Run 325 wind direction of 295 degrees refers

to wind towards 295 degrees (southeast) from east in a counter-

clockwise sense or from direction 115 degrees ( northwest) in the

same sense.









































Note: (1) Local time. (2) True direction, direction
referred to the east counterclockwise (see
figure 9). (3) Sampling rate =400 sps.
(4) Oil slick experiment.


Table 1
of the Cases Selected for Detailed Data Analysis


Number of data


Sumnma ry
(1)
Date Time Run ID


9/27 15:50 325

9/27 16:36 326

9/28 9:55 327

9/28 14:41 328

10/2 13:34 102

10/11 5:26 106

10/11 7:29 110

10/18 22:05 107

10/19 10:33 108

10/19 14:13(4) 105

11/19 20:02 217

11/19 20:53 104


Wi nd
speed (m/s) direction (deq)(2)
7.7 295

7.5 291

3.0 163

3.5 249

6.3 150

5.9 39

4.3 54

5.4 329

11.3 50

12.7 48

4.5 300

5.0 273


648,335

426,466

261,239

482,092

719,328

632,079

339,725

494,665

548,259

158,369

513,334

724,662


































~East








Fig. 9 Definition Sketch of true direction. The true direction ea
is the angle of Va referred to the east counterclockwise.a















CHAPTER IV
DATA ANALYSIS



The digitized data gave digital representation of voltages for

each of the sensors used. Before useful quantities could be obtained

the voltages had to be converted to the physical quantities they were

measuring by applying their calibration relationships. Following these

steps, various calculations were conducted to achieve specific goals

of this study. In the following the various algorithms used in the

study are discussed.



4.1 Spectral Analysis

For a time series y(t) of duration T, sampled at N times, with

time interval a so that T=Na, the corresponding discrete Fourier

transform is (Newland, 1975)

N-1
Yn N j exp(-i 2nj n/N) (4.1)
j=0

where

yj = y(jn), i2 = -1 and n = 0, 1, ..., N-1.

Its inverse transform is


N-1
y, = z Yn exp(i 2;rj n/N) (4.2)
Sn=0









with j =0 .. -

In this study, a FFT routine (Krogh, 1970) was called to

calculate Yn for a given discrete time series y.. This routine
was capable of calculating the inverse Fourier transform shown in

equation (4.2).

An algorithm suggested by Welch (1967) was used to estimate

the properties of the energy spectrum for this study. In brief, a

time series was divided into K, segments, each with length L, and

the discrete Fourier transform for each Kth segment was performed

according to

L-1
YK= yr W.jW exp(-i 2nj n/L) (4.3)
n Lj=0

where W is the window function given by


L12



The estimate of the spectrum for the



I T (4.5)
z W.
j=0


The final estimate of energy spectrum was obtained by averaging

the In's given in equation (4.5).

The discrete Fourier transform technique was extensively used

in this study such as in the cross-spectrum analysis, the demodu-

lation of the wave slope signals, and the calculation of the

orbital velocity. The energy spectral estimates by the segment

averaging method was used to determine the slope spectrum, the

orbital velocity spectrum and the wave height spectrum.








Various K and L values were used for the different calculations.

They will be stated in the text under related discussions. The

equivalent degrees of freedom (EDF) using this procedure is

2 x K (Welch, 1967).



4.2 Directional Spectral Estimates

Waves, as they occur in nature, are random and two-dimensional,

in appearance and, as such, have to be treated as a statistical

phenomenon. The current meter used during the experiment served a

dual purpose: (1) to detect the component of orbital current due to

the long surface waves, and (2) to measure the tidal component of

current. The capability to measure both the long waves orbital

velocity and the low frequency (f < 10-2 Hz) tidal fluctuations was

reported by 01son (1972) and Winant and 01son (1976).

In this study, we use a high-pass filter to obtain the component

of current due to waves; the cut-off frequency was set at 0.02 Hz.

Figures 10 and 11 show typical time series of the sea surface

displacement, n, and the current components u and v. The corres-

ponding spectra for n and u are shown in figure 12 (K=10 and EDF=20).

The close agreement between the spectra Enn and Euu suggests that
electromagnetic current meter provides reasonable measurement of the

orbital velocity of surface waves. It is noted that the current

meter was located at 4.75 m below the mean sea level. The orbital

velocity due to waves at this level are attenuated to less than 20%

of the surface value for waves with frequencies greater than 0.3 Hz.

Figure 13 gives exact attenuation values of various wave frequencies.

Hence it was anticipated that the current meter would not detect

orbital velocity components with frequencies greater than 0.3 Hz.




















































0.

b. The

Fig. 10


o


20



6


20.Tiilie (sec) 30.

velocity after removal


1


VI


.. ~11


~


1o.

The u-comlponentt of the orbital


do. 5Do

of the mean tidal current.


10. 20Time (sec) 3o. do.

v-component of the orbital velocity after removal of the mean tidal current.

Sam~ple time series measured by the electromagnetic current: meter for Run 108.



























30. 4(
TIME ISEC1


U,-OawN WIND SLOPE


U. 10. 20. 30. 40.
TIME (SECI
b. The down-wind slope record.

Fig. 11 Sample time series for sea surface displacement and slope for Run 108.


L~_


t


0. 50.


UITER SURFACE ELEVATION


~-/~


1.0

0. 10. 2(

a. Sea surface displacement.






.2


-.o

11 I




-.3











































O .2 .4

f(Hz)


Fig. 12 Frequency
gives. the
gives the


spectra for n7 and u for Run 108. The solid line
surface displacement spectrum. The dashed line
u-component of the orbital velocity/ spectrum.






































20










0 0.1 0.2 0.3 0.4
FREQUENCY (HZ)


Fig. 13 Attenuation of the orbital velocity. u/uo is the ratio of
u-component of orbital velocity at 4.75 m below the mean
sea level to that at the sea surface.









The sampling rate for the current meter signal was set at 5 sps which

was sufficiently high for wave frequencies less than 0.3 Hz.

The tidal current was considered as the D.C. component of each

of the 15 min record lengths used. These values were compared

favorably with those provided in Tidal Tables. A summary of the

cases analyzed is given in Table 2. It is provided here as a guide

for the subsequent analysis described in what follows.

The sea surface displacement signal is subject to inherent

limitations due to the wave follower mechanical frequency response.

The latter was found to be 1.0 Hz. The sea surface displacement time

series were sampled at 25 sps. Because the current meter was

stationed at 3.66 m away from the displacement sensor in the

horizontal direction, a direct measure of the sea surface at the

position of the optical sensor was not available. The method

followed was to calculate the directional wave height spectrum from

u, v andn time series and then to infer the surface orbital current

at the location of the optical sensor. The latter was then used

for demodulation analysis of surface wave slopes. The technique

used for computing wave directional properties is described in

Appendix A.



4.3 Determinations of Surface Wave Slope

Short wave amplitudes are small in nature; the slopes are

tuned by the wavenumbers which are large for short waves. It is

therefore easier to measure the slopes of short waves instead of

their wave heights.

The laser-optical sensor was used to determine the surface normal

A as discussed before. The optical receiver used a photo-diode


















Run ID Wave Characteristics Tidal Current

H13 Tpa Direction of () ut et
(m) (sec) Dmeneae (m/s) (degrees)


325 1.06 5.12 314 .16 247

326 1.05 5.39 332 .26 246

328 .59 5.69 310 .60 244

102 .45 5.39 238 .54 60

106 1.04 6.34 12 .72 51

107 1.59 7.88 347 .33 272

108 1.23 6.02 5 .29 217

217 .83 4.88 320 .35 238

104 .85 4.65 238 .60 36


Table 2
Observed Values of the Significant Wave Height,
Dominant Wind Direction and Tidal Current


Note: (1) H1/3 is the significant wave height

(2) Tpa is the period of the spectral peak

(3) true direction defined as before.









to determine the position of refracted laser beam. The latter was

calibrated to give the coordinates in terms of v and 4. For a

fixed deflection angle, v, the calibration voltages were recorded

in the azimuth angle, 4, with increments of 450. The procedure

was repeated for deflection angles from 00 to 300 in increments of So

Figures 14 and 15 show the calculated results after the

application of the calibration curve derived from the best fit of

the calibration signal for y and S. The rms errors were found to

be 0.980 for y and 5.70 for i. Using equation (3.3), the slope

component along a specified direction is determined. A sample of

the down-wind slope is given in figure 11b. The corresponding slope

spectrum (with EDF=100) is shown in figure 16. The probability

density function derived in the standard procedure (Bendat and

Pierso1, 1971) is shown in figure 17.



4.n Demodulation Analysis

As mentioned earlier, the encounter frequency of short waves

deviates from the intrinsic frequency as the medium is swept by the

orbital velocity of the long waves. The slope signals measured with

the laser-optical sensor contain the Doppler effect due to wind

drift, orbital velocity of long waves and tidal current. This

Doppler effect must be removed in order to determine the dynamic

properties of these modulated short waves. Removing the Doppler

effect is referred to as "demodulation."

The demodulation procedure proposed by Evans and Shemdin

(1980) was modified for this study. We first discuss the demodu-

lation procedure used for the simple case of a monochromatic wave


























$20-





10





O I I 20 30
7 (Degrees)


Fig. 14 Calibratiion results for thec deflection annle; r is the
calculated value.





























8 180-




90-





O 90 180 270

S(Degrees)

Fig. 13 Calibration results for thle azimiu~th angle; p is the
calculated value.




















1-2




10

1-












1-5
10- 10 110
FRQECY(z

Fi.1 h onwn lp petu o u 0 ; EF=10























0.2 P



*. *





-3 -2 -1 1 2
NORMALIZED SLOPE

Fig. 17 Probability density function (PDF) of the sea surface slope summarized fromt
40,000 points of Run 108. The open circles and solid points denote the PDF
for down-wind slope and the cross-wind slope, respectively. The normalized
slope is the ratio of the slope to the standard deviation. The solid curve
is a Gaussian distribution with the same variance. The variance of down-wind
slope is 0.0342 and that of cross-wind slope is 0.0310.







46

train with direction 6, wave number ke and amplitude A The sea

surface displacement at point ( x1,x2) and time t is


nt(x1 ,x2, t) = Ap cos (ky cose)x1 + (kl sine)x2 at .(4.6)

The magnitude of the horizontal component of the surface orbital

velocity at point (0,0) can be approximated by


u (0, 0, t) = u (t) = un~ coth(k h), (4.7)

where h is water depth (=18 m),iqi is the radian frequency and k,
is the wavenumber. The dispersion relation is given by


o, = gki tanh k h .(4.8)

The medium travel speed along the direction of the short wave

is given by


Vm = ut oet +O ut coset (4.9)

where 49 is the angle between the short wave and the long wave, at
is the angle between the tidal current and the short wave.

The distance travelled by a short wave between two adjacent

sample points is given by


6x = |Vs + m|* a (a.10)

where Vs is the resultant phase speed and a is the sampling period.
This relationship was found by Reece (1978) to be valid under

laboratory conditions where the modulation of a wind-generated
wave spectrum by a mechanically-generated monochromatic wave was

investigated. Theoretical considerations of the interaction of









sinusoidal waves with current (Phillips, 1977) suggest that waves in

an opposing current can only exist if

a < /4U ,
where o is wave radian frequency in absence of current, g is

gravitational acceleration and Um is mean current. The results of
Reece (1978) do not confirm this requirement suggesting that consider-

ations based on monochromatic waves are not necessarily valid in

wind-generated waves where the interaction of wind with a spectrum of

waves plays the dominant role. Further evidence on this point is

given by Wright (1976) who demonstrated experimentally that the

induced wave breaking of short waves by long waves is different in

character from that predicted by Banner and Phillips (1974) based on

hydrodynamic considerations of monochromatic waves. While considerable

research is in progress to provide insight on the interaction of wind

generated short waves by long waves,the validity of equation (4.10) for

the moment cannot be disputed. Hence, this equation is adopted for

the demodulation analysis. It is recognized that the field conditions

in MARSEN are different from the laboratory conditions from which this

relationship was derived. The results derived from the demodulation

analysis are therefore subject to the validity of equation (4.10) under
field conditions.

In the presence of the wind, the resultant phase speed is given

by Shemdin (1972)

Vs = Cs + aUa (4.11)


where Cs is the phase speed of the short waves given by









Cs = (g/ks + rks/p)2 (4.12)


Ua is the wind speed, a is determined experimentally to be 0.03
(Huang, 1979). In equation (4.12) r is the sea surface-tension, and

p is the sea water density. The gravitational acceleration in

equation (4.12) is modified to give

g' = g eg (4.13)

where the second term on the right hand side is usually small

under field conditions compared to the first and is neglected in

this study. The demodulation procedure consists of computing an

equivalent spatial distance for a given time series from equation

(4.10). By using interpolation, a demodulated slope signal time

series is derived in the spatial domain. The demodulated time

series is then band-passed for a selected wavelength band in the

wavenumber domain. The band-passed result is transformed back to

time domain for cross-correlation with the surface displacement

time series.

For the field data, the sea was composed of many sinusoidal

waves. We assume that, for each frequency component, there is a

dominant wave direction. We estimate the sea surface orbital

velocity by summing u~ calculated for each frequency component

according to equation (4.7). The detail algorithm for demodulation

is described in Appendix B.

Three wavelength bands were selected for demodulation. These

were 3 cm, 8 cm and 23 cm which correspond to radar wavelengths

for X, c and L band radars respectively.















CHAPTER VI
EXPERIMENTAL RESULTS



Most of the accessory information for the runs is presented in

Tables 1 and 2. The runs cover a range of wind speeds from 3.0 to

12.7 m/s. Figure 18 shows the sea surface displacement spectrum Elf)

where two peaks appear at two significantly different frequencies.

The dominant peak of Elf) at f = 0.39 Hz corresonds to the wind-generated

wave, the secondary peak at f 0.19 Hz corresponds to the swell.

Cases like this will be identified as "mixed sea." In contrast, the

sea surface displacement spectra shown in figures 19 and 20 represent

"well-defined peak" cases and are typical of wind sea conditions.

The sea conditions of the 12 runs are summarized in Table 3.

The principal results of this study are presented in various

ways to bring out different points. We first present the general

features of the wave slope spectra observed in the field. These will

be followed by the results of the modulation of mean square slopes

by long waves and the probability density function of wave slopes.

The chapter is concluded by presentation of the calculated hydrodynamic

modulation levels.



5.1 Characteristics of Wave Slope Spectra

Figure 21 illustrates the general features of the down-wind slope

spectra at four different wind speeds from 3.5 m/s to 11.3 m/s.














0.10
.39 I
0. 09-

0. 08-

0. 07

0.06-

S 0.05-

S0. 04-

0. 03 ,-0.186Hz

0. 02-

0. 01

0.00 Il
0. O 0. 1 0. 2 0. 3 0. 4
FREQUENCY (Hz)

Fig. 18 An example of the sea surface displacement spectrum for
"mixed sea" case. Run ID = 102, Ua = 6.3 m/s.













0. 35I
0. 176 Hz

0. 30--



0. 25--



7 0. 20 -



0. 15--



0. 10--



0. 05--


0.00 I I
0. O 0. 1 0. 2 0. 3 0. 4 0. 5
FREQ UENCY ( Hz )

Fig. 19 Sea surface displacement spectrum of Run 328, a "well-defined
peak" case. Ua = 3.5 m/s.














0. 195 Hz
0. 8-


0. 7-


0. 6-





S0.4


0. 3


0. 2


0. 1-


0. O
0. 0 0. 1 0. 2 0. 3 0.4 0. 5
FREQUENCY (Hz)


Fig. 20 Sea surface displacement spectrum of Run 325, a "well-defined
peak" case. Ua .7ms



















Run ID Ua(m/s) Mean Square Wave Slope Sea Condition
down-wind cross-wind

327 3.0 .0104 .0141 mixed sea

328 3.5 .0094 .0101 well-defined peak

110 4.3 .0223 .0209 mixed sea

217 4.5 .0259 .0189 mixed sea

104 5.0 .0309 .0382 mixed sea

107 5.4 .0447 .0429 mixed sea

106 5.9 .0347 .0267 mixed sea

102 6.3 .0098 .0117 mixed sea

326 7.5 .0218 .0317 well-defined peak

325 7.7 .0159 .0314 well-defined peak

108 11.3 .0342 .0310 well-defined peak

105 12.7 .0240 .0206 well-defined peak


Table 3
Mlean Square Wave Slopes and Sea Conditions




















































F -'10 0 an m 2


f (Hz)


Fig. 21 The down-wind wave slope spectra.









The equivalent degrees of freedom (EDF) for each spectrum is 100;

the Nyquist frequency is selected to be 200 Hz which is high enough

to avoid the possible signal aliasing. Within the allowable error

range, the wind dependence of the high frequency portion is evident.
The frequency-weighted down-wind wave slope spectra for the same

cases are shown in figure 22. It is noted that the shape of the

spectrum varies from about f- for low wind speed case to about f 1
for high wind speed case. The frequency-weighted down-wind wave slope

spectrum normalized by that at the peak frequency is further illustrated

in figure 23. It is expected that the high frequency part (f/fm > 1)
can be approximated by


f~( )~l=f (5.1)


where f, is the peak frequency of the down-wind wave slope spectrum
and the function F(U ) is to be determined from the shape of the down-

wind wave slope spectrum. The air friction speed U, is related to

the wind speed Ua through the drag coefficient CD, i.e.

U: = CD Ua (5.2)

Figure 24 illustrates the best logarithmic fitted F(U,) for "well-
defined peak" cases as


F(U,) = 0.88 logl0(U,/42.5) (5.3)

where U, is in cm/s.
From the relation between the sea surface displacement spectrum

and the wave slope spectrum in the gravity range


S(f) f4 Elf)


(5.4)

























































IE100 n n


/m/s






-- 7.7
/~-.--.- 7.5
.- 3.5


f (Hz)


Fig. 22 The frequency-weighted down-wind wave slope spectra.

































cnc




m/s
/ ___11.3

/ -~- -- 7.7








JO 10 0 1 0 10' 103

f/fm


Fig. 23 The normalized down-wind wave slope spectra.





























0/ U


F(U*)=0.88 log,,j4.




o Well-Defined Peak
*Mixed Sea


- F U,)


/'





0.6 L'
10


U, (cm/s )


Fig. 24 Best fitted F(U ) from the cases of "well-defined peak."









Mitsuyasu and Honda's result (1974) shown in .equation (2.2) can be

reduced to

f S(f) l og10U,
~-( )( f (5.5)

This is similar to our results shown in equation (5.1). Although

the f- dependence in the wave slope spectrum is true for waves in

the gravity and capillary ranges, the frequency dependence of that for

waves in the capillary-gravity range is not clear. Furthermore, the

dispersion relation of the high frequency wave is affected by the

coexisting current and hence equation (5.4) is no longer valid in

this situation. These can also account for the deviation from f-

in our proposed spectral form shown in equation (5.1).

The mean square slopes for the down-wind and cross-wind components

are presented in order of increasing wind speed in Table 3. The mean

square wave slopes defined as the sum of the wave slopes along two

orthogonal directions are illustrated in figure 25. It is noted that

"mixed-sea" cases show high values of the mean square wave slopes.

All "well-defined peak" cases except one associated with oil-slick

experiment (Ua = 12.8 m/s) show increasing mean square wave slopes
with higher !rind speeds.



5.2 Modulation of Mean Square Wave Slopes by Lonq Waves

To investigate the modulation of the wave slope signal by long

waves, we first calculate the cross-correlation function C(T) of the

local mean square down-wind wave slopes, s2, and the sea surface

displacement 7 from the equation













































10 12


Cox and Munk (1954),Clean Sea Surface
Cox and Munk (1954), Slick
This study Well Defined Peak
This study, Mixed Sea


Fig. 25 Mean square wave slope as a function of the wind speed.


I 1 1 I


o




O~ e
o '



00

o-


2 o o


0.08 >-


0.06E


0.04H -


0.02H


0.000


2 4 6 8

U, (m /s)










C~ I = 2(t)n(t+,)dt (5.6)


where T(= 327.68 sec) is the averaging period. An example of the

cross-correlogram is shown in figure 26. The periodic feature shown

in the figure is typical for all cases studied.

The phase lead ema in degrees is calculated by
maxo
4 mx x 3600 (5.7)
max Tpa

where rma is the lag of the first peak in the cross-correlogram,

Tpeak is the peak period of the waves. It is interesting to note that

emax is a function of wind speed as shown in figure 27. The normalized
cross-correlation function, defined as the cross-correlation function

normalized by the product of the standard deviation of local mean

square down-wind wave slope and that of sea surface displacement, gives

the strength of correlation for each case. The peak values of the

normalized cross-correlation function, "max, are plotted against the

wind speeds in figure 28. Neither "well-defined peak" cases nor

"mixed sea" cases show any significant trend.



5.3 Probability Density Function of Wave Slopes

The probability density functions for down-wind and cross-wind

wave slopes are calculated from 40,000 data points. Figures 17, 29

and 30 show the probability density functions of "well-defined peak"

cases at various wind speeds. Figure 31 illustrates a probability

density function in a "mixed sea" case.






























-2










-4


O 2 4 6

T (sec)


Fig. 26 The cross-correlation function of mean square down-wind
wave slope and sea surface displacement for Run 325.
The lag of the first peak in the cross-correlation
defines Ima












I I 1 1 1 1 11


Well-Def ined Peak
Mixed Sea


100


20E-


2 1 1 Il I l lI
2 3 4 5 6 IC


Ua (m/s)
Fig. 27 The phase lead emax as a function of wind speed Ua.
























I .0



0.8-




x, 0.6 o

0.






0. 8 -01





Ua (m/s)

Fig. 28 Peak value of the normalized cross-correlation function as a
function of wlinnd speed. The open circles and solid points
denote p values of "well-defined peak" cases and "mixed-
sea" cas 0~ respectively.











































-3 -2. -1. O i. 2. 3.

NORMALIZED SLOPE



Fig. 29 Probability density function for Run 328. The notation used
is the same as that in figure 17; U = 3.5 mn/s; the variances
are 0.0094 for down-wind wave slopesaand 0.0101 for up-wind
wave slopes.





































.2-







-3. 2. 1. O I. 2. 3.

NORMALIZED SLOPE




Fig. 30 Probability density function for Run 325. The notation used
is the same as that in figure 17; Ua .7ms the variances
are 0.0159 for down-wind wave slopes and 0.0314 for cross-wind
wave slopes.








67





























o


4-3 o2 o o 3

NORA .ZE 0


Fig. 3 Probbilit densiy funtion O R 12 h oai ue




|- ,o aea hti iue1; ,=63ns h aine
ar .09 or *onwn lpsad0017frcoswn lp









All cases show similar distributions; cross-wind slopes are

relatively peaked at the origin compared to the down-wind slope

distributions. The down-wind slope distribution does not show

obvious skewness in the windward direction; this is not consistent

with the observations by Cox and Munk (1954). In contrast, some cases

reveal (figures 17, 30 and 31) double peaks in the down-wind

probability density functions.



5.4 Modulation of Short Waves by Long Waves

To study the radar imaging mechanism, we need to investigate the

hydrodynamic modulation level for a particular wavelength band.

This work was done by carrying out the cross-correlation analysis on

the demodulated short wave signals and the sea surface displacement.

Table 4 illustrates the general characteristics of the short

wavelength bands elected for the modulation study.

If we compare equation (2.31)with equation (2.35), we have the

following analogies between measurements of hydrodynamic modulation

and those of radar modulation:

Radar Modulation Hydrodynamic Modulation

P F

p F(O


v nI
MradarM

Because the line of sight speed v is of the same order as on~ and

the short wave energy denstiy is proportioned to the short wave slope

square. Equation (2.36) can be generalized to























C
U



UI ~
5-















UC
0 CD
.oC
a

















yN











LC I



OU

.0














r-


r

.C

O



Y

01
>r

Z






*r-



as


o -
NU,



5- 0
.0 0
.C

Om

.OC

cr
1.

Or










- $.
"Z


T
C





01
n



(U)










II

L.





.C












vi


O 1.0

O 2


X 0 -3


M n10)0















CO N


(Un 1mC











E,
M = 1 sni (5.8)
52 k~ nn1

whreE5 is the cross-spectrum of the demodulated mean square slope

signal and the sea surface displacement, Enn is the auto-spectrum of
the latter and ki is the long wavenumber.

Figures 32 to 34 illustrate the hydrodynamic modulation levels

of the three wavelength bands selected at three different wind speeds.

It is noted here that the hydrodynamic modulation level is the lowest

for the 8 cm wave at the lowest wind speed. The wind-dependence

for various wavelength bands is shown in figures 35 to 37.

The most striking feature is that the lower the mean wind speed the

higher the modulation level for each wavelength band.

The modulation level for one "mixed sea" case, shown in figure 38,

shows much higher values than those in "well-defined peak" cases under

similar wind speed conditions.





















25.000a



20. 000



15. 000



10. 000



5. 000



0.000


o 23 cm
O 8 cm
n 3 cm


00


0B


0. 10 0. 20 0. 30 0. 40 0. 50
FREQUENCY (Hz)



Fig. 32 Hiydrodynamlic modulation level for Run 328. Ua = 3.5 m;/s.























25. 000



20. 000-
O 23 cm
O 8 cm

15. 000-



10. 000-



5. 000 O




0. 10 0. 20 0. 30 0. 40 0. 50
FREQUENCY ( Hz )


Fig. 33 ;iydrodynamic modulation level for Run 325. Ua = 7.7 m/s.























































Fig.


25. 000



20. 000-
O 23 cm
o 8 cm

15. 000 t- a





oo

O 0




o. 000
0. 10 0. 20 0. 30 0. 40 0. 50
FREQUENCY (Hz)



341 Hydrodynamic modulation level for R~un 108. Ua = 11


L


.3m/s.


















25. 000 -~



20. 000-
n n 3.5 m/s
o 7.7 m/s
O 11.3 m/s
15. 000-




10. 000 -




5. 000 ~- O o n n


coBaaeagBoo o 8a

0. 000
0. 10 0. 20 0. 30 0. 40 0. 50
FREQUENCY (Hz)



Fig. 35 Hydrodynamic modulation level for 23 cm waves.




















25. 000




20. 000
a 3.5 m/s
o 7.7 m/s
O 11. 3 m/ s
15. 000 a




10. 000-









0. 000
0. 10 0. 20 0. 30 0. 40 0. 50
FREQUENCY (Hz)




Fig. 36 Hydrodynamic modulation level for 8 cm waves.



















25. 000I




20. 000
a 3. 5 m /s
o 7.7 m/s
O 11.3 m/s
15. 000 t- O




10, 000-




5. 000-



0. 000co
0. 10 0. 20 0. 30 0. 40 0. 50
FREQUENCY (Hz)


Fig. 37 Hydrodynamic modulation level for 3 cm waves.















40 _L oa un
A 8 cm
3 cm










20-





10-






1. 1 0. 2 0. 3 0. 4 0. 5
FREQUENCY (Hz)

Fig. 38 Hydrodynamic modulation level for a "mixed-sea" case.
Run ID = 102, Ua = 6.3 m/s.














CHAPTER VI
DISCUSSION OF RESULTS



The results presented in the previous chapter are scarce in

company. Besides the observation of Cox and Munk (1954) and those of

Evans and Shemdin (1980) we know of no other direct field measurements

of wave slopes. In fact, this study is the only in situ field investi-

gation of the hydrodynamic modulation in a two dimensional wave field.



6.1 Wave Slope Spectra

The principal finding from the down-wind wave slope spectra is the

dependence of high frequency waves on wind speed. Under laboratory

conditions the wind speed dependence of the down-wind slope spectrum is

reported by various investigators, the last being Lleonart and

Blackman (1980). The conclusions derived from the laboratory experi-

ments all support a f- law for high frequency waves (equation (2.5)).

The same investigation does not provide converging views on the wind

dependence of the high frequency waves as stated in section 2.2. The

proposed spectral form shown in equation (5.1) is seen to agree well

with the spectral form (equation (5.5)) suggested by Mitsuyasu and

Honda.

The mean square wave slopes measured by Cox and Munk (1954) from

sun glitter have compared with our results, as shown in figure 25. All

"well-defined peak" cases show the same trend as those reported by

Cox and Munk. The results from the "well-defined peak" cases fall








within the limit of data scatter and in agreement with those of Cox and

Munk. The "mixed sea" cases show mean squared slopes that are higher

than those of "well-defined peak" cases and those of Cox and Munk.

This peculiar behavior is currently under further investigation.



6.2 Modulation of Mean Square Wave Slopes

The modulation of mean square slopes is measured by the cross-

correlation of the mean square wave slope and sea surface displacement

as shown in equation (5.6). The cross-correlogram reveals the presence

of the dominant frequency which is close to that at the peak of the

surface displacement spectrum. This result suggests that our assumption

of the cyclic change in the energy spectrum made in equation (2.29)

is reasonable.

The local mean square slope leads the long wave with the angle

max varying from 30 to 1500 with wrind speeds varying from 3.0 m/s to
12.7 m~/s. Evans and Shemdin (1980) reported ema to vary from 300 to

450 with a wind speed Ua = 5 +_ 1 m/s. Our emax suggests a strong

variation of ena with U .
When the radar incidence angle is small (<200), the radar back-

scatter is from the entire wave spectrum rather than from the Bragg

waves only (Valenzuela, 1978). Our cross-correlation analysis pre-

sented here should be useful for determining radar modulation at such

small incidence angles. The normalized cross-correlation function

which gives the relative strength shows no evidence of dependence

on wind speed (see figure 28).








6.3 Probability Density Function

All the probability density functions computed are found to

follow the Gaussian distributions; this is consistent with

Wu's laboratory results (1971). The cross-wind wave slope distribution

is slightly peaked, this follows the field observation by Cox and

Munk (1954).

However, the double peak in the probability density function of

down-wind wave slopes (see figures 17, 30 and 31) might be caused by

the peak downward profile of capillary waves as noted by Schooley

(1958). The field investigation by Cox and Munk (1954) from sun

glitter probably averaged out these fine scale features of waves.



6.4 Hydrodynamic Modulation Level

Few direct measurements of the short wave modulation by long waves

have been made in the field. In fact this is the first attempt to

investigate the short wave slope modulation in a two-dimensional field

setting. The only comparable measurements are those of Wright et al.

(1978) who used frequency modulations of backscattered microwaves to

obtain the wave orbital speed and backscattered power modulation to

obtain the amplitude modulation of the (short) scattering waves. By

using the optical sensor and other supporting measurements, it is

hoped that the actual hydrodynamic modulation levels can be measured

directly rather than inferred from radar backscatter.

The maximum modulation level predicted from the relaxation model

is -(ks/F(0)(aF(0/aks) +1 which is higher for the steeper spectral

shape. The modulation levels shown in figures 32 to 34 with values 3 to

25 are larger than -(ks/F(0) )(;F(0)/Bks) t1 in general. However, the









wind speed dependence of the modulation level illustrated in figures

35 to 37 agrees well through the inference of the relaxation model

and the wind speed dependence on the spectral shape (section 5.1).

That is the lower the wind speed, where the down-wind slope spectral

shape is steeper, the higher the modulation level as predicted from

the relaxation model.

The power dependence of the high wavenumber (frequency) energy

spectrum will make (ks/F(0)(aF(0/aks) independent from the wavenumber
of the short wave and therefore, according to the relaxation model,

the modulation level is independent of the selected wavelength band.

Our results show that the modulation levels for 3, 8 and 23 cm waves

are of the same order.

Similar to the radar modulation transfer function reported by

Wright et al. (1980), the increase in the modulation level with

decreasing wave frequency is opposite to that predicted by the

relaxation model if the relaxation rate is assumed constant.

Figures 39 to 41 show comparisons of the hydrodynamic modulation

levels for 8 cm and 23 cm waves with the modulus of the radar transfer

function for X band (Xr = 2.3 cm) and L band (Ar = 23 cm) radars.

The hydrodynamic modulation level, |M is smaller than the modulus

of radar modulation transfer function, |Mradarj. The radar modulation
transfer function measured by Wright et al. includes the possible

scattering effect of the intermittent breaking of short waves when

they are saturated at the long wave crest. Our wave follower

measurement, due to the inherent restrictions, averaged out this fine

scale wave breaking. Further studies are needed to investigate the

contribution of the small wave breaking on the radar backscatter.




















7E.


20

O 23cm

r' 15 L band
-o O X band



10~ -o









0.10 0.20 0.30 0.40 0.50
FREQUENCY (Hz)




Fig. 39 Comparison of hydrodynamic nodulation level and modulus of
radar modulation transfer function for Ua = 3.5 m;/s.



















25



20

O 23cm

S15~ L bnd


IO-on

OM





5-n O
OL--~O

0.10 0 20 0.30 0 40 0.50
FREQUENCY (Hz)


Fig. 40 Comparison of hydrodynamic modulation level and modulus of
radar modulatiion transfer function for Ua = 7.7 m/s.







841















20-

O 23 cm
8 3c
r 15 L band
~ AX band








sC


0.10 Q20 0.30 040 0.50
FREQUENCY (Hz)


Fig. 41 Compiiarison of hydrodynamlic modulation level and mnodulus of
radar modulation transfer function for Ua = 11.3 m/s.









At the low wind speed (figure 29) |M| and |Mradar| are of the
same orders*, this implies that hydrodynamic modulation is strong

enough to constitute a major radar imaging mechanism for ocean waves.













CHAPTER VII
CONCLUSIONS AND RECOMMENDATIONdS



7.1 Conclusions

The cases investigated in this study covered a wind speed range

from 3.0 m/s to 12.7 m/s. Within this range the important conclusions

are reported below.

(i) The slope of the high frequency region of the normalized

down-wind wave slope spectrum is wind-speed dependent.

(ii) Mean square wave slopes in the "well-defined peak" cases

are consistent with observations reported by Cox and

Munk (1954). However, the mean square slopes from the

"mixed sea" cases show higher values than those observed

by Cox and Munk (1954).

(iii) The mean square down-wind slope, which is proportional

to the short wave energy, reveals a modulation induced

by the dominant long wave. The phase lead of this

modulation, relative to the dominant long waves, increases

with the wind speed.

(iv) The hydrodynamic modulation level, obtained by the

demodulation technique, decreases with the increasing wind

speed. This tendency is consistent with the relaxation

model results (Keller and Wright, 1975).

(v) The measured hydrodynamic modulation magnitudes are of the

samie order as the radar modulation transfer function for









the low wind cases, the former are smaller than the latter

for the high wind speed cases. This implies that the

hydrodynamic modulation at low wind speeds is strong

enough to constitute an important mechanism in radar

imaging of ocean waves.

(vi) The decreasing trend of the hydrodynamic modulation

level with increasing wave frequency agrees with radar

modulation transfer function results.



7.2 Recommendations

Future work in the line of the present research is described as

follows:

(i) In this experiment the laser-optical sensor is the sole

instrument that provides the wave slope components along

two specified directions. Slope components thus obtained

could be contributed by waves coming from other directions.

To make more direct comparison with the data measured by

radar which only responds to the ocean wave coming from

one particular direction, it is required to design a

hardware system that is capable of detecting the

directional properties of short waves.

(ii) Although the demodulation procedure developed in this

dissertation is complete in its present form, alternative

analyses can be pursued conveniently in the frequency

domain. However, the high frequency waves encountered

are modulated by the unsteady underlying current and







88

therefore the demodulated spectrum can only be estimated

locally. This requires use of analog methods.

(f ii) To date, the radar modulation study has excluded the

possible scattering effect due to the intermittent short

wave breaking. Studies of the drop-out patches in the

received optical sensor signal can be used to study short

wave breaking characteristics.












APPENDIX A
ALGORITHM FOR ESTIMATING THE WAVE HEIGHT DIRECTIONAL SPECTRUM


Various methods for the estimation of the directional wave height

spectrum were proposed for different measured parameters (Longuet-Higgins
et al., 1963; Panicker, 1971; Forristall et al., 1978). For the MARSEN

Noordwijk tower experiment, the current meter was located at a horizontal

distance of x0 = 3.66 m from the location where the surface displacements
were measured. Here we proposed an algorithm to estimate the wave height

directional spectrum E(f,B).
As stated above, we have the surface displacement measurement which

is specified as


n(0,0,t) = jt Amj cos(-2nfmt m~j). (A-1)


Here A is the amplitude of the component wave, fn is the circular

frequency which is related to the wavenumber km as

(2nif ) = gkm tanh k h,

t is the time and emj is the random phase uniformly distributed on
the interval (0,2n). The u-current is specified as


u(x0,0,t) = u(t) = J Amj casej Rm cos (kmx0 cos ej

2nfm mQj), (A-2)

and the v-current component is given by








v(x ,0,t) = v(t) = E A sine Rm cos(k xg casej


-2n~1fm mj ) (A-3)

where
cosh km zg
R =2f 0 ,(A-4)
Rm m sfinh k h

and

AZ. = 2E(f ,e .) af, re (A-5)

In equation (A-4), zO is the vertical distance of the sensor above the
sea bed, h is the depth of the water (h = 18 m).

The cross-correlation function R(r) defined as the mean of the

product of two functions separated by a lag r can be derived for three
measurements as follows:
A2
R (7), = 7 ,_ml Rm cos a cos(k xg cos e 2*f 1) (A-6)


A2
R V(r) = m a Rn sin 6 cos(k xU cos e 21Tf Tl) (A-7)

A2
R (7~) = "m R2 sin e cos e cos(2nf r) (A-8)

A2
R ~(r) =I mi cos(2nfr) (A-9)

A2
R (T), = E~ mj~ R~ sin2 i0.Cos(27f I) (A-10)


Rv(') = mJ 2 ICOS' 6. cos(2fnf T) (A-11)


Strictly speaking, equations (A-9) to (A-11) are auto-correlation
functions. For the infinitesimal increment of f and 6, we can change




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