FIELD INVESTIGATIONS OF THE SHORT WAVE
MODULATION BY LONG WAVES
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
This dissertation is dedicated to Chi-Mei.
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
ACKNOWLEDGEMENTS . . . . . . . . . . .
LIST OF FIGURES . . . . . . . . . . . .
LIST OF TABLES . . . . . . . . . . . .
ABSTRACT . . . . . . . . . . . .
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...
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
A Algorithm for Estimating the Directional Wave Height
Spectrum .. . .. .. . .. . . . 89
B Demodulation Procedure . .. ... .. ... .. 96
REFERENCES .. . .. . .. .. .. . .. .. .. 99
BIOGRAPHICAL SKETCH . . .. - - . . . 103
LIST OF FIGURES
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
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
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
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
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
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
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
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.
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
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
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
Elf) = 0.1393 f-b (2.2)
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
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
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
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
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
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
+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)
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
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
-(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)
In the relaxation model, the relaxation rate p is introduced such
F(1) = (1) eik x
f(1) can be solved by substituting equation (2.28) into equation (2.27)
(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
-a) + ip] (2.301)
k (0)i k 1
N = (-aa + iu)
i92 + 02
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
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
O 1 2 3
Fig. 2 Hydrodynamic modulation transfer function for short gravity
waves from relaxation model..
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
line of sign speed. V(t) can be detected directly from the instan-
taneous Doppler shift of the backscattered signal.
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.
------ - 'ci-~
--r- L- ----~-
Fig. 4 The wave follower in the wave following mode.
Fig. 5 Close view of the optical sensor mounted on the wave follower.
End ess Cable`
Fig. 6 Schematic diagram of the wave follower.
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
(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
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).
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
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.
of the Cases Selected for Detailed Data Analysis
Number of data
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
speed (m/s) direction (deq)(2)
Fig. 9 Definition Sketch of true direction. The true direction ea
is the angle of Va referred to the east counterclockwise.a
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)
Yn N j exp(-i 2nj n/N) (4.1)
yj = y(jn), i2 = -1 and n = 0, 1, ..., N-1.
Its inverse transform is
y, = z Yn exp(i 2;rj n/N) (4.2)
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
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
YK= yr W.jW exp(-i 2nj n/L) (4.3)
where W is the window function given by
The estimate of the spectrum for the
I T (4.5)
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.
20.Tiilie (sec) 30.
velocity after removal
The u-comlponentt of the orbital
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.
U,-OawN WIND SLOPE
U. 10. 20. 30. 40.
b. The down-wind slope record.
Fig. 11 Sample time series for sea surface displacement and slope for Run 108.
UITER SURFACE ELEVATION
0. 10. 2(
a. Sea surface displacement.
O .2 .4
Fig. 12 Frequency
spectra for n7 and u for Run 108. The solid line
surface displacement spectrum. The dashed line
u-component of the orbital velocity/ spectrum.
0 0.1 0.2 0.3 0.4
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
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
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
O I I 20 30
Fig. 14 Calibratiion results for thec deflection annle; r is the
O 90 180 270
Fig. 13 Calibration results for thle azimiu~th angle; p is the
10- 10 110
Fi.1 h onwn lp petu o u 0 ; EF=10
-3 -2 -1 1 2
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.
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
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
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.
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
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. 03 ,-0.186Hz
0. O 0. 1 0. 2 0. 3 0. 4
Fig. 18 An example of the sea surface displacement spectrum for
"mixed sea" case. Run ID = 102, Ua = 6.3 m/s.
0. 176 Hz
7 0. 20 -
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. 0 0. 1 0. 2 0. 3 0.4 0. 5
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
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
Mlean Square Wave Slopes and Sea Conditions
F -'10 0 an m 2
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)
IE100 n n
Fig. 22 The frequency-weighted down-wind wave slope spectra.
/ -~- -- 7.7
JO 10 0 1 0 10' 103
Fig. 23 The normalized down-wind wave slope spectra.
o Well-Defined Peak
- F U,)
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
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
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
2 o o
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
4 mx x 3600 (5.7)
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.
O 2 4 6
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
I I 1 1 1 1 11
Well-Def ined Peak
2 1 1 Il I l lI
2 3 4 5 6 IC
Fig. 27 The phase lead emax as a function of wind speed Ua.
x, 0.6 o
0. 8 -01
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.
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
-3. 2. 1. O I. 2. 3.
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
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
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
X 0 -3
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.
o 23 cm
O 8 cm
n 3 cm
0. 10 0. 20 0. 30 0. 40 0. 50
Fig. 32 Hiydrodynamlic modulation level for Run 328. Ua = 3.5 m;/s.
O 23 cm
O 8 cm
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.
O 23 cm
o 8 cm
15. 000 t- a
0. 10 0. 20 0. 30 0. 40 0. 50
341 Hydrodynamic modulation level for R~un 108. Ua = 11
25. 000 -~
n n 3.5 m/s
o 7.7 m/s
O 11.3 m/s
10. 000 -
5. 000 ~- O o n n
coBaaeagBoo o 8a
0. 10 0. 20 0. 30 0. 40 0. 50
Fig. 35 Hydrodynamic modulation level for 23 cm waves.
a 3.5 m/s
o 7.7 m/s
O 11. 3 m/ s
15. 000 a
0. 10 0. 20 0. 30 0. 40 0. 50
Fig. 36 Hydrodynamic modulation level for 8 cm waves.
a 3. 5 m /s
o 7.7 m/s
O 11.3 m/s
15. 000 t- O
0. 10 0. 20 0. 30 0. 40 0. 50
Fig. 37 Hydrodynamic modulation level for 3 cm waves.
40 _L oa un
A 8 cm
1. 1 0. 2 0. 3 0. 4 0. 5
Fig. 38 Hydrodynamic modulation level for a "mixed-sea" case.
Run ID = 102, Ua = 6.3 m/s.
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
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)
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
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.
r' 15 L band
-o O X band
0.10 0.20 0.30 0.40 0.50
Fig. 39 Comparison of hydrodynamic nodulation level and modulus of
radar modulation transfer function for Ua = 3.5 m;/s.
S15~ L bnd
0.10 0 20 0.30 0 40 0.50
Fig. 40 Comparison of hydrodynamic modulation level and modulus of
radar modulatiion transfer function for Ua = 7.7 m/s.
O 23 cm
r 15 L band
~ AX band
0.10 Q20 0.30 040 0.50
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.
CONCLUSIONS AND RECOMMENDATIONdS
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.
Future work in the line of the present research is described as
(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
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.
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)
cosh km zg
R =2f 0 ,(A-4)
Rm m sfinh k h
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:
R (7), = 7 ,_ml Rm cos a cos(k xg cos e 2*f 1) (A-6)
R V(r) = m a Rn sin 6 cos(k xU cos e 21Tf Tl) (A-7)
R (7~) = "m R2 sin e cos e cos(2nf r) (A-8)
R ~(r) =I mi cos(2nfr) (A-9)
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