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TRANSFORMATION OF WAVES
ACROSS THE SURF ZONE

Galo Padilla Teran

SoP?SrGRADUArE SCHOOL
MOWTERfy, CALIFORNIA 93943-5002

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESIS

TRANSFORMATION

WAVES

ACROSS

THE

SURF

ZONE

Galo

Padilla Teran

•

March

1981

Thesis

Advisor:

E.B

. Thornton

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Transformation of Waves Across the Surf
Zone

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Galo Padilla Teran

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Naval Postgraduate School
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Wave transformation model Shoaling

Waves

Surf Zone

Distributions

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Comparisons between the empirical distributions and
theoretical distributions, and between measured and
theoretical rms wave heights, are made. It is found
that Goda's model over-predicts the tails and under-
predicts the peaks of the empirical distributions, and
that the calculated rms wave heights are too large
compared with measured values .

The range of breaking, and the coefficients used in
the breaking criteria by Goda, are modified in order to
obtain a model which better fits the distribution of
observed hieghts , and which matches the model and
observed rms wave heights. The results are quite good,
with error envelope for predicted rms wave heights less
than 20%. Linear shoaling theory is applied to the
model and found to be as good as applying nonlinear
theory .

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Transformation of Waves Across the Surf Zone

Galo Padilla Teran
Lieutenant, Ecuadorean Navy
Ecuadorean Naval Academy, 1970

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY

from the

NAVAL POSTGRADUATE SCHOOL
March 19 81

1 KjUi^

ABSTRACT

Goda's (1975) model, describing wave transformation from
deep water to across the surf zone, is compared with a large
amount of wave data obtained from experiments conducted at
Torrey Pines Beach, San Diego, California. Goda's model
simulates wave breaking by truncating the Rayleigh distribu-
tion in order to estimate the wave height distributions across
the surf zone; wave heights are shoaled by applying nonlinear
theory. Comparisons between the empirical distributions and
theoretical distributions, and between measured and theoreti-
cal rms wave heights, are made. It is found that Goda's
model over-predicts the tails and under-predicts the peaks
of the empirical distributions, and that the calculated rms
wave heights are too large compared with measured values.

The range of breaking, and the coefficients used in the
breaking criteria by Goda, are modified in order to obtain
a model which better fits the distribution of observed
heights, and which matches the model and observed rms wave
heights. The results are quite good, with error envelope
for predicted rms wave heights less than 20%. Linear shoal-
ing theory is applied to the model and found to be as good
as applying nonlinear theory.

TABLE OF CONTENTS

I. INTRODUCTION 12

II. THEORETICAL BACKGROUND 14

A. RAYLEIGH DISTRIBUTION 14

B. TRUNCATED PROBABILITY DISTRIBUTIONS 17

1. Collins Distribution 18

2. Battjes Distribution 18

3. Kuo and Kuo Distribution 19

4. Goda Distribution 20

5. Summary 23

III. EXPERIMENT 24

A. INSTRUMENTS 24

B. DATA ANALYSIS 25

IV. RESULTS 29

A. TYPICAL SPECTRA 29

B. HEIGHT STATISTICS 29

C. COMPARISON OF EMPIRICAL WITH

MODEL DISTRIBUTIONS 30

D. COMPARISON OF RMS WAVE HEIGHTS 33

E. WAVE HEIGHT DISTRIBUTIONS USING

CURRENT METERS 35

F. COMPARISON OF MODEL AND MEASURED

CUMULATIVE DISTRIBUTIONS 36

V. CONCLUSIONS 37

BIBLIOGRAPHY 56

INITIAL DISTRIBUTION LIST 59

LIST OF TABLES

I. Truncated Probability Densities. The dotted
lines represent the original Rayleigh dis-
tributions and the heavy lines represent

the modified distributions 39

II. Wave Height Statistics (W and C represent

wave staffs and current meters respectively) 40

III. Measured and Calculated rms Wave Heights
Obtained with Goda ' s Model and Modified

Goda's Model 41

LIST OF FIGURES

1. Cross-section of surf zone showing instrument
spacing and elevations relative to measured
waves on 20 November 19 78 at Torrey Pines

Beach, California 42

2. Definition sketch of zero-up-crossing wave heights - 43

3. Typical spectra measured during the experiments 44

4 . Empirical distribution of wave heights compared
with those predicted with Goda's model, starting
in deep water (P7) and going into shallow

water (W21,W38) , 20 November 1978 45

5. Empirical distributions of wave heights compared
with those predicted with Goda's model, starting
in deep water (W21) and going into shallow

water (W38,W41), 17 November 1978 46

6. Empirical distributions of wave heights compared
with those predicted with the modified Goda's

model, 20 November 1978 47

7. Empirical distributions of wave heights compared
with those predicted with the modified Goda's

model, 17 November 1978 48

8a. Range of measured and Rayleigh root-mean-square

wave heights 49

8b. Range of measured and Rayleigh significant

wave heights 49

9. Correlation of measured rms wave heights with

calculated Goda's rms wave heights 50

10. Comparison of the changes of Hj-^ with the Goda's
model applying nonlinear (heavy line) and linear
shoaling (light line) . Upper figure illustrates
the unmodified model; lower figure illustrates

the use of the modified coefficients in the model — 51

11. Percentage error of predicted (modified model)
compared with measured rms wave heights 52

12. Empirical distributions of wave heights obtained
from current meters (C23, C37 and C40) compared
with predicted wave heights calculated with the
modified model, 17 November 1978 53

13. Comparison of measured cumulative exceedance
distributions with predicted distributions
(modified model) , 20 November 19 78 (W38) and

17 November 1978 (W41) 54

14 . Comparison of measured cumulative exceedance
distributions with predicted distributions
(modified model) , 20 November 1978 (W21)

and 17 November 1978 (W38) 55

LIST OF SYMBOLS

a Root-mean-square (rms) amplitude

A Goda's breaking criteria coefficient

C Speed of energy propagation

E Energy density

g Acceleration due to gravity

h Local depth below still water level

S Sea surface elevation

H, Breaking wave height

H Deep water wave height

H Wave height parameterizing truncated Rayleigh
distribution

H Root mean square wave height

rms n 3

H Transfer function that relates the velocity spectrum
components to the kinetic energy

H Transfer function that relates potential energy to
the kinetic energy

k Wave number

K Goda's breaking criteria coefficient

KE Kinetic energy

K Shoaling coefficient

L Deep water wave length

m Lowest moment variance of the frequency spectrum

p Probability density function of wave heights

p Probability density function of unbroken waves

S Horizontal velocity spectrum, x-component

S Horizontal velocity spectrum, y-component

T Wave period

X Ratio wave height to deep water wave height

X, Higher limit of breaking range

X« Lower limit of breaking range

z Measurement elevation
m

an Deep water incident angle

a. Incident wave angle at breaking

6 Bottom slope

6 Delta function

Y Ratio of breaking wave height to depth of water at
breaking

v Root mean square spread of the noise about the mean
frequency

p Water density

ACKNOWLEDGMENTS

The author wishes to express his appreciation to Dr.
Edward B. Thornton, Professor of Oceanography at the Naval
Postgraduate School, Monterey, California, as Thesis
Advisor, for his guidance, method and systematic assistance
in the preparation of this study. The assistance of Ms.
Donna Burych, Computer Programmer in the Oceanography Depart-
ment is gratefully recognized.

I. INTRODUCTION

The evaluation of an irregular group of shoaling waves
as they approach and pass through the breaker zone is a com-
plex process which requires special measurements and analy-
sis considerations . The usual approach to shallow water
wave transformations is to predict, from a single "represen-
tative" set of deep water parameters, the wave height, the
wavelength and the frequency at specific shallow water depths,
using linear shoaling theory. The primary objection to this
approach is that a single set of deep water wave parameters
does not realistically represent the distributional charac-
teristics of naturally occurring sea surface waves. A
secondary objection arises from the use of linear transfor-
mations which become inadequate when applied through the
surf zone (Wood, 1974).

Wave heights in deep water, having Gaussian surface ele-
vations, are described by the Rayleigh distribution (Longuet-
Higgins , 1952). Waves propagating toward shore can increase
in height due to shoaling effects, refraction and wave inter-
actions, and eventually reach a depth where they start break-
ing. The energy dissipation due to breaking has been simu-
lated (Goda, 1975) by truncating the tail of the Rayleigh
distribution .

Experiments were conducted at Torrey Pines Beach, San
Diego, California, during November 1978. Sea surface eleva-
tions, pressures and velocities were measured at closely

spaced locations in a line extending from 10m depth to
inside the surf zone. This thesis applies Goda ' s model to
the measurements in order to examine the shoaling and trans-
formation of wave heights and their probability density func-
tions (pdf 's) from deep water to breaking and across the
surf zone to the shoreline.

II. THEORETICAL BACKGROUND

A. RAYLEIGH DISTRIBUTION

The Rayleigh distribution was shown theoretically by
Longuet-Higgins (1952) to apply to deep water wave heights
on the assumption that the sea waves are a narrow-banded
Gaussian process. Barber (1950) had earlier presented empiri-
cal evidence that the Rayleigh distribution agreed with the
measured distribution of waves. On the assumption that the
wave height is twice the wave amplitude, the wave height
probability density is then represented by

p(H) = 2H/H* exp(-H2/H^ ) CD

rms rms

where H is the rms wave height.

rms 3

Using pressure records in the Gulf of Mexico, Longuet-
Higgins (1975) observed that the Rayleigh distribution fits
the observed distribution reasonably well in "fairly deep
water" . He found that there is a slight excess of waves
with heights near the middle of the range and a deficit at
the two extremes. Since much of the high-frequency portions
of the wave records were filtered out by the pressure trans-
ducer, Longuet-Higgins (197 5) suggested that the narrow band
approximation may not be as applicable for the unfiltered
records. In shallow water with much steeper waves, the
Rayleigh distribution can again be expected to be less
applicable due to the non-linearities.

Since the Rayleigh distribution theoretically did not
apply to broadband wave spectra, Goda (1970) numerically
simulated wave profiles, where the amplitudes were specified
by various theoretical spectra of varying bandwidth and the
phase was random. He then examined the simulated records for
surface elevations, crest-to-trough wave heights and zero-
up-crossing wave heights. He found that, using the zero-up-
crossing determination of wave heights, the Rayleigh distri-
bution is a good approximation irrespective of the spectral
bandwidth. Tayfun (1977) , in studying the transformation of
deep water waves to shallow water waves, showed that the
Rayleigh distribution for wave amplitude was generally
applicable to all bandwidths .

The Rayleigh distribution is applied correctly only to
low waves in deep water (Longuet-Higgins, 1975), since it is
assumed that the contributions from different parts of the
generating area are linearly superposable . Under this assumpt-
tion, the distribution clearly should not hold for waves
approaching maximum height, i.e., close to breaking, as in
the surf region or even in the open sea with whitecaps. It
has been found by several authors (Chakrabarti and Cooley,
1977; Forristal, 1978) that the theoretical Rayleigh dis-
tribution over-predicts the maximum wave in the tail compared
with large wave observations. Forristal (1978) attributed
the differences to the non-linear, non-Gaussian and skewed
nature of the free surface.

Tayfun (19 80) examined non-linear effects by consider-
ing an amplitude-modulated Stokian wave process with the
restriction that the underlaying first order spectrum is
narrow band. The surface displacements were found to be
non-Gaussian and skewed, and wave heights distributed ac-
cording to the Rayleigh probability law, particularly for low
and medium wave height ranges. On the basis of the results
obtained, Tayfun (1980) concludes that the non-Gaussian
characteristics of the free surface do not directly result
in reducing maximum wave heights in a manner consistent with
field observations and that a more plausible mechanism is
wave breaking, which is a non-linear effect not directly
accounted for in the analytical wave models currently available

Longuet-Higgins (1980) analyzed the effects of non-
linearity and finite bandwidth on the distribution of wave
heights to explain the differences with observations found
by Forristal. He found that the reason for the discrepancy
could be accounted for by the presence of free background
"noise" in the spectrum, outside the dominant peak, and that
it was not due to a finite-amplitude effect. Longuet-Higgins
concludes that the distribution of wave heights even in a
storm is well described by the Rayleigh distribution, pro-
vided the rms amplitude, a, is estimated from the original

— 1/2
record and not from the frequency spectrum as a = (2m_)

The effect of finite bandwidth is estimated from a model

assuming low background noise linearly superposed on a very

narrow (delta function) spectrum. For narrow bandwidths ,

he obtains the formula

a2/2m0 = 1 - 0.734 v2 (2)

where v is the rms spread of the noise about the mean fre-

2
quency. Values of v corresponding to Pierson-Moskowitz

(broad-band) spectra also give results in close agreement

with observation. Therefore, the Rayleigh distributions

calculated in this paper are parameterized using the rms

wave height.

B. TRUNCATED PROBABILITY DISTRIBUTIONS

In concept, waves are described by the joint distribu-
tion of height, period (or equivalent wavelength) and direc-
tion. To simplify the analysis, all authors assume a very
narrow band frequency spectrum and a narrow directional spec-
trum, so that all the wave heights of the distribution are
associated with a single mean frequency and mean direction.
Therefore, starting in deep water, the waves are described
by the unaltered single-parameter Rayleigh distribution,

(with the implied assumptions. The deep water wave heights
are transformed into shallow water waves using shoaling
theory in which frictional dissipation is neglected. Even-
tually the waves reach such shallow water that they start
to break, with the largest waves breaking furthest offshore
first. Wave breaking is simulated by truncating the tail
of the Rayleigh distribution.

1 . Collins Distribution

Collins (1970) was the first to apply the technique

of a truncated distribution to describe the effects of wave

breaking, using a sharp cut-off with all broken waves equal

to H, which results in a delta function at H, . Collins does
b b

not give an explicit formula for his distribution, but it
would be the same as the described by Battjes (1974) (see
below) . He used linear shoaling theory and the breaking
criterion after Le Mehaute and Koh (1967

Hb/HQ = 0.76 tan1/76(H0/L0)"1/4 (3)

where tan 3 is the bottom slope and Hn and L_ are the deep
water wave height and length, respectively. For waves break-
ing at an angle a, the bottom slope is actually tan 3 cos a. ,

1/2
and H. should be replaced by H~ cos / a,.. The various dis-
tributions, and how they are truncated are showed schemati-
cally in Table I .

2 . Battjes Distribution

Battjes (1974, 1978) again used a sharp cut-off of
waves and applied the breaking criterion based on Miche ' s
formula for the maximum height of periodic waves of constant
form,

Hb z 0.88/k tanh (y/0.88 kh) (4)

where y is an adjustable coefficient. In shallow water (4)
reduces to

Hb = y h . (5)

He uses linear theory to shoal the waves. The probability
density for breaking wave heights is given by:

p(H) = H/2H2 exp[-l/2(H2/H2) ] , for 0 <_ H <_ Hb (6)
p(H) = exp[-l/2(H2/H2) ]5(H -Hb) , for H>Hb (7)

where H is the wave height parameterizing the truncated
Rayleigh distribution by Battjes. All waves that have
broken, or are breaking, assume the height prescribed by

(7); this results in a delta function at the truncation
height, H, , of the distribution (see Table I).
3 . Kuo and Kuo Distribution

Kuo and Kuo (19 74) investigated the effect of break-
ing on wave statistics using a conditional Rayleigh distri-
bution sharply truncated, specified by the breaking wave
height simply proportional to local water depth, equation

(5) . The conditional probability density function of wave
heights, p, (H), is calculated using the following equation,

p (H) = p(H/0 ^H < Hb) = g(H) , for 0 < H < Hb

b jj

/ p(H)dH
0

= 0 , for H > Hfa .

(8)

Describing the conditional probability in this manner re-
sults in the proportional redistribution of probability-
density associated with the broken or breaking waves over the
range of H. This removes the delta function at the breaking
wave height, H, , previously described by Collins and Battjes.
Table I shows the original Rayleigh distribution with dotted
lines, and the modified Rayleigh distributions in heavy lines
after applying the cut-off to the tails using the breaking
criterion. The distribution by Kuo and Kuo is more realis-
tic but still results in a sharp cut-off of the distribution
at the breaking heights.
5 . Goda Distribution

Goda (1975) derived a more realistically truncated
distribution, qualitatively anyway, by requiring a gradual
cut-off of the distribution. He uses a shoaled Rayleigh
distribution to describe the unbroken wave heights at shore-
ward locations prior to applying his cut-off which is given
by

p (H) = 4H/K2H2 exp(-2H2/K2H2) (9)

o so so

where K is the shoaling coefficient. Goda (1975) calculated

s 3

the wave shoaling using the nonlinear theory of Shuto (1974) :
which dictates the following:

0 < gHT2/h2 < 30: Small Amplitude Theory (10)

30 < gHT2/h2 < 50: H h2/7 = constant (11)

50 < gHT2/h2 <_ °°: Hh5/2 [ VgHT2/h2 -2/3] = constant. (12)

Goda assumes that wave breaking occurs in a range of wave
heights between H2 and H.., with varying probability. The
probability density function of unbroken waves only is
expressed as:

Pr(X) = pQ(X) ; for X < X2 (13)

X — X

Pr(X) = Pq(X) -x _x2 P0(\) ; for X2 < X < Xx (13)

Pr(X) = 0 ; for X± < X (15)

where, normalized wave heights are defined by X = H/H ,
X, = H,/H and X_ = H2/H . The artifice of spreading breakers
over a range partly represents the inherent variability of
breaker heights, and partly compensates for the simplifica-
tion of using a single wave period in the estimation of
breaker height. Broken waves generally have some height
smaller than X,. Since no theory is available for describing
waves after they have broken, the heights of broken waves
are assumed to be redistributed across the range and to be
proportional to the unbroken waves as was done by Kuo and
Kuo (1975) . Therefore, the conditional probability density
function for all heights is calculated by

Pr(X)

p(X) = — ■=-*■ (16)

Xl
/ p (X)dx
0 r

where :

X. 2V2

1 9 -a X,

/ p (X)dX = 1 - (1+a X (X -X )}e L , (17)

0 r . ± z

is a constant of proportionality applied to p (X) to normal-
ize the pdf. In equation (17), the constant, a, is equal to
/2/Ks.

The breaker height is estimated using the following
formula, which is an approximate expression for Goda's
breaker index (1970) based on laboratory data,

H L H

g£ = A^{l-exp[-1.5 ^^(1+K tanpS)]} , (18)
o o o o

where tan 3 denotes the bottom slope and the coefficients
are assigned the following values for best-fitting to the
index curves,

A = 0.17, K = 15, and p = 4/3.

The range of breaker height, X, - X_ , is calculated
by assigning the following values for A:

A = 0.18 for X ,

A = -j A = 0.12 for X2

The upper limit of A., was selected by considering the varia-
bility of breaker heights, whereas the lower limit of A2 was
chosen simply as two-thirds of A, . The coefficients used

are based on matching laboratory data taken on a 1/10 and
1/50 beach slope and several field experiments. The Goda
model is applied here to the experimental data described
below.

5. Summary

The common idea of these studies is to cut-off the
portion of wave height distribution beyond the breaker
height, which is controlled by the water depth and other
factors. The methods differ in the techniques of cut-off
and the formulae used to define breaker heights.

III. EXPERIMENT

Experiments were conducted at Torrey Pines Beach, San
Diego, California, during November 1978, as part of the
Nearshore Sediment Transport Study. At this site there is
a gentle sloping, moderately sorted, fine-grained sandy beach,
The beach profile shows no well-developed bar structure and
is remarkably free from longshore topographic inhomogenei-
ties. Winds during the experiments were light, and variable
in direction. Shadowing by offshore islands and offshore
refraction, limits the angles of wave incidence in 10m depth
to less than 15°. During the experiments, significant off-
shore wave heights varied between 60 and 160 cm. The condi-
tion of nearly normally incident, spilling (or mixed plung-
ing-spilling) waves, breaking in a continuous way across the
surf zone, prevailed during most of the experiments.

A. INSTRUMENTS

An extensive array of instruments was deployed to study
nearshore wave dynamics . Measurements described here are
from sensors located on an offshore transect from 10 m depth
to across the surf zone (Fig. 1) . The sensors were of three
types: pressure (P) , current (C) and surface-piercing-staff
(W) .

The pressure sensors were Stathem temperature-compensated

2
transducers with dynamic range of either 912-2316 g/cm or

2
912-3720 g/cm . They were statically precalibrated and

postcalibrated by being lowered into a salt-water tank and
were found quite linear; the gains differed by less than 2%
between calibrations.

Current meters were two-axis , Marsh-McBirney electro-
magnetic, spherical (4 cm diameter) probes, with a three-
pole output filter at 4 Hz. Precalibration and postcalibra-
tion of current meters showed little change in replicate
runs with steady or oscillating velocity fields. The uncer-
tainty associated with using a single gain factor for all
frequencies is roughly estimated at ±5% in amplitudes (10%
in variances) .

The wave staffs were dual resistance wires with low
noise, high resolution, and good electronic stability. The
accuracy of the wave staffs was about ±3% based on repeata-
bility of gain calibrations measured in the laboratory and
in situ.

B. DATA ANALYSIS

Sea surface elevation and wave velocity components were
retrieved from sensors by telemetering to shore and there
recorded on a special receiver/ tape recorder, described in
detail by Lowe et a_l. , (1972) . The sampling rate was 64
samples/s which was reduced to 2 samples/s by digital low-
pass filtering. Record lengths of approximately 68 minutes
from each data set were analyzed.

It was desired to examine only the sea-swell band of
frequencies between 0.05 to 1.0 Hz (20 to 1 s periods). The

data were first linearly detrended to exclude effects of the
rising and falling tides. The da-.a were then high pass
filtered with a cut-off frequency of 0.05 Hz (20 s period).
The high pass filter used a Fast Fourier Transform algorithm
to obtain the amplitude spectrum of the entire 68 minute
record. The Fourier coefficients corresponding to 0 to
0.05 Hz were used to synthesize a low frequency time series
which was subtracted from the wave record. The limiting high
frequency (Nyquist) was 1.0 Hz.

The energy density spectra were calculated in a similar
manner for wave and velocity measurements using the Fast
Fourier Transform (FFT) algorithm. A cosine-squared taper
data window was applied to the time series to minimize leakage

The highest maximum and lowest minimum of the surface
elevation within a period interval defined, respectively,
the crest and trough of a wave. A wave height H is defined
as the total range of £ (t) in that interval, the time between
two consecutive zero-up-crossings of c(t) (see Fig. 2).
Since the average wave period was about 14 sec, the total
number of waves in the 68 min. record was about 300, which
gives reasonable wave statistics. The height statistics of
mean wave height H, root mean square wave height H ,
significant wave height H, /3 (average of the heights of the
1/3 highest waves) , and H, ,.- (average of the heights of the
1/10 highest waves) , are calculated from the ordered set of
wave heights.

The pdf and cumulative distributions of wave heights
were calculated. The heights were normalized using the
deep water root mean square value. Theoretical probability
distributions were calculated using the Goda model and com-
pared with measured distributions.

A deep water reference wave height was calculated by-
measuring the energy using current meter C9 located at about
4 m depth and backing the energy out to deep water. Kinetic
energy spectra were calculated from the measured horizontal
velocity spectra, S (f) and S (f ) ; linear theory transfer
functions were used to integrate the spectra over the water
column, so that the average kinetic energy is

<KE(f)> = |H„_(f) |2[S (f) +S (f)] (19)

J\Ij u v

2
where |H (f) |~ is the transfer function that relates the

JS.il

velocity spectrum components to the kinetic energy,

i it t*\ i2 1 sinh 2kh , orn

lHKE(f)l = 4k p — zrrj— ; (20)

cosh k(h + z )
m

where z is the measurement elevation. Guza and Thornton
m

(1980) showed, for these same experiments, that using linear
theory transfer functions to calculate average kinetic energy
gave reasonable results. To first order in energy, the
average potential energy equals the average kinetic energy.
The potential energy in deep water is obtained by applying
linear shoaling transformation

<PE(f)>-0 ^ wa. ^ = |H(f) |2<PE(f)>. , (21)

deep water ' s ' 4m

where H (f) = VC /C , is the linear shoaling coefficient.

s g0 g
The rms wave height is related to the PE and KE by

<PE> = ~ p g H2 = <KE> = / <KE(f)>df , (22)
o rms _

from which the deep water rms wave height, denoted hence-
forth by H , can be found.
2 o

To take advantage of the large number of current meters,
current data were used to infer wave heights . The velocity
signals were convolved using linear wave theory to obtain
surface elevations. The complex Fourier spectra of the
horizontal velocity components U(f), V(f) were first calcu-
lated and vectorially added. The complex surface elevation
spectrum, X(f), was calculated applying the linear wave
theory transfer function, H(f)

X(f) = 5(f) -V.(f) (23)

where

H(f) = Si£h *£ - r (24)

oj cosh k(h + z )
m

The complex surface elevation spectrum was then inverse
transformed to obtain the surface elevation time series from
which the wave height distribution is calculated. The entire
68 min . record was convolved at one time in order to minimize
the end effects which result in spectral leakage.

IV. RESULTS

A. TYPICAL SPECTRA

A broad range of wave and weather conditions were en-
countered during the experiments. Typical velocity spectra
for two days (Fig. 3, lower panel) include an example of
very narrow band spectra calculated for November 20 for the
current meters C22x and C23x which straddled the mean breaker
line. In shallow water depths, the waves generally become
more "peaky", resulting in increased energy at the harmonics.
The presence of strong harmonics in the spectra indicates
the importance of nonlinearities of the waves in shallow
water. The spectral energy level decreases at all frequen-
cies except at the very lowest from the deeper instrument
C22x (heavy line) to the shallower instrument C23x due to
breaking. The other typical spectra (Fig. 3, upper panel)
are an example of combined sea and swell with a narrow band
of energy at swell frequencies, but with broad band energy
at higher frequencies.

B. HEIGHT STATISTICS

The wave height statistics for six days, inferred from
wave staff and current meter data, are presented in Table II.
The statistical parameters listed are: root-mean-square
heights, H , significant wave heights, H..,-, average of
the heights of the 1/10 highest waves, H1/1Q, and the maximum
height, H . These parameters were obtained from 68 min.

records. The reference wave height H , obtained by backing

the energy measured at current meter C9 out to deep water,

the frequency peak of the spectrum and the depth for each

instrument, are also presented. Most of the wave staff

measurements were made inside the surf zone, after the waves

have started breaking.

Shoaling effects are observed in the data of Table II.

The H , H, ,-,, and H, ,,Af increase as depth decreases,
rms I/-J l/±u

until they reach the breaking point after which the wave
heights decrease as the depth decreases, due to breaking.
The range of depths of the instruments is from about
570 cm to 40 cm. The average peak frequency (peak frequency)
varied little during the experiments and was about 0.07 sec
(Table II) . The relative depth for the waves at the peak
frequency is h/L < 1/25 in all cases so that they can be
considered shallow water waves.

C. COMPARISON OF EMPIRICAL WITH MODEL DISTRIBUTIONS

The Goda model is compared with measured data. The model

is first run using Goda ' s original coefficients. This model

results in over-prediction of the H . The coefficients

r rms

are changed to optimize the model's description of the wave

height distribution qualitatively, and H quantitatively.

The H parameter was calculated from the second moment of
rms c

the wave height distribution and is, therefore, a more sensi-
tive parameter to describe the shape of the distribution
(particularly the tail) than, say, first moments such as the
mean H, H^, *1/1Q.

A comparison (Fig. 4) was made between the empirical
wave height distribution normalized with the reference deep
water wave height and Goda ' s model distribution for November
20. Starting in deep water, it is observed that the model
fits quite well the empirical distribution obtained from the
pressure sensor P7. It is clearly seen that the wave heights
in deep water are essentially Rayleigh distributed. In
shallow water the model over-predicts the tails and under-
predicts the peaks. The smaller the depth the greater the
errors (Figs. 4 and 5). Due to this over-prediction at the
tails of the distribution the rms wave heights obtained by
the model are larger than the measured ones.

To obtain a better agreement between the measured and
the calculated rms wave heights, the higher limit of breaking
and the range of breaking of the model are changed. Goda
(1975) calculated the shoaling using the nonlinear theory
of Shuto (1974) (see equations 10, 11 and 12) . For all the

ranges of wave heights, frequencies and depths used here,

2 2
the values of gHT /h are calculated and in all cases analyzed,

fall in the third category, equation (12) . This law was used

to calculate the nonlinear shoaling coefficient K . Goda

assumed a range of breaking between HI and H2 that takes

into account the variability of breaker heights and the use

of a single frequency. The variable breaker height H, is a

function of the frequency, depth and bottom slope (see equation

18) . The values of the coefficients used by Goda in his

breaking criterion are the following: Al = 0.18, A2 = 2/3 Al ,

K = 15 and p = 4/3, empirically assigned to give the best-
fit with observed wave heights. The coefficients used here
to give a qualitatively better fit with the measured dis-
tributions and quantitatively fit with H are: Al = 0.136.

^ ■* rms

A2 = 1/2 Al , K = 20; p was left equal to the previous value.
The most sensitive coefficient is Al which was determined
first. Values of Al were calculated for various distributions
by first obtaining the higher breaker limit (XI) from the
measured empirical distributions as indicated by the maximum
value of the distribution, and then calculating Al using equa-
tion 18 . The values of Al were then averaged to give a
single representative value for all distributions. The
coefficient A2 was chosen as 1/2A1, which results in a more
symmetrical distribution as indicated by the results. The
coefficient K, which weights the slope of the beach, was
determined by trial-and-error testing of Goda's model for a
variety of values looking for the best fit of all the distri-
butions. The model is not very sensitive to changes in K.
The comparison (Fig. 6) of the empirical distribution
in deep water for November 20 with the distribution obtained
applying the model used the new coefficients. In deep water,
there is no notable difference from the original model's
results (Fig. 4). In shallow water, the predicted values
obtained with the modified model fit much better the
empirical distributions than those obtained applying the
original Gcda model.

As stated earlier, the coefficients Goda originally speci-
fied were based on matching laboratory data for a two beach
slopes of 1/10 and 1/50. The model was then applied to some
field data, but unfortunately the beach geometry (slopes)
were not given. The Torrey Pines Beach is approximately
1/50 at the beach face and the wave climate is characterized
by long period (-14 sec) swell. The reason for differences
in the model coefficients needed to fit this data set com-
pared with Goda's suggested coefficients is not known.

D. COMPARISON OF RMS WAVE HEIGHTS

The model calculated H values and measured values of

rms

H calculated directly from the wave heights are used to
test how well the model works. Table III shows the measured,
Rayleigh, and the calculated Goda and modified Goda root-
mean-square wave heights corresponding to the wave staffs
and current meters for six different days.

Goda's model-predicted H values are. in general,
r rms ' 3

larger than the measured ones (Fig. 9) . In an attempt to

explain the differences, H values obtained from the
r rms

model were plotted against depth, deep water wave height

and Ursell number; but no meaningful correlation was obtained

with any of these variables.

Measured H and calculated values assuming the wave

rms a

heights are Rayleigh distributed so that H = /8m were

3 ■* ' rms o

compared (Fig. 8a). Measured and assumed Rayleigh signifi-
cant wave heights were also compared (Fig. 8b) . The comparisons

show that the agreement between measured and Rayleigh sta-
tistics are good for both small and large wave heights,
inside and outside the surf zone. The good comparisons with
Rayleigh statistics suggests that the wave heights, although
decreased by breaking, are still more nearly Rayleigh-dis-
tributed than cut-off Rayleigh, as suggested by Goda.

The change of the rms wave heights with depth, between

the measured H and the calculated ones obtained by the
rms x

model first applying nonlinear (heavy line) and then linear
shoaling (light line) were compared (Fig. 10) . The model with
nonlinear shoaling clearly over-predicts the values, while
the model with linear shoaling gives reaonable values, com-
pared with measurements. The nonlinear shoaling "blows up"
in very shallow water (<30 cm) and should be ignored.

Measured and the calculated rms wave heights with depth
obtained by applying linear and nonlinear shoaling using the
modified coefficients were also compared (Fig. 10, lower
panel) . This figure illustrates that the modified Goda model
fits better the data than the original does. The majority

of the measured H fall between the two curves obtained with

rms

nonlinear and linear shoaling. Based on the choice of coeffi-
cients, applying linear shoaling to Goda's model can give
as good, or better, results as applying nonlinear shoaling.
The rms wave height values, obtained using the modified
model coefficients, were plotted against depth, deep water
wave height and Ursell number; no obvious correlation could

be noted. It is found that the new predicted H values

rms

compared with measured H have an error of less than +20%

r rms

at all depths (with the exception of one anomalous point)
(Fig. 11) .

E. WAVE HEIGHT DISTRIBUTIONS USING CURRENT METERS

As described earlier, the surface elevations were derived
by linearly convolving the velocity records and wave height
distributions calculated. The basis for applying this analy-
sis is the earlier work of Guza and Thornton (19 80) where
they showed, for this same data set, that linear theory spec-
tral transformations could be used to calculate surface ele-
vation standard deviations either from pressure meters or
current meters with less than a 20% error, and typically less
than 10%. Examples of the derived wave height distributions
are considered for November 17 (Fig. 12) . These measurements
were made just outside the surf zone, at about the breaker
point and inside the surf zone (current meters C23, C37 and
C40 respectively). In general, the model overestimates the
velocity derived wave height distributions more than the
direct measurements . The reason for the discrepancy is that
linear wave theory underestimates the surface elevations in
convolving the velocities, particularly in the crest region
of the waves . In other words , linear theory does not account
for the finite amplitude of these highly nonlinear waves.

F. COMPARISON OF MODEL AND MEASURED CUMULATIVE DISTRIBUTIONS

The cumulative exceedance of wave height distributions
normalized with rms deep water wave height were calculated
applying the modified Goda model (using nonlinear shoaling)
and plotted with the measured cumulative distribution for
comparison. The cumulative exceedance distribution empha-
sizes information in the tail of the distribution. For an
example in shallow water, inside the surf zone (116 cm depth) ,
there is a good agreement between the measured and predicted
distributions with a slight underprediction by the model in
the tail (Fig. 13) . With sensors located just outside the
surf" zone, under-prediction of the tail is larger and over-
prediction in the middle range occurs (Fig. 14). In general,
there is a better agreement between the two distributions well
inside the surf zone, e.g., wave staffs W38 and W41, than
those (W21 and W29) which were generally either at breaking
or just outside the surf zone, the most nonlinear wave
region.

V. CONCLUSIONS

It is confirmed that the wave heights in deeper water
(7 m) are Rayleigh distributed.

Goda's model, using the empirical coefficients originally
suggested on the basis of laboratory and poorly specified
field measurements, over-predicts the tail of the distribution
and under-predicts the peaks. As a consequence, the predicted
rms wave heights are larger than the measured. As the depth
decreases, the errors of the predicted distributions increase.
To get a better fit with the measured wave height distribu-
tions, the coefficients in the breaking criterion used in
Goda's model were modified. The values for Goda's breaking
criterion giving the best fit to the measured data of this
s tudy are :

Al = 0.136, A2 = 1/2 Al and K = 20 .

The percentage of error between the measured and the pre-
dicted H values from the model with these coefficients is
rms

les than ±20%.

Linear shoaling was found to be as good as nonlinear
shoaling in applying Goda's model across the surf zone.

Good comparisons were obtained between empirical and
Rayleigh-derived statistics. This indicates that the wave
heights, although decreased by breaking, are still more
Rayleigh-distributed than the cut-off Rayleigh-distributed
as suggested by Goda.

Goda's model was tested here with a large amount of
field data for a variety of wave conditions on a 1/50 beach
slope. Further comparisons should be made for a variety of
beach slopes and wave climates in order to test the general
applicability of the model.

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TABLE II. WAVE HEIGHT STATISTICS
(all values in cm)

Date

Inst

H
o

H
rms

Hl/3

Hl/10

H
max

Nov

W41

42.5

.0703

33.2

44.4

53.2

74.3

W38

125

42.5

.0703

44.8

61.5

73.4

98.1

W21

177

42.5

.0703

50.6

71.8

90.9

131.3

W29

225

42.5

.0703

56.7

81.3

113.8

188.3

Nov

W21

170

66.1

.0632

61.0

85.5

107.4

140.4

Nov

W41

44.8

.0729

37.8

50.7

59.4

79.5

W38

141

44.8

.0729

52.0

72.1

86.9

111.5

W21

197

44.8

.0729

54.1

76.2

94.1

139.2

W29

209

44.8

.0729

52.0

72.8

93.7

154.0

Nov

W3 8

116

52.4

.0666

35.9

49.3

57.9

83.8

W21

153

52.4

.0666

48.6

68.1

84.2

120.3

W29

182

52.4

.0666

73.1

110.2

143.0

202.7

Nov

C42

52.4

.0703

18.0

27.0

33.8

49.1

C39

102

52.4

.0703

35.6

50.5

59.8

82.6

C37

116

52.4

.0703

36.5

53.6

62.1

81.8

C36

142

52.4

.0703

40.1

60.4

69.8

89.5

C23

147

52.4

.0703

51.6

80.2

98.0

118.9

C22

188

52.4

.0703

60.8

96.1

127.7

159.3

C19

250

52.4

.0703

68.9

105.1

149.0

210.4

C15

355

52.4

.0703

58.7

88.5

125.5

217.9

C09

571

52.4

.0703

55.8

84.0

115.2

192.4

Nov

W3 8

36.4

.0639

10.1

14.5

18.4

26.3

W21

36.4

.0715

26.7

36.6

43.5

55.8

Nov

W41

55.4

.0757

33.8

45.8

55.4

109.1

W21

195

55.4

.1552

61.4

87.2

103.9

131.7

W29

198

55.4

.0756

63.9

91.3

117.2

166.3

TABLE III. MEASURED AND CALCULATED RMS WAVE HEIGHTS
OBTAINED WITH GODA'S MODEL AND MODIFIED
GODA'S MODEL (all values in cm)

RMS WAVE

HEIGHTS

SIG.

HEIGHTS

Date

Inst.

Me as .

Ray.

Goda

Mdf . G .

Meas .

Ray

Nov 4

W41

33.2

33.7

34.6

30.9

44.4

46.9

W38

44.8

46.2

55.9

47.5

61.5

63.4

W21

50.6

52.3

67.3

51.1

71.8

71.6

W29

56.7

58.2

65.3

58.5

81.3

80.2

Nov 10 W21 61.0 64.9 85.3 64.5 85.5 86.3

Nov

W41

37.8

37.9

48.6

34.7

50.7

53.5

W38

52.0

52.2

66.1

53.4

72.1

73.5

W21

54.1

55.6

70.3

56.0

76.2

76.5

W29

52.0

53.8

69.8

58.1

72.8

73.5

Nov

W38

35.9

36.2

50.3

43.9

49.3

50.8

W21

48.6

50.3

68.3

44.9

68.1

68.7

W29

73.1

67.5

77.6

53.3

110.2

103.4

Nov

C42

18.0

18.2

21.4

18.0

27.0

25.4

C39

35.6

36.1

54.0

38.5

50.5

50.3

C37

36.5

36.3

60.3

43.9

53.6

51.6

C36

40.1

40.0

70.1

53.8

60.4

56.7

C23

51.6

50.0

71.5

55.7

80.2

73.0

C22

60.8

57.5

79.1

54.9

96.1

86.0

C19

68.9

65.0

78.3

67.5

105.1

97.4

C15

58.7

60.9

72.0

64.0

88.5

83.0

C09

55.8

60.7

64.0

55.6

84.0

79.0

Nov 24 W38 10.1 10.7 26.7 24.4 14.5 14.3
W21 26.7 27.5 44.6 32.7 36.6 37.8

Nov 18

W41

33.8

35.5

33.4

31.4

45.8

47.8

W21

61.4

59.7

59.6

50.8

87.2

86.8

W29

63.9

61.7

80.7

57.5

91.3

90.4

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Figure 2. Definition sketch of zero-up-crossing
wave heights .

NOV V FILE 1
C23X C31X

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

FREQUENCY (HZ)
Figure 3 . Typical spectra measured during the experiments

EMPIRICAL DISTRIDUTIO-i OP WOVE HEIGHTS
N0V50 T07

1.5

1.0

0.5

0.0

1.5

1.0

o.s

0.0

1.5

1.0

0.5

0.0

M/HO

Figure 4 . Empirical distribution of wave heights compared
with those predicted with Goda's model, starting
in deep water (P7) and going into shallow water
(W21, W38) , 20 November 1978.

EMPlRICflL DISTRIBUTION OF u?)VE HEIGHTS
N0V17 H2!

I.S

1.0

0.5

0.0

1.5

1.0

0.5

0.0

1.5

1.0

O.S

0.0

1. 2.

M/HO

Figure 5. Empirical distribution of wave heights compared
with those predicted with Goda's model, starting
in deep water (W21) and going into shallow water
(W38, W41) , 17 November 1978.

EMPIRICAL DISTRIBUTION OF ufivE hFICmTS
NOV20 P07

Figure 6

1.5

t.o

0.5

0.0

t.s

1.0

o.s

0.0

1.5

1.0

0.5

0.0

FksJl

M/hO
N0V20 W?l

H/HO

Empirical distribution of wave heights compared
with those predicted with the modified Goda s
model, 20 November 1978.

EMPIRICAL D15TMCUU0M C? wfivE hCICmTS
N0VI7 H21

1.5

1.0

0.S

0.0

t.S

1.0

o.s

0.0

1.5

1.0

O.S

0.0

H/HO
N0V17 Mm

H/HO

Figure 7. Empirical distribution of wave heights compared
with those predicted with the modified Goda's
model, 17 November 1978.

MEASURED HRMS AT HAVE STAFFS V5 RAUEICH HRMS

100.

80.

? 60.

t HO.

20.

o«c

IM4T

otnxicm

■l»04

M4I

«}•

i«

ITT

J/l

■

»0» 10

■21

HOT IT

u«l

"it

i*i

l»T

w.'9

!?1

♦

»o».-a

IS J

1*1

M'rt

•

Mill

«*•

Iff

0. 20. 40. 60. 80.
HEASURED HRMS

100.

Figure 8a.

Range of measured and Rayleigh root-mean-
square wave heights.

MEASCREO Hl/3 AT HAVE STAFFS VS RATLEICH Ml/3

100.

80.

- SO.

t »o.

20.

OftM

MSI

ocrimcm

■

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IJI

ITT

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•

■Oil*

HMI

III

Hit

III

20. MO. 60. 80.
MEASURED HI/3

100.

Figure 8b.

Range of measured and Rayleigh significant
wave heights .

MEASURED HRMS AT WAVE STAFFS VS GODA'S HRMS

100.

80.

60.

5 io.

20.

DATE

INST

0EPTHICM)

N0V04

W<Jl

W38

125

W2I

177

W29

225

N0V10

H21

170

NOV 1 7

H38

1M1

H21

197

H29

209

♦

N0V20

H38

116

W21

153

W29

182

N0V24

M38

W2I

♦

N0V18

mil

H2I

195

U29

198

0. 20. 40. 60. 80.
MEASURED HRMS

100.

Figure 9 . Correlation of measured rms wave heights
with calculated Goda's rms wave heights.

CHANCE OF HRMS WITH DEPTH

3:
oc

SHOALING

NONLINEAR

LINEAR

NOV

♦

NOV

2. 3. 4.
DEPTH/HO

X
V.

iX
X

SHOALING

NONLINEAR

LINEAR

NOV

♦

NOV

Figure 10.

2. 3. 4.
DEPTH/HO

Comparison of the changes of Hrms with, the Goda's
model applying nonlinear (heavy line) and linear
shoaling (light line) . Upper figure illustrates
Goda's original model; lower figure illustrates
the use of the modified coefficients in applying
Goda's model.

PERCENTAGE OF ERROR OF HRMSMOO.) VS DEPTH

2:
sc

r
1

0.6

0.1

0.2

0.0

-0.2

z »

0 K

OflTE

INST

depth ichi

N0V04

U4 1

U38

125

M21

177

U29

225

N0V10

W21

170

NOV 17

MMI

W38

141

M21

197

W29

209

N0V20

U38

116

W2I

1S3

W?9

182

N0V24

H38

M21

•

N0V18

W21

195

H29

198

0. 50. 100. 150. 200.

DEPTH

250. 300,

Figure 11.

Percentage error of predicted (modified
model) compared with measured rms wave
heights.

EMPIRICAL DISTRIBUTION OF HAVE HEIGHTS
N0V17 C23

1. 5

1.0

O.S

0.0

1.5

1.0

0.5

0.0

1.5

1.0

0.5

0.0

Figure 12.

Empirical distributions of wave heights
obtained from current meters (C23, C37 and
C40) compared with predicted wave heights
calculated with the modified model,
17 November 1978.

PROBABILITY GREATER THAN STATED VALUE

P(H)

OEPTH IN CM
□ 116

N20 W38

PROBABILITY GREATER THAN STATED VALUE

P(H)

DEPTH IN CM
□ 92

Ni7 wm

4.0

Figure 13.

Comparison of measured cumulative exceedance
distributions with predicted distributions
(modified model) , 20 November (W38) and
17 November 1978 (W41) .

PR0BA81LITT GREATER THAN STATED VALUE

P(H)

DEPTH IN CH
□ 153

N20 M21

4.0

PROBABILITY GREATER THAN STATED VALUE

P(H1

OEPTH IN CM
N17 H38

Figure 14.

u.O

Comparison of measured cumulative exceedance
distributions with predicted distributions
(modified model) , 20 November (W21) and
17 November 1978 (W38) .

BIBLIOGRAPHY

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Institution of Electrical Engineers, London, 22 pp.,
1950.

Battjes, J. A., "Computation of Set-Up, Longshore Currents,
Run-UP and Overtoping Due to Wind-Generated Waves , "
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Battjes, J. A., "Probabilistic Aspect of Ocean Waves,"

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412, 1970.

Chakrabarty, S.K. and R.P. Cooley, "Statistical Distribu-
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March 20, 1977.

Forristal, G.Z., "On the Statistical Distribution of Wave
Heights in a Storm," Jour, of Geophysical Research,
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Goda, Y., "Numerical Experiments on Wave Statistics with
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Japanese)

Goda, Y. , "Irregular Wave Deformation in the Surf Zone/'

Coastal Engineering in Japan , Vol. 18, pp. 13-26, 1975.

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of Sea Surface Elevations, Pressures and Velocities,"
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pp. 1524-1530, March 20, 1980.

Kuo, C.T. and S.J. Kuo, "Effect of Wave Breaking on Statis-
tical Distributions of Wave Heights," Proc . Conf. on
Coastal Engineering, June 1974.

LeMehaute", B. and R.C.Y. Koh, "On the Breaking of Waves

Arriving at an Angle to the Shore , " Journal of Hydraulic
Research, Vol. 5, No. 1, 1967.

Longuet-Higgins, M.S., "On the Distribution of the Heights
of Sea Waves: Some Effects of Nonlinearity and Finite
Bandwidth," Jour, of Geophysical Research, Vol. 85,
No. C3, pp. 1519-1523, March 20, 1980.

Longuet-Higgins, M.S., "On the Statistical Distribution of
the Heights of Sea Waves," Jour, of Mar. Research,
Vol. 11, No. 3, pp. 245-266, 1952.

Longuet-Higgins, M.S., "On the Joint Distribution of the
Periods and Amplitudes of Sea Waves," Jour, of Geo-
physical Research, Vol. 80(18), pp. 2688-2694,
20 June 1975.

Longuet-Higgins, M.S., "The Effect of Nonlinearities on
Statistic Distributions in the Theory of Sea Waves,"
Jour, of Fluid Mech. , Vol. 17, pp. 459-480, 1963.

Lowe, R.L., D.L. Inman, and B.M. Brush, "Simultaneous Data
System for Instrumenting the Shelf," Proc. Conf.
Coastal Eng. 13th, pp. 95-112, 1972.

Shuto, N., "Nonlinear Long Waves in a Channel of Variable
Section," Coastal Engineering in Japan, Vol. 17,
pp. 1-12, 1974.

Tayfun, M.A. , "Linear Random Waves on Water of Nonuniform
Depth," Ocean Engineering Report No. 16, Part II,
Department of Civil Engineering, University of Deleware,
Newark, DE, August 1977.

Tayfun, M.A. , "Narrow-Band Nonlinear Sea Waves," Jour, of
Geophysical Research, Vol. 85, No. C3, pp. 1548-1552,
March 20, 1980.

Thornton, E.B., J.J. Galvin, F.L. Bub and P.D. Richardson,
"Kinematics of Breaking Waves Within the Surf Zone/'
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July 1976.

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International Symposium on Ocean Wave Measurements and
Analysis, Vol. I, pp. 774-789, 1974.

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12763 Teran

Transformation of

vaves across the surf
zone.

*, • 193704

Thesis

T27-63 Teran

c.l Transformation of

waves across the surf

zone,

thesT2763

Transformation of waves across the surf

3 2768 002 03444 9

_, DUDLEY KNOX LIBRARY

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