Waves on Beaches and Resulting Sediment Transport. Proceedings of an Advanced Seminar, Conducted by the Mathematics Research Center, the University of Wisconsin, and the Coastal Engineering Research Center, U. S. Army, at Madison, October 11–13, 1971

460 Pages·1972·7.606 MB·English

by  R. E. Meyer (Eds.)

ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION
Waves on Beaches
and Resulting Sediment Transport
Edited by R. E. Meyer
Proceedings of an Advanced Seminar
Conducted by the Mathematics Research Center
The University of Wisconsin, and the
Coastal Engineering Research Center
U. S. Army, at Madison, October 11 -1 3, 1 971
Academic Press
New York · London 1972
COPYRIGHT © 1972, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY
INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT
PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
111 Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1
LIBRARY OF CONGRESS CATALOG CARD NUMBER: 72-7348
PRINTED IN THE UNITED STATES OF AMERICA
Preface
This book represents the Proceedings of an Advanced Seminar held in
October, 1971, aiming to offer a coherent, interdisciplinary view of the state
of physical research in coastal oceanography and the direction in which this
subject is moving. The articles range from wave refraction to littoral erosion,
and the authors from geologists to mathematicians.
The success of the Seminar was due to many persons, including the active
and enthusiastic members of the audience; the chairmen of sessions, Drs.
R. G. Dean, I. R. Hershner, T. Y. Wu, D. L. Inman, and R. L. Miller; members
of the program committee, Drs. C. J. Galvin, D. L. Harris, R. E. Meyer,
L. B. Rail, and M. C. Shen; and the Directors of the two institutions which
jointly conducted the Conference. Mrs. Gladys Moran conducted the efficient
preparations for it and Mrs. Carol Chase prepared this book.
This conference was supported by the United States Army under Contract
No. DA-31-124-AROD-462.
Vll
Characteristics of Wave
Records in the Coastal Zone
D. LEE HARRIS
CONTENTS
1. Introduction 1
2. The Spectrum of Wave Record 2
3. Useful Transformation of Wave Spectra 12
4. Wave Gage Response 14
5. Wave Spectra from Pt. Mugu, California 14
6. Wave Record Analysis in the Time Domain 27
7. Photographic Wave Records 35
8. Long Term Distribution Functions 40
9. Summary 47
1. INTRODUCTION
Wave recordings are examined to evaluate the quality
of wave data available from instruments and photographs and
to determine the extent to which the record analyses confirm
or contradict speculation about wave characteristics published
before many instrumental wave records were generally avail-
able.
The theoretical study of waves was well advanced
before the first recordings of ocean waves were made. For
the most part, theory has remained well ahead of observa-
tions. Accurate wave observations are difficult to obtain and
to the best of my knowledge no one has yet proposed an ac-
ceptable primary standard for wave observations under normal
field conditions. When observations and their interpretation
are both difficult, the experimenter is inclined to rely rather
1
D. LEE HARRIS
heavily on theory in analyzing his data. The theoretician
often relies on the published analyzed data in refining the
theory. This feedback may result in having an excessive
research effort being devoted to the study of relatively rare
phenomena of no special importance while the study of other
more important and equally challenging problems, is neg-
lected.
Analyses of records obtained in a more or less con-
tinuous effort to record waves at several locations in a near-
shore environment are presented in this paper. The primary
objective is to show a range of conditions that actually
occurs and to show the phenomena that need to be explained.
Locations of the observation sites are shown in Figure 1.
Only a few years ago, the oceanographer was happy
to obtain any kind of a record that looked something like
ocean waves. As late as 1944, an oceanographer, who is a
leader in the field today, wrote of wave gages "there is no
need for extreme precision in the design or calibration of
instruments, nor in the manner in which observations are to
be carried out. " That statement is underlined in the original
report. Although extreme precision could not be used in 1944,
the application of many more recent theories requires precise
measurement of wave characteristics and it is unlikely that
the author of the above quote would repeat it today. Yet, we
are still lacking a widely accepted fundamental standard for
wave gaging in the filed.
It is necessary to consider the response character-
istics of wave gages and a logical method for analyzing wave
records before a consideration of the records themselves be-
comes very meaningful.
2. THE SPECTRUM OF A WAVE RECORD
The most satisfactory procedure today for wave record
analysis for many problems is the energy-spectrum analysis.
This is true because even a casual glance at the sea surface
will often reveal a number of more or less independent wave
trains. These can be described mathematically by an ex-
pression like
2
r y
A Buzzards Bay
r-^^k Gilgo Beach
A Jones Beach
Long Branch
Atlantic City
hesapeake Beach
Bay Bridge
Virginia Beach
m Nags Head
>
Wrightsville Beach <
*—Frying Pan Shoals m
3 Holden Beach J3
Mil! Mission Βογ V( Savannah Tower Oom
M Huntington Beach 33
1 Venice Pier \. 1 S ^ Daytona Beach Ό
1 El Segundo
1 Point Mugu LEGEND Palm Beach
1 Port Hueneme [· Lake Worth
Point Conception • Active
A Inactive
Figure 1, Location of recording wave gages operated by CERC at some time
between 1948 and 1971.
D. LEE HARRIS
N
h(x, y, t) = /. A cos[k (xcosG +
> i ? u n L n n
(1.1) n=l
+ ysin6 ) - σ t - φ 1
1 n n TnJ
where h is the departure from the mean water surface, A
the amplitude of the n'th component, k the wave number,
n
θ the angle between the direction of propagation and the
η
x axis, σ the frequency and φ the phase along lines in
η η
a vector x - t space where
σ\ = k (xcos0 + ysinG )
t n n n
When records from stationary gages are considered,
the dependency on x and y may be omitted to obtain
N
(1.2) h(t) = Yu.. A nC OS(Œ nt - φY n )
n=l
or the equivalent expression
N
h(t) = Y. (a coscr t + b sina t)
H n n n n
n=l
2 2 2
(1.3) A = a + b
n n n
tan φ =b /a
Ύη n n
If the σ were known a priori, the a and b
n n
could be obtained from the record by standard regression
techniques as in the analysis of tide records. In the anal-
ysis of wave records, however, the σ are not generally
η
known, and may be selected in any convenient manner. The
procedure which minimizes the arithmetic is a Fourier expan-
sion of the record. That is, we write
4
WAVE RECORDS
N
(1.4) h(t) = Σ [a cos(2mTTt/D) + b sin(2rmrt/D)]
m=l
where D is the duration of the record. It may be assumed
that the resulting values of A, where
2 2 2
(1.5) A = a + b ,
m m m '
will be large for 2πιπ/ϋ approximately equal to some value
of σ , and that other values of A^ will be small. In
η
general, it should not be assumed that any particular value
of σ is identical to any of the 2mTr/D.
η
According to G. I. Taylor (19 38) this concept was
first discussed by Lord Rayleigh in a study of the kinetic
energy of sound waves. Rayleigh called the function A^
an energy spectrum since it is proportional to the kinetic
energy of the waves when the function being analyzed is the
velocity of a particle.
The term "energy spectrum" is also appropriate for
(1. 5) when the calculation is based on a wave gage record
for it can be shown (Kinsman, 196 5, Chapter 3) that the aver-
age potential energy of a wave system is proportional to the
variance of the water surface elevation and that the variance
is one half of the sum of the squares of all of the coefficients
obtained in (1.4). The widely used expression "power spec-
trum" is not appropriate for discussing the spectrum of dis-
persive waves, for power is the energy flux. The flux can
be computed from the energy spectrum by multiplying each
value by the appropriate group velocity, but this step is
rarely carried out.
The standard deviation of the record is a convenient
measure of the characteristic wave height wherever the vari-
ance is computed. This parameter is often called the root
mean square wave height (Draper, 1967).
The most efficient procedure available for computing
the spectrum parallels the original proposal by Raleigh, and
is based on a Finite Fourier Transform (FFT) of the record
(Bingham, Godfrey and Tukey, 1967, and Tukey, 1967).
5
D. LEE HARRIS
This procedure generally gives more resolution in
frequency space than can be readily used. Therefore, the
variances computed for individual harmonics, often referred
to as lines, are summed to form bands in the form
m
^ 2
E(f ) = λ A , p = 1, 2, 3, .... p
m
P
(1.6)
f = (m . + m )/2D
P P+l P
The increment in frequency (fp+j - f ) is usually
p
taken as a constant, but variable increments are useful in
some problems.
Energy density spectra can be obtained by dividing
each term in (1.4) by the appropriate spectrum increment.
When a constant increment is used, the actual division is
not essential.
Several widely used conventions for displaying the
spectrum calculations are shown in Figure 2. All five graphs
are displays of the same spectrum.
Figure 2A shows the variance (energy) as a linear
function of the frequency. This display has the feature that
a unit of area under the curve is proportional to a unit of
energy anywhere on the graph. It represents the information
in the record as accurately as possible but it confines most
of the information to a small part of the graph.
Figure 2B displays the product of frequency and energy
as a function of the logarithm of the frequency. It retains the
characteristic of having a unit of area equal to a unit of ener-
gy and gives added emphasis to the high frequency compo-
nents.
Figure 2C displays the logarithm of energy as a func-
tion of the logarithm of frequency. This form abandons the
concept of unit energy per unit area but it is useful in looking
for universal functional relations. The sloping line on the
right indicates the exponent of -5 in the spectrum of the equi-
librium range as proposed by Phillips (19 58). Most of the
6