Depth-limited significant wave height : a spectral approach

Ro ee Gvi1e A

TR 82-3

Depth-Limited Significant Wave Height:
A Spectral Approach

by

C. Linwood Vincent

TECHNICAL REPORT NO. 82-3

AUGUST 1982

Approved for public release;
distribution unlimited.

U.S. ARMY, CORPS OF ENGINEERS
COASTAL ENGINEERING
RESEARCH CENTER

Kingman Building
Fort Belvoir, Va. 22060

Reprint or republication of any of this material
shall give appropriate credit to the U.S. Army Coastal
Engineering Research Center.

Limited free distribution within the United States
of single copies of this publication has been made by
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The findings in this report are not to be construed
as an official Department of the Army position unless so
designated by other authorized documents.

AA

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4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

Technical Report

6. PERFORMING ORG. REPORT NUMBER

8. CONTRACT OR GRANT NUMBER(s)

DEPTH-LIMITED SIGNIFICANT WAVE HEIGHT:
A SPECTRAL APPROACH

7. AUTHOR(s)

C. Linwood Vincent

10. PROGRAM ELEMENT, PROJECT, TASK
AREA & WORK UNIT NUMBERS

A31592

12. REPORT DATE
August 1982
13. NUMBER OF PAGES

23

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Department of the Army

Coastal Engineering Research Center (CERRE-CO)
Kingman Building, Fort Belvoir, Virginia 22060

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Department of the Army

Coastal Engineering Research Center

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19. KEY WORDS (Continue on reverse side if necessary and identify by block number)

Depth-limited wave height Spectral waves Wind wave energy

20. ABSTRACT (Continue om reverse sida if necessary and identify by block number)

A theoretical equation that describes the region of a wind wave spectrum
above the frequency of the spectral peak in a finite depth of water is used to
develop a method for estimating depth-limited significant wave height. The
theoretical background for the equation, along with supporting field and labora-
tory data, is given. The method indicates that significant wave height, defined
as four times the standard deviation of the wave record, is approximately
proportional to the square root of the water depth.

FORM
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PREFACE

This report presents a method for estimating depth-limited significant
wave height of an irregular wave field. The work was carried out under the
U.S. Army Coastal Engineering Research Center's (CERC) Wave Estimation for
Design work unit, Coastal Flooding and Storm Protection Program, Coastal
Engineering Area of Civil Works Research and Development.

The report was prepared by Dr. C. Linwood Vincent, Chief, Coastal Ocean-
ography Branch, under the general supervision of Mr. R.P. Savage, Chief,
Research Division. J.E. McTamany prepared the computer integration scheme;
W.N. Seelig and L.L. Broderick provided laboratory data.

Technical Director of CERC was Dr. Robert W. Whalin, P.E., upon publica-
tion of this report.

Comments on this publication are invited.

Approved for publication in accordance with Public Law 166, 79th Congress,

approved 31 July 1945, as supplemented by Public Law 172, 88th Congress,
approved 7 November 1963.

f/f cf
ED E. BISHOP

Colonel, Corps of Engineers
Commander and Director

_ CONTENTS

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI)
SYMBOLS AND DEFINITIONS

if. ENTRODUGDION irae ot cuter toil see etree teh oy Late fetes a, Wee staroig ates erm Svan US
II THEORETICAL BACKGROUND

III FIELD EVIDENCE FOR THE FINITE-DEPTH SPECTRAL FORM .

IV FORMULATION OF DEPTH-LIMITED SIGNIFICANT WAVE HEIGHT, Hy .
V FIELD AND LABORATORY EVIDENCE FOR DEPTH-LIMITED SIGNIFICANT

HEIGHT, Hg

VI DISCUSSION

VII SUMMARY

wo N

-

TCI RV NIU, GID) 6 6 6 0 6 6 60,0 6,6 6.6050 5 6160060 0 6

TABLES

Normalized form regression analysis (average of percent variance
in regression of normalized form against frequency, f,
explained by f) .

Average slope, X10°3, against f

Variation of Hp with depth for ocean, large lake, and small
lake generation cases

FIGURES

Location of the XERB buoy and the wave gages at CERC's Field Research

Facility, Duck, North Carolina, during the October-November 1980
ARSLOE experiments

Comparisons of wave spectra at various depths to f % and £ ° laws
Selected storm spectra at different water depths

Depth-limited significant wave height, Hg, as a function of water
depth and cutoff frequency

Plot of R= (a/0.0081) !/2 as function of peak frequency of spectrum
and windspeed, U Bi Rrcill Guess crag e OI Wa Pan mse tauicet 8

Variation of significant wave height, Hg, with the square root of
water depth 5

Estimate of He for laboratory conditions

Variation of wave height with square root of depth, 25 October 1980,
Duck, North Carolina

wo ~~ sw DD

2.

15
18
22
23

11
11

22

10
13

14

16

17

19

20

21

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT

U.S. customary units of measurement used in this report can be converted to
metric (SI) units as follows:

inches

square inches
cubic inches

feet

square feet
cubic feet

yards
Square yards

cubic yards

miles
square miles

knots
acres

foot-pounds

millibars
ounces

pounds

ton, long
ton, short

degrees (angle)

Fahrenheit degrees

0.0283
0.9144
0.836

0.7646

1.6093
259.0

1.852
0.4047

1.3558

1.0197
28.35

453.6
0.4536

1.0160
0.9072
0.01745

3/9)

x 102

centimeters
square centimeters
cubic centimeters
centimeters
meters
Square meters
cubic meters
meters
Square meters
cubic meters

kilometers
hectares

kilometers per hour
hectares
newton meters

kilograms per square centimeter

grams

grams
kilograms

metric tons
metric tons
radians

Celsius degrees or Kelvins!

1To obtain Celsius (C) temperature readings from Fahrenheit (F) readings,
use formula: C = (5/9) (F -32).

To obtain Kelvin (K) readings, use formula:

KS (O/)) GF S82) sP ZI Solo

SYMBOLS AND DEFINITIONS
E total variance in wind sea, often called energy
E(£) variance density, often called energy density
EB depth-limited value of total variance

Ef) upper bound on energy density in a frequency, f

F variance density spectrum in wave number space

f frequency

f. low-frequency cutoff

45 peak frequency of the spectrum

H depth-controlled wave height (spectral)

Hy depth-limited wave height (monochromatic)

Hy depth-limited wave height (irregular sea)

Huo zero-moment wave height, also called significant wave height
H ax largest individual wave

Hi/3 significant wave height

h depth

k wave number

R transcendental function of dimensionless frequency wu,

U windspeed

a Phillips' equilibrium coefficient

1 3.1415

C) dimensionless function describing deviation from deepwater equilibrium
range

wh dimensionless combination of g, f, andh

DEPTH-LIMITED SIGNIFICANT WAVE HEIGHT: A SPECTRAL APPROACH

by
C. Linwood Vineent

I. INTRODUCTION

Research into the shape of wind wave spectra in finite-depth water has
suggested an expression for the upper limit on the energy density as a func-
tion of depth and frequency (Kitaigorodskii, Krasitskii, and Zaslavskii, 1975).
In this report this expression is integrated over the part of the spectrum
expected to contain energy to estimate a limit on the energy, E, in the wind
wave spectrum and to define a depth-limited significant wave height, Hp? _

Hp = 4.0(B) 1/2 (1)

More precisely, the quantity estimated is the variance of the sea surface to
which E is directly related. Following convention, E and E(f) denote
energy and energy density spectrum although the true units of computation are
length squared and length squared per hertz. The term zero-moment wave height,
Hyo will be used to denote 4.0(E) [2 H,/3 is the average height of the one-
third highest waves. Hg denotes values of Hy, that are depth limited. In
deep water, Hy is approximately Hj /3, but this is not necessarily true in
shallow depths. H, refers to the depth-limited monochromatic wave. The vari-
ation of Hp with depth, h, is investigated and compared with the mono-
chromatically derived depth-limited wave height, Hyg. Because Hy, and H)/3
are about equal in deep water, they are both frequently called significant
wave height.

This report briefly reviews the theoretical development of the limiting
form for spectral densities as a function of water depth and presents field
evidence supporting this form. The simple derivation of the depth-limited
energy and significant wave:height is then given, followed by field and labora-
tory data evaluating the prediction equation. Unless otherwise noted, the
developments of this report are restricted to wave conditions described by a
wave spectrum of some width such as an active wind sea or a decaying sea.

II. THEORETICAL BACKGROUND

Phillips (1958) suggested that there should be a region of the spectrum
of wind-generated gravity waves in which the energy is limited by wave steep-
ness. Phillips derived an expression for the limiting density in deep water:

Em(£) = ag?£ °(2m)7* (2)

where a was considered to be a universal constant. Field studies reviewed

by Plant (1980) demonstrated that equation (2) adequately describes the part

of the wind sea spectrum above the peak frequency of the spectrum. However,
Hasselmann, et al. (1973) indicated that the equilibrium coefficient a is

not constant but varies systematically with wave growth leading the authors

to speculate that resonant interactions in the spectrum force the spectrum to
evolve to the form of equation (2). Toba (1973) suggested that the equilibrium
range form might be proportional to ner in order to remove the variation

of oa.

Kitaigorodskii, Krasitskii, and Zaslavskii (1975), using Phillips' (1958)
expression for the steepness limited form of a wave spectrum, F, in terms of
the wave number modulus,

F(k) = k73 (3)

solved the transformation of equation (3) to a frequency spectrum in finite-
depth water. The finite-depth form, E_(f,h), was shown to be equal to the
deepwater form (eq. 2) times a dimensionless function, aw),

ag?f£7>
BCE) = eye OG) (4)

Kitaigorodskii, Krasitskii, and Zaslavskii suggested a value of 0.0081 for a.

The function © requires an iterative procedure for solution and is
defined as

Mi IO) Wot
Op) = RO? (a) | + ——2#—_4_ (5)
Sinh (207 RCu, )

with wy = w(h/g)!/2 (6)

where w= 27f£ and R(w,) is obtained from the solution of

2 = 7
Rw, ) eat R(u,)) 1 (7)
The dimensionless parameter w, related the frequency and depth to the devia-
tion from the deepwater form. Phen Ww, is greater than 2.5, © is approxi-

mately 1, when On is zero © is zero. When On is less than 1

(uw) = wp /2 (8)

For w, less than 1, a combination of equations (8) and (4) leads to the
expression

i Cagn) = aghf~3/(2(21)2) (9)

Thus in the shallow-water PATE the bound on energy density in the wave
spectrum is proportional to f° compared to f° in deep water, and depth
is included linearly.

Resio and Tracy (U.S. Army Engineer Waterways Experiment Station, personal
communication, 1981) have analyzed the resonant interactions and derived
equivalent expressions to equations (3) and (4) on the basis of similarity
theory. The conclusion of their theoretical study is that the role of the
wave-wave interactions in both deep and shallow water is to force the spectrum
to evolve to the form of equation (4). Their theory may be distinguished from

that of Kitaigorodskii, Krasitskii, and Zaslavskii (1975) in that their
coefficient a is expected to vary with wave conditions and not remain a
universal constant.

III. FIELD EVIDENCE FOR THE FINITE-DEPTH SPECTRAL FORM

Prior to Kitaigorodskii, Krasitskii, and Zaslavskii (1975), Kakimuma
(1967) and Druat, Massel, and Zeidler (1969) had noted that the shape of the
spectrum in shallow water deviated from Phillips’ (1958) form. Kitaigorodskii,
Krasitskii, and Zaslavskii cited evidence supporting the f ~ form, as did
Thornton (1977) and Gadzhiyev and Kratsitsky (1978). Ou (1980) provided
laboratory evidence for equation (4). A review of spectra collected at the
Coastal Engineering Research Center's (CERC) Field Research Facility (FRF) at
Duck, North Carolina, and at other gages in shallow water supports a near
£-° spectral slope in depths less than 10 meters for large wave energies.

These findings indicate a further evaluation is needed of how well the
equation fits observed spectra. During the Atlantic Remote Sensing Land and
Ocean Experiment (ARSLOE) conducted in October and November 1980 at the FRF,
North Carolina, wave spectra were collected in 36 meters of water about 36
kilometers offshore of the CERC facility (Fig. 1), using the National Ocean
Survey's directional buoy, XERB, with accelerometer buoys in depths of 25, 18,
and 17 meters of water located at distances of 12, 6, and 3 kilometers offshore
along a line from the facility to the XERB. In addition, data from Baylor
gages at seven locations in 1.5- to 9-meter depths along the FRF pier were
collected. On 25 October 1980 a large, low-pressure system generated waves
with significant heights up to 5.0 meters. Data were collected continuously
at the XERB during the period of high waves and spectra at all gages were
computed every 20 minutes.

As a test the observed spectra, E(f), were normalized to the following
forms

a. (£) = £°E(£) (21)7*/22 (10)
ana (£)) = £°E(£) (2m)*/g? e(u,) (11)
Sale) = ESE (6))//en (12)

Equation (10) is an estimate of the equilibrium coefficient as a function of
frequency if the spectra follow the deepwater form. Likewise, equation (11)

is an estimate of the coefficient if the spectra follow the proposed finite-
depth form, and equation (12) is an estimate of the coefficient if the proposed
shallow water (w, less than 1) holds over most of the spectrum. If any of
these forms fit a spectrum then the corresponding function @(f) should be
constant with frequency. Therefore in a regression of f against a(f), f
should explain no variance; consequently, the degree of fit to the spectrum

by each of the three forms can be estimated by how poorly f explains variance
in the regression and how flat the slope with f is. The regressions were
performed over the region from the spectral peak to twice the spectral peak
and the results are tabulated in Tables 1 and 2.

75°40
a +

fa ATLANTIC

+

NMouticol Miles

=f

sin oO
ro. 500.0 ___—-.-55
4S 7

Ke

J

“10. ~1.5

N
SSS 100.0
O

Base Line (000.0) h , i meee Meters
0 50 100 150 200
Figure 1. Location of the XERB buoy and the wave gages at CERC's Field
Research Facility, Duck, North Carolina, during the October-
November 1980 ARSLOE experiment.

10

Table 1. Normalized form regression (analysis average

of percent variance’ in regression of
normalized form against frequency, f,
explained by f).

Deepwater | Finite-depth | Shallow-water
form? limit form

Isince the proposed form is supposed to remove variation
with f£, a high explained variance with f indicates that
the form does not fit the spectra well.

265 E(£) (27)7*/g?
3£5 E(£) (2n)7*/g2 © (up)
4_3 E(£)/g,

Table 2. Average slope, X107%,
against f°.

4

-3

Islope a is unit change of
a per hertz. Region of the
spectrum analyzed in regression
analysis is about 0.1 hertz.

it

Using the criteria established aboye, data summarized in Table 1 indicate
that in all cases either the finite-depth form or the £-° limit appears to
fit the spectra better than the deepwater form. This is because f consist-
ently explains less variance in these regressions than in the regressions
against the deepwater form. In a regression analysis under an assumption of
normally distributed variates, the hypothesis of zero correlation is rejected
for the number of frequency components from f, to 2fp) if the regression
coefficient is greater than 0.632 at a 5 percent level of significance. This
translates to a value of 40 percent for the values in Table 2. Table 1 indi-
cates that the average R* for the regressions in the deepwater form are
always greater than 40 percent, suggesting that there is correlation with f.
The average finite-depth form value is less than 40 percent for all but two
(655 and 615) of the gages, suggesting a tendency for no correlation with f.
The shallow-water limit results suggest zero correlation except for gages XERB,
655, and 615. Table 2 indicates that the slopes are, in general, lower as
well. Plots of f£°E(£) and f°E(f) show that the spectra appear to more
closely follow a £ 2 slope (Fig. 2).

The results of the regression analysis for the gages at depths greater
than 9 meters appear to be more closely fit by a £7” form than the results
at 9 meters and at shallower gages. The observed spectra at the shallower
gages tend to be less than the proposed upper limit. It is thought that
refraction, bottom friction, and massive breaking must dominate the spectra
in and around the peak, suppressing the values below the proposed limiting
value. This would indicate that in very shallow water, the proposed form may
be conservative. Plots of storm spectra at different gage sites are compared
to the limiting form in Figure 3.

The variation of the equilibrium coefficient a computed over the range
fp to 2fp) varies based on gage and time (as represented by sea and swell
conditions), with a for the sea conditions being larger. Additionally,
there appeared to be a tendency for a to increase slightly from deep to
shallow water. On occasion a calculated at the peak of the spectrum exceeded
the value of 0.0081. However, when’'the a value at the peak was compared to
the a value averaged over the frequencies from f. to 2f., it was evident
that the average value was much less than the value at the peak.

The field evidence from a variety of sources supports the conclusion that
the maximum energy densities above the peak frequency of the spectrum can be
approximated by equation (4), which in the shallow-water limit approaches
equation (9). Evidence from Ou (1980) and the data in this report suggest
that the coefficient a may not be a universal constant. There is also
evidence that once very shallow depths are reached, other mechanisms can
dominate spectral shape in the vicinity of the peak; the deviation, however,
is such that equation (4) appears to be an overestimate.

IV. FORMULATION OF DEPTH-LIMITED SIGNIFICANT WAVE HEIGHT, He

Since equation (4) provides an estimate of the upper limit on energy
density in water depth h as a function of frequency and wave generation
condition as expressed by the coefficient a, it is possible to estimate the
upper bound on the depth-limited wave energy, Ey, if a low-frequency cutoff
value, f,, is known. E, can simply be estimated by

12

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14

foe)

En = Ie Em(£,h)df = Pa ag? £75 O(u,)/(20)* dé. (13)

The depth-limited significant wave height (spectral) is then

Hy = 4.0 (B,)1/? (14)
In shallow water, Hp is expected to be different from Hj )/3, but how differ-
ent is uncertain. Although H 1/3 has a long tradition of use in coastal engi-
neering, the wave height Hp defined in equation (14) appears to be a more

consistent parameter because it is directly related to the energy of the wave
field.

Figure 4 provides curves of Hg as a function of cutoff frequency, f,,
and depth, h, for a= 0.0081. If a is different an estimate of Hg for
that a can be made by

Hp = 76a/0.0081)1/? (15)

where HF is Hp estimated with a of 0.0081.

Clearly the cutoff frequency and the value of o are crucial for obtaining
estimates of Hg. An examination of storm spectra indicates that the spectral
peak is quite sharp. Consequently, a reasonable choice for f¢ would be about
90 percent of fp. If there is evidence of more energy on the forward face of
the spectrum, f, could be estimated by using a lower percentage. The param-
eter a can be obtained by fitting equation (4) to observed data if available.
For field engineers, most often this may not be possible in which case a can
be estimated by knowledge of the peak frequency, fp, and windspeed, U,
through the relationships developed by Hasselmann, et al. (1973). The values
of fp and U can be obtained from hindcasts or measurements. Figure 5
provides values of (a/0.0081) !/2 as a function of Ep and U.

When the primary frequency components containing the major part of the
energy are in shallow water, as determined by the condition wh <1, then Em
is given by equation (9). This can be integrated analytically to give an
estimate of H for a = 0.0081

1 =
He = = (agh) 1/2 feat (16)
Equation (16) has the remarkable consequence of suggesting that Hg defined
as 4.0(E)!/2 varies with the square root of depth when the primary spectral
components are depth limited. The monochromatic depth-limited wave height,
Hg, varies linearly with h.
V. FIELD AND LABORATORY EVIDENCE FOR DEPTH-LIMITED SIGNIFICANT HEIGHT, Hp

In order to test the applicability of equations (15) and (16) in predicting
Hg in shallow water, laboratory data taken by Seelig and Broderick (1981) in

is)

Hg (m)

) 2 4 6 8 10 12 14 16 18 20
Depth (m)

Figure 4. Depth-limited significant wave height, Hg, as a function
of water depth and cutoff frequency. Curves are calculated
for a= 0.0081. Hy, is plotted for lower limit of 0.8 h.

AS la/lO,ool) '“?
ine)

O 0.1 0.2 0.3 0.4
Peak Frequency , fp (Hz)

Figure 5. Plot of R= (a/0.0081)?/2 as function of peak frequency
of spectrum and windspeed U. Data based on JONSWAP wind
sea relationships (Hasselmann, et al., 1973). Coefficient
R is used to adjust curves in Figure 4 to account for
variation in oa.

17

a flume 44 meters long and 0.45 meter wide with a maximum water depth of 0.6
meter and a bottom slope of 1:30 at one end were examined. Seelig and Broderick
ran a variety of spectral shapes and energies. Figure 6 is a plot of H
calculated, as in equation (1), from a Fourier analysis of their wave data
against ni/2, Typically, the wave appear to shoal with decreasing depth,
thereby increasing in height until a point is reached at which the wave height
decreases linearly with the square root of depth. Figure 7 is an estimate of
Hp, based on equation (15), for two forms of fc. A plot of the maximum
individual wave, Hpax, is plotted as is the monochromatic breaking limit which
Hmax appears to follow. Hg is much less than the monochromatic breaking

limit in this case. Figure 8 provides plots of Hp versus hi/2 for wave data
at FRF on 25 October 1980. The value of h is estimated by an average of
profiles before and after the storm and poe udes the tide and the wave setup.

The curves are approximately linear with h!

VI. DISCUSSION

An examination of the characteristics of spectral shape in shallow water
has led to a method of estimating the upper bound on wave energy as expressed
by a depth-limited wave height. It is shown that in the shallow-water limit
this leads to an approximate variation of Hg with the square root of depth.
Frequently, the monochromatic limiting value Hg is used to provide an upper
bound on the wave height in shallow water. This report indicates that such an
approach can significantly overestimate the significant wave height. The tra-
ditional method of estimating wave conditions in shallow water has been to
obtain an estimate of H)/3 in some depth of water, then refract and shoal
it into the shore. At some point H,/3 becomes larger than Hg, in which
case H)/3 is set to Hg. This report indicates, however, that the wave
height Hp, which is directly related to the wave energy, varies with nl/2
and is normally much less than Hg. Consequently, when the energy in the sea
is of concern, Hp should be used rather than Hg. If the maximum individual
wave that can occur is of concern then Hg is appropriate.

The method in this report also indicates that the maximum significant wave
height, He, in shallow water in lakes and bays can be different than that in
the open ocean because the cutoff frequency, f,, in the smaller water bodies
is normally much higher than f, for large ocean storms. Table 3 provides
esimtates for Hp as a function of h for an ocean, a large lake, and a small
lake for the same windspeed, U, of 25 meters per second but for different
frequencies. Longer waves in an ocean are expected to develop than in small
lakes; consequently, f,. is higher in the short fetch cases. The coefficient
a increases in short fetch cases, but it enters Hg through a square root
relationship.

Estimates of depth-limited wave conditions have traditionally been based
on linearity of wave height and depth. This linear relationship is well estab-
lished for monochromatic waves by both laboratory and theoretical studies.
Extensions to irregular wave conditions have relied on this linear relationship
but with a coefficient of about 0.4. Figure 8 is a plot of this variation for
25 October 1980 and shows that in slope and magnitude this form is a poor
predictor. The method in this report is based on a theory about spectral shape
and appears to be a better predictor. It should be noted, however, that
evaluations of the newer method must account for variations in a and fc
as wave conditions change. Hence, simply plotting Hp versus h or nl/2 for

18

Wave Height(cm)

Figure 6.

3 4 5
Square Root of Depth (m'/2)

Variation of significant wave height, Hp, with the square
root of water depth. After a region of shoaling, wave height
drops off linearly with the square root of water depth.
Differing slopes are due to variations in a and f,.

Wave Height (cm)

Figure 7.

Monochromatic Hg = 0.8h

coe
H max
th oe
pee 4 x x

—
_-_—

IW SS
RSS
NS

15 30 , 45 60 75 90
Woter Depth (cm)

Estimate of Hp for laboratory conditions (from Seelig and

Broderick, 1981). He is estimated using two estimates of

f., anda line Hy = 0.8 h is also provided. Hyg at the

toe of the 1:30 slope is 13.4 centimeters with f£.-1 = 1.47
A F ; Pp

seconds. A linear shoaling curve is also shown.

20

e Data, 25 Oct. 1980, 1215
Offshore Hy ~ 4.2m, h=25m

e Data, 25 Oct. 1980, 0915
Offshore Hs ~ 4.4m, h=25m

2
Square Root of Depth (m'/2)

Figure 8. Variation of wave height with square root of depth, 25
October 1980, Duck, North Carolina. Solid line is based
on measured a and f,. Dashline represents estimated
band on monochromatic theory with H = 0.5 h.

|

Table 3. Variation of Hp with depth for ocean, large
lake, and small lake generation cases.

Small lake,

4

2¢ = 0.08, £, = 0.07, (a/0.0081)!/2 = 1.20.
3¢ = 0,12, f. = 0.11, (a/0.0081)!/2 = 1.37.
*£, = 0.16, f£, = 0.14, (a/0.0081)1/2 = 1.44.
SLarger than Hj.

one gage will show considerable scatter because of the time variation of «a
and f,. The evaluations of the method in this report have removed this
constraint by using a series of gages across the nearshore zone.

The use of the method at the beginning of this report was restricted to
spectra of some breadth such as storm seas. It is clear that nearly mono-
chromatic waves follow the linear depth relationship, yet it is increasingly
clear that irregular waves do not. A question of major importance not yet
resolved is how wide must a spectrum be before the waves follow the relation-
ships in this report. Equally important is the isolation of the physics of
wave motion that determine these differences. In a shoaling monochromatic
wave, nonlinearities arise which force the development of harmonics in the wave
frequency and tend to broaden the spectrum, yet the absence of other wave
components may reduce the transfer energies by resonant interactions. If the
bottom slope is sufficiently steep, the evolution of the swell waves may be
markedly different from irregular waves which may more easily exchange energy
due to resonant interactions.

VII. SUMMARY

A method for estimating depth-limited significant wave height, Hp,
based on a theoretical form for the shape of shallow-water storm wave spectra
was presented. The method requires an estimate of the peak frequency of the
wave spectrum, fp; knowledge of the Phillips’ equilibrium coefficient, a;
and water depth, h. A method for estimating a based on information about
the peak frequency of the sea spectrum is also given. The results indicate
that the depth-limited significant wave height, Hg, based on the energy of
the sea state is generally less than the depth-limited monochromatic wave
height, Hg. The depth-limited wave height defined as 4.0(E)!/2 appears
to be related to the square root of depth.

22

LITERATURE CITED

DRUAT, C.Z., MASSEL, S., and ZEIDLER, B., "Investigations in Wind-Wave Struc-
ture in the Surf Zone by Methods of Spectral Characteristics," Instytut
Budownictwo Wodnego Polska Akademiia Nauk-Gdanik, Rozprawy Hydrotechniczne -
Zeszyt, No. 23, 1969, pp. 71-80.

GADZHIYEV, Y.Z., and KRATSITSKY, B.B., "The Equilibrium Range of the Frequency
Spectra of Wind-Generated Waves in a Sea of Finite Depth," Izresttya, Atmos-
pherie and Ocean Physics, USSR, Vol. 14, No. 3, 1978, pp. 238-242.

HASSELMANN, K., et al., "Measurements of Wind-Wave Growth and Swell Decay
During the Joint North Sea Wave Project JONSWAP," Deutsches Hydrographischs
Institut, Hamburg, Germany, 1973.

KAKIMUMA, T., "On Wave Observations off Heizu Coast and Takahama Coast,"
Bulletin No. 10B, Disaster Prevention Institute, Kyoto University, Japan,
1967, pp. 251-272.

KITAIGORODSKII, S.A., KRASITSKII, V.P., and ZASLAVSKII, M.M., "Phillips Theory
of Equilibrium Range in the Spectra of Wind-Generated Gravity Waves,"
Journal of Physteal Oceanography, Vol. 5, 1975, pp. 410-420.

OU, S-H., "The Equilibrium Range in Frequency Spectra of the Wind-Generated
Gravity Waves," Proceedings of the Fourth Conference on Ocean Engineering
in Republic of China, 1980.

PHILLIPS, O.M., "The Equilibrium Range in the Spectrum of Wind-Generated Waves,"
Journal of Flutd Mechanics, Vol. 4, 1958, pp. 426-434.

PHILLIPS, 0.M., The Dynamics of the Upper Ocean, Cambridge University Press,
London, 1972.

PLANT, W.J., "On the Steady State Energy Balance of Short Gravity Wave
Systems,'' Journal of Physteal Oceanography, Vol. 10, 1980, pp. 1340-1353.

SEELIG, W., and BRODERICK, L., "Effects of Wave Steepness and Water Depth on
Laboratory Irregular Waves," unpublished laboratory memorandum, U.S. Army,

Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va.,
1981.

THORNTON, E.B., "Rederivation of the Saturation Range in the Frequency Spectrum
of Wind-Generated Gravity Waves," Journal of Phystcal Oceanography, Vol. 7,
Jan. 1977, pp. 137-140.

TOBA, Y., "Local Balance in the Air-Sea Boundary Process II, Partition of Wind

Stress to Waves and Currents," Journal of the Oceanographic Society of Japan,
Vol. 20, 1973, pp. 20-25.

23

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