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 this Center. Additional copies are available from:

Nattonal Technical Information Service ATTN: Operations Dtviston

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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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UNCLASS IFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO, 3. RECIPIENT’S CATALOG NUMBER TR 82-3 F

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

15. SECURITY CLASS. (of this report)

9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of the Army

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

11. CONTROLLING OFFICE NAME AND ADDRESS

Department of the Army

Coastal Engineering Research Center

Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office)

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Approved for public release; distribution unlimited.

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17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)

18. SUPPLEMENTARY NOTES

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.

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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 compared to 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

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NMouticol Miles

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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 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

= 0.08, £, = 0.07, (a/0.0081)!/2 = 1.20. = 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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