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 5285 Port Royal Road Springfield, Virginia 22161 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 0 0301 0090102 1 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) UNCLASSIFIED 15a. DECL ASSIFICATION/ DOWNGRADING SCHEDULE Approved for public release; distribution unlimited. 16. DISTRIBUTION STATEMENT (of this Report) 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. FORM DD , AA SE) 1473 ~—s EDITION OF 1 NOV 65 tS OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) Py) wt hea ve f ye Oech aie Ne War ‘a % eee "Chyaysia aan Nive Ho ai ie EN ae ea : He i \ ae Tao ta eres on ~~ Se eet: if orcacronenl te mere airs yitehe Seen Noew Ay, wavew Peas ssen {i Cano Lae ese Fa etiad i Rue Ry ane tae eas Le : ieee cee aN A RR ae et me ee fore) BE fe EER EO RD betel Bo gastoS are rstoavew | +8353 3 } a ae o oe “ia ape i i a pete nr eh all A ba spurte ae RNA i a mg i ee SY siceagi ts ’ TEAS! Te 0 RATT SP Bas io 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 “OUTT MPT JF 9Yq wWorz SuTIeTASp ATJUPOTJTIUZIS oTTYM meT e-F 943 02 TeTTe1ed ATasoTo s10ow aq 03 puay eazoeds oui g ydea3 ut -‘o8uea 2440U-81°O 03 -T°O 943 UT MET sty Worz BUTIeETAP etjoeds 9y YIM zZIA10Y ZG 02 ['Q Worz Bue Aduenbezz ayq 12900 meT ss B MOTTOF OF umoys ote e1JI0ds tzadeep oy y ydea3 uy * SolTouopuodop gat DEO a8 Aejdstp 03 poztytewizou eram O861 42990390 ¢Z uo ‘euT{oIe9) yIION ‘yong 3e Peq.eTTOo eBajoeds aAeM ‘smeT o-3 pue ,_j 0} syzdep snotazea qe e1j00ds oaem JO suostieduog -°Z oan8t1q *TequozTAoYy eq PTNoys 3/(F)q ys Aq peztTeuszou (J)a wnajoeds paanseou ayy ‘setouenbeizy jo a8uei owos isao ptoy 04 meT azomod yz ue 104, (S/1) Aduanbasy ($/1) Aduanbary os°0 LE*0 S2°0 2t'0 00°*0, os*0 ze€"0 Se'0 21°0 00°0 AS] OGL + We GIi9+ Sens 6e'0 (2%™9) / (( (N) 4) ® (4) 3) 907 (oem) / (( (N) &*3) ¥ (5) 3) 907 wisi OF9) v WG Goo ¥ LS ‘8- 1S"@—- WGZ Oll@ wi6é G29 oO ga'g- LL'g- , IUapuadagq ¢-3 | 99uapuadag 20inss )22Uapuadaq . _y | 22uapuadaq a gh'e-