ZY, CA : NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS COMPARISON OF WAVE CELERITY THEORIES WITH FIELD DATA by Michael R. Syvertsen March 1985 Thesis Advisor: E. B. Thornton Approved for public release; distribution unlimited T206806 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Whit Out Entered) imRART.KAVALKWKBADWOBBCBOOL T'ONTrlREY.CA 93940 REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM t. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4. TITLE (and Subtitle) Comparison of Wave Celerity Theories with Field Data 5. TYPE OF REPORT & PERIOD COVERED Master's Thesis; March 1983 6. PERFORMING ORG. REPORT NUMBER 7. AUTHORf*> Michael R. Sy vert sen S. CONTRACT OR GRANT NUMBERf*; >• PERFORMING ORGANIZATION NAME ANO ADDRESS Naval Postgraduate School Monterey, California 93940 10. PROGRAM ELEMENT. PROJECT, TASK AREA a WORK UNIT NUMBERS II. CONTROLLING OFFICE NAME ANO ADORESS Naval Postgraduate School Monterey, California 93940 12. REPORT DATE March 1983 13. NUMBER OF PAGES 40 pages 14. MONITORING A3ENCY NAME a AOORESSf'/ dlllatent /rem Controlling Olllce) IS. SECURITY CLASS, (of this report) UNCLASSIFIED 15«. OECLASSIFI CATION/ DOWN GRADING SCHEDULE :«. DISTRIBUTION STATEMEN r (ot thle Report, Approved for public release, distribution unlimited. 17. DISTRIBUTION »T>TEMEN T (of the eoetrect entered In Slock 20, It dltlerent from Report) It. SUPPLEMENTARY NOTES This research partially supported by ONR Contract NR388-114 19. KEY WORDS (Cor,tlnua on revert* eld* If noceaemty and Identity by block number) Wave Speed, Wave Celerity, Phase Speed, Wave Speed Theory, Linear Wave Theory, Bore Wave Theory, Solitary Wave Theory, Hyperbolic Wave Theory, Cnoidal Wave Theory 20. ABSTRACT (Continue) on revetee tide II neceteiy and Identity by block number) Three independent wave celerity data sets , measured on natural beaches , are compared with linear, bore, solitary, and hyperbolic wave theories. In the range of relative water depths ( . 006 < h/T < 13 cm/s ) and wave heights (.!■£ H/T< 3 cm/s ) tested, hyperbolic wave theory, which is an assymptotic DD I JAN 73 1473 COITION OF 1 NOV 83 IS OBSOLETE S/N 0102- LF- 014- 6601 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Bnterec UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (WtlMl Dmta £ntar«d) form of cnoidal theory' in shallow water, agreed most closely with measured wave celerities. Linear wave theory also gave satisfactory results; but bore and solitary wave theories overestimated the observed wave speeds. It is concluded that the observed waves are weakly dispersive in amplitude and that care must be taken to apply the theories only in their regime of validity. S N 0102- LF- 014- 6601 UNCLASSIFIED SECURITY CLASSIF'CAT'C* OF THIS »»3*"»»!«r» w*<« Snww.' Approved for public release; distribution unlimited Comparison of Wave Celerity Theories with Field Da*; by Michael R. Syvertsen Lieutenant, United States Navy E.S. , University of Washington, 1977 E.A., University of Washington, 1977 Submitted in partial fulfillment of the requirements for the d=grae of MASTEF CF SCIENCE IN METEOROLOGY AND OCEANOGRAPHY from the NAVAL POSIGRADUATE SCHOOL Karch 1933 at / c.l ABSTRACT Three independent wave celerity data sets, measured on natu- ral teaches, are- compared with linear, bore, solidary, and hyperbolic wave theories. In the range of relative water depths (.006 < h/T < 13 cu/s') and wave heights (.1 < H/TZ< 3 cm/s ) tested, hyperbolic wave theory, which is an assymp- totic form cf cncidal thecry in shallow water, agreed mcst closely with measured wave celerities. Linear wave thecry also gave satisfactory results: but bore and solitary wave theories overestimated thc observed wave speeds. It is con- cluded that the observed waves are weakly dispersive in amplitude and that care mist b^ taken to apply the theories only in their regime cf validity. TABLE OF CONTENTS I. INTRODUCTION 9 II. EXPERIMENTS 14 A. SEVEN WILE EEACH, AUSTRALIA 14 B. TCRREY PINES BEACH, CALIFORNIA 15 C. IEADEETTER EEACH, CALIFORNIA 17 III. RESULTS 23 IV. CONCLUSIONS 29 LIST OF REEESENCES 36 INITIAL DISIEIEUTION LIST 3 LI SI OF FIGUFSS Figure 1. Regions of Applicability 30 Figure 2. Measured Wave Speed vs. Linear Theory .... 31 Figure 3. Measured Wave Speed vs. Solitary Theory . , . 32 Figure 4. Measured Wave Speed vs. Bora Theory 33 Figure 5. Measured Wave Speed vs. Hyperbolic Theory . . 34 LIST OF TABLES TABLE I. Statistical Comparison of Wav= Theories . . 35 ACKNOWLEDGEMENT The tireless efforts cf Ms. Donna Eurych and Ms. Andree Web- ster in makir.g Tcrrey Pines and Leadbetter beach data usable were noteworthy and greatly appreciated. Had it not been for their ready answers tc perplexing problems, many hours would have been spent needlessly. I also thank, the many courteous and industrious computer operators at the N. E. Church computer center. To Prof. E. B. Thornton: for all your time, efforts, suggestions and leadership, THANKYCO, from the bottom cf my heart. This thesis would net have been started *ere it not for your knowledgeable insight and direction. Lastly, I want to thank my loving wife Ee~ty, for your patience and understanding during our two year ten- ure and especially during the final few weeks of thesis preparation. I. INTRODUCTION Considering all the various theories available to describe wave celerity, surprisingly little work has been oriented toward comparing wave theories with data, particu- larly field data. Wave celerities acquired in the labora- tory were tested against wave theories by LeMehaute et al (1968); they concluded that cnoidal wave theory as proposed by Keuiegar and Patterson (1940) , with equations and tables by Masch and Wiegel (1961), gave a 'best fit* solution. Other laboratory studies have suggested appropriate wave celerities to he described by solitary wave theory (Ippen and Kulin, 1955) and (Kishi and Saeki, 19t>7), Stokes thecry (De, 1955), hyperfcolic/cncidal theory (Iwagaki, 1968). Cel- erities measured in the field have been compared with linear theory (Thornton and Guza, 1982), and bore theory (Bradshaw, 1982) and (Suhayda and Pettigrew, 1977). The validity of various theories depends upon the relative dep^-.h (h/L) , where h is the water iepth and L is the wavelength, and the relative wavaheicht (H) measured in terms of a wavelength (H/L) or depth (H/h) . In this study, various celerity formulae given by pro- gressive viave theories are tested against celerities measured in the field. To test the theories against data, the phase speed equations require seme combination of water depth, naveheight, and wave period (T) . Several relatively large field data sets did not provide one or the other of these parane-ers and could not be used. Data which repre- sents reflected waves froa a sieep beach such as a data set collected at Fort Ord Beach, California (Salienger et al, 1983) could net te used- Also, another large data set, col- lected by University of California, Berkeley, California (Moffitt, 1953) using photographic methods, was no-1 included in the study. This is because the waveheight was omitted in the measurements. Waveheight must be included tc test higher order theories, especially if amplitude dispersion is to be examined. Five wave theories were compared with wave celerities measured in the field in water depths from .07 to 10.0 meters, fcr wave height- from .12 to 1 . 0 meter, and for periods varying between 4,7 and 18.3 seconds. Stokes (third order) theory as presented by Hunt (1953) proved unsuitable due to the relatively shallow water depths encountered. The four remaining theories tested are: linear theory; solitary theory as modified by Laitone (1959); bore thecry as given 10 by Keller ft al (1960), and hyperbolic theory of Iwagaki (1968), which is an assy mp to tic form for cnoidal theory in shallow water. The data hav = been plotted as a function of waveheight and water depth versus period in Figure 1. This char-, adapted from LeMshaute (-1976), attempts to quantify regimes of applicability. For example, the diagram indicates that the data dees not fall in the Stokes (third order) theory regime where exp arime rtat ion proved to be true. Additicr- 2 3 ally, the Drsall (Ur) parameter (Ur= HL /h ) , plotted as a dashed line, is commonly used to parameterize the nonlinear waves in shallow water (Thornton and Guza, 1982). For Stokes theory to be valid, the Ursell parameter should be small, which is not the case for most of the data consid- ered. In the case of very long waves in shallow water, the Ursell parameter becomes meaningless since it is directly proportional tc the square of the wavelength. To determine the range of validity for various theories, the underlying assumptions are examined. All the theories presented assume: the mction t c be irrc tational, the fluid to be incompressible, no mean current flow, and normal wave incidence. As a consequence of the wave theory assumptions, 11 the Ursell parameter can now be quantified and is included for comparison purposes. The assumptions for the various theories and oheir celerity equations are as fellows: LINEJ.F: c - (gk tanh kh)1/2 0) Assumes: H/h << 1, H/I << 1, h/L << 1, and Ur >> 1 for shal- low water, h/I >> 1 ard Ur « 1 for deep water. SOLITAFY: c - /gh 1.0 +I(|) 20 Kh' J (2) Assumes: H/h < 1, h/L < .1, Ur=0(1). BORE: c - /gh (l.o + £)1/2 (l.o + fjr)172 (3) Assumes: H/b = C(1), h/I << 1f Ur >> 1 (very shallow water). 12 HYPEREOLIC (l o + I!) (I)2 i I (I - I) - Ml u.u + Kh; th> ^ K ^K A; 20 f J (tt) Assurces: fi/h < 1, h/L < .1, Ur >> 1, where K is defin=d below. Iwagaki (1968) linearized the computationally diffi- cult Jacobian elliptic function into a modulus K, where: v T/g7TT /T*/Hxl/2 K I kp 1.0 - a ( H v n "1 m h; J (5) in which a=1.3, n = 2 and m=.5 for H/h ^ 0.55 and a=0.54, n=1.5 and m=1 for H/h > 0.55 This approximation greatly simplified the analysis. 13 II. EXPERIMENTS Three field data se:s are used to compare with various theories. Experiments conducted at Torrey Pines B^ach and Leadbetter Beach, California were both part of the Nearshcre Sediment Transport Study (NSTS) (Seymour and Duane, 1976). These beaches were chosen for their relatively simple beach plan, both unbarred, and essentially straight and parallel nsarsfcore contours. The third data set was from Seven Mile Baach, Australia. A., SEVEN KILE EEACH, AUSTRALIA The Seven Mile Beach (Shoal haven Bight) experiment on the south coast cf New South Wales, Australia, was conducted in early 1982 by Eradshaw (1982) . He examined the relation- ship between bore velocity, height, and water depth by ana- lyzing movie camera pictures using stakes driven into the sand as references. This technique examines each individual wave by counting the number of movie frames between the four reference stakes to deteriine wave speed while reading bcra height and water depth directly from the incrementally marked stakes. 14 The beach is composed of fins quartz sand and has multi- ple cffshci€ tars. The beach slcpe outside was 0.04, while, the inner surf zone had a slope cf 0.03. The waves gener- ally troke en the outer tars and then reformed and propa- gated as teres inside the surf zone. The outside breakers ranged from 1.0 to 1.5 meters. The data are limited to only shallow water bores inside the treaker line, where tore heights were from 0.14 to 0.30 meters in water depths cf 0.07 to 0.42 meters. Data points are plotted as circles ir Figure 1 and generally reside in the very shallow water, snail waveheight regime. Cf the 27 data points available, three points were discarded because bore height greatly exceeded the water depth (H/h > 3). Fxact pericd/f reguency data was not available for the 27 runs; however, during the two-day experiment, an 3 tc 12 second period was observed. Hence the mean (10 second) period is chosen for computational purposes. This assump- tion is adeguate since only shallow water bores were consid- ered and they are essentially no ndispersiv e. 3. ICRR2Y fINES ESACE, CALIFORNIA Field irea eurements were made at Torrey Pines Beach near San Diego, California in August 1978. A detailed 15 description cf the experiment can be found in Gaza and Thcrntcn (1980), Guza and Thornton (1982), and Thornton and Guza (1982). Tcrrey Pines is a gentle sloping (.02) beach and is composed cf fine grain sand. The waves were gener- ally narrow banded and approached "he shore at a near normal angle. Directional properties of the waves were measured using a linear, five pressure sensor array in 10 meters depth. The angle of swell approach was limited to the maxi- mum and minimum of +/- 15 degrees due to sheltering by off- shore islands and coastline restrictions, but the angle was generally less than +/- 5 degrees. Refraction analysis shewed that the predominant swell waves (T = 13 ssc.) start- ing at 15 degrees in 10 meters dep-h results in angles cf incidence cf 6.5 degrees in 3 meters depth and 4.9 degrees in 1 meter depth (Thornton and Guza, 1982). The 92 data pcirts, plotted as + 's in Figure 1, are gen- erally restricted to the shallow water regime. Waveheights ranged from small to 2.0 meters, water depths at sensor locations from 0.27 to 7.C meters, and mean periods from 9.0 to 18.0 seconds. The experimental domain included spilling or mixed spilling and plunging breakers. 16 C. LEADBEITEP EEACH, CALIFORNIA The Leadtetter Beach, Santa Barbara, California, experi- ment was conducted during the period of 30 January to 23 February 1980. Experimental details are described in Gable (196 1). leadtetter is a relatively straight, steeper slop- ing (.05) teach composed of fire to medium, well-sorted sand. A series of sterns, resulted in abnormally large waves and teach erosion for the period commencing 4 Febru- ary, 1990. The storms formed what has bean described as a •50 year' storm event. Measured wave heights during the experiment ranged from 0.18 to 1.9 meters, with sensor depths up to 10 meters, and periods from 4.7 re 16 seconds. Breaking wave types were of both the spilling and plunging variety. The 83 Santa Barbara data points, plotted as tri- angles in Figure 1, exhibit the largest waveheights and water depths examined. Leadbettsr Beach has an east-west orientation which is counter to the north-south orientation of the California coastline. Ihe predominant northwest ocean swell entering the narrow gap between Foint Conception and the Channel Islands would have to refract nearly 90 degrees tc approach normal to the beach. As a consequence, the ocean swell 17 waves approach at a relatively large, well focused angle from the west. At other times, storm waves generated inside the Channel Islands appicached at large angles from the east. Therefore, the wave angularity has to be considered in the wave c=lerity calculations. Incident offshore sea-swell was measured using a fcur pressure senscr square array with 6 meter legs located in a water depth of 8 meters. From these sensors, the mean inci- dence angle at the teak frequency was determined for the nine data sets. The wave rays were then manually refracted, using Snell's Law, from the pressure array location to a point where the bottom contours could be considered straight and and parallel to the beach. Shoreward of this point, the incident wave angles were calculated using a constant refractive coefficient. Deep wafer angles varied between +/- 20 degrees and were refracted into 4 meters of water from there into the shallower water region of the current senscrs. For example, wave angles of 2J degrees in 8 meters depth, period cf 12.0 seccnds, resulted in refracted angles of 10 degrees in 3 meters decreasing to 5 degrees in 1 meter of water. 13 Spectral methods kere utilized to determine mean celeri- ties and wave heights usir.g current and pressure sensor data from San Die go and Santa Barbara. The celerity and wave height calculations are a r. average ever many waves. Since the record lengths were 34 minutes and the mean period (T) was afceut 12 seconds, approximately 170 waves ar= averaged for the spectral estimate cf wave celerity. This is in con- trast to the camera methods of Bradshaw (1982) where each individual wave was photographed as it passed the reference stakes and waveheight and water depth were read directly. Pressure and current meter data were telemetered to shore where they were digitally recorded at a rate of 64 samples/second. The data were averaged to 2 samples/second which results in a Nycuist frequency of 1.0 Hertz. Records were then compiled into 4C96 da ta points. By breaking the series into 256 point records, the phase, kinetic energy, and coherency spectra were computed with 32 degrees cf freedom. Celerity spectra were calculated from the phase spectra measured between adjacent pairs cf current meters located in a line normal to the teach. The actual celerity (Cx ) was computed using: C (f) = 2lTfAx (6) 19 where, (ax) is the distance between sensors, (*) is the phase difference between sensors, and (f) is the frequency. It was shewn by Thornton and Guza (1982), for the Torrey Pines data, that the waves can be considered frequency non- dispersive such that the celerity spectra is constant, at least across the energetic region of the spectrum. Hence, a mean celerity, representative of the entire spectrum, was choser at that value corresponding to the peak frequency in the energy spectrum and is used in the data comparison here. Also, the frequency at the spectral peak generally coincided with the maximum coherency. Bence, the Tcrrey Pir.es Beach celerity data were calcu- lated using (6), where f - f(peak). At Santa Barbara the wave angularity was taken into account. Since the celerity spectrum was calculated using instruments in a line normal to the beach, only the x component of celerity is measured directly. The tctal 'mean1 phase speed is calculated using: c„(f ) ) = — P C = C(f_) = (7) cos a (f ) P where, 0."?5 (Suhayda and Fettigrew, 1S77). Conceivably, Figure 1 should be modified in the very shallow water regime tc account for the three fccre categories. 23 The data fcr Torrey Fines and Leadbetter Beaches also depart from the H/h = 0.78 slopa. At Torray Pines Beach, Thornton and Guza (1982) fcund waves insida -he surf zona at saturation, H(EHS)/ h = 0.44. A possible difference is that tha H(PMS) waveheight, calculated for Torray Pines and Lead- better Beach data do not recessarily correspond tc tha indi- vidual waveheight used in the solitary wave criteria. Individual waveheight is also used for Seven Mile Beach data . The measured phase speeds from tha three experiments are compared with the four wave theories in figures 2-5. Tha solid line in figures 2 thru 5 denotes a perfect fit line (slope=1.0), the dashed is the 'cast fit1 linear equation line, and the dctted lines are the 95% confidence intervals. Table 1 is a syncpsis of a linear, least squares, regression comparing treasured values versus -he theoretical celerity. The eguaticns cf 'best fit* lines are given as well as the correlation coefficient (Corr). All data pcints are unbi- ased (given equal weight) when computing Table 1 values. The 953 confidence intervals (dotted lines) have been placed on Figures 2-5 for statistical purposes. Any pre- diction afccut an individual Y (actual) associated with a 24 given X (theory) will be most meaningful near the mean of X. Hence, the 95?? confidence limits are bowed inward (toward the regression line) near the -mean of X. The curvature associated with the confidence lines is slight due to the significant number (199) cf data points. A perfect fit of the data tc a theory consists of a regression line whose slcpe is 1.0 ard intercept is 0.0. The regression equation which is the best approximation cf the perfect fit is presumed to be the most accurate. It is apparent frcm the 'best fit1 (dashed) lines of figures 2-5 that hyperbolic wave thecry most closely approximates the perfect scluricn (solid line) , while linear wave theory is only slightly less accurate. Bo"-',- theory appears tc give a near constant cver-es timaticn of phase speed while solitary thecry does well at the lew end (<500 cm/sec) cf the spec- trum. Statistically, the errors tend to nullify sach ether when the data sets are conbined. Tnis is a recognized prop- erty cf a simple linear regression and becomes quite evident as the number cf data poirts (N) apprcach.es infinity. While the equations of the least squares linear regres- sion suggest that hyperbolic wave theory may have a slight edge; the same cannct he said if each beach is analyzed 25 separately. Torrey Eines Beach is best explained by bcre theory, Seven Mile Beach ty solitary theory, and Leadfcetter Beach by hyperbolic theory. We also notice at Seven Mile Beach that ncne cf +he theories did particularly well at predicting wave speeds. This should be no surprise since 15 cf the 2k data points fell outside the regime cf the wave theories considered. As can be seen from Table 1, a definitive conclusion based on the correlation coefficient would be difficult. Considering the teach tctals, ail four wave theories have comparable correlations; however, the regression equation for fcypertclic theory has the best slope and intercept com- bination, as further shown in Figure 5. There appears to be a wide rang a of scatter associated with the Santa Barbara data for which there is no simple explanation. Beth, Torrey Pines and Leadbetter 3each, data sets were siiilarily collected and analyzed. While it is true that wave refractr.cn diagrams were required for the Santa Barbera data, there is no reason to believe that there were significant errors in the computations. The offshore sensors from Lsadbetter Beach displayed little scatter (Fig- ures 2-5) in the data; since these sensors were subject to 26 the largest angles of incidence as well as the largest cel- erities. There must be some other mechanism to explain the widespread scatter at the intermediate wave speeds. Guza and Thornton (1982) have suggested that the pres- ence cf surf teat may be responsible for the differences between measured and theoretical celerities. Surf beat is a long reflected wave with a period cf the order cf several minutes, wavelength of the order of the surf zone width, and a maximum amplitude cf the order cf the swash at the beach face, decreasing offshore. Surf beat causes variations in the water depths, particular at the an tin odes of the reflected wave, as perceived by the shorter sea-sweii waves. This night explain seme of the scatter of the data at Lead- better since the phase speed is depth dependent and sensor placement could have coincided with antinodal activity which would result in a lew frequency modulation of the water depth above the current meters. Surf Dear also induces cn- offshcre velocities, particular at the nodes of the reflected waves, which can also affect the phase speed of the sea-swell waves. Sensors placed near the nodes cf long waves could experience large amplitude excursions in the waves' horizontal velocities. These periodic differences in 27 the leng wave induced velocities could result in a dcppl. shifting ci the phase spectra and could account for scatt in Lsadbetter data. 28 IV. CONCLUSIONS fiyparfcclic theory, which is an assymptotic form of cnoi- dal theory in shallow water, appears to give best agreement whsr. all three data sets are collectively analyzed and com- pared. LeMehaute et al (1968) simiiariiy concluded cnoidal theory gave the best comparison with laboratory data , while Dean 0965) found cnoidal theory is valid particularly in deep water hut is lacking for shallow water waves. Linear theory, on the ether hand, yields almost as good a compari- son and is computationally easier. The data did net allow for a proper test of wave theories in the deeper water and larger waveheight regime. Ecre ar3 solitary wave theories generally cverpr edic-ed the treasured celerities while hyperbolic and linear theory tended tc slightly underpredic1-. . The differences are attributable tc the relative amplitude being dispersive, as predicted ty various -Theories. Eore and solitary theory are sorcngly anpiitude dispersive, hyperbolic theory is weakly dispersive, and linear theory has no amplitude dispersion. Therefore, it is concluded that the waves are best categor- ized as weakly amplitude dispersive. 29 10 10' DEPTH/ T**2 (CM/SEC/SEC) Figure 1. Regions of Aoplica bility Torrey Pines" (♦) Seven Mile Beach (o) Leadbe" ar ( a ) 30 o LD O «■■ ' O o m CD *-* o C") • LJ (_) in tn rn "v 21 « f ") o If) Cs. CD O LJ • n LJ in Q_ CD cn o LJ o 10 U) LD Cl- in (_) Q_ o in O •*r LJ o cm o LD CO to en n LJ 2Z o LT) 00 O O LO •—1 o o ID A ..A'AA •'A At "^A PERFECT FIT BEST FIT 957. CONFIDENCE 50.0150.0 250.0 350.0 450.0 550.0 550.0 750.0 850.0 950.0 1050.0 LINEAR THEORY (CM/SEC) 'igure 2. Measured Wave Speed vs. Linear Theory Tcrrey Pinss (+J Seven Mile Beacn (o) L-= a abetter ( a ) 31 PERFECT PIT BEST PIT 95*/. CONFIDENCE 50.0150.0 250.0 350.0 450.0 550.0 650.0 750.0 850.0 950.0 1050.0 SOLITRRY THEORY (CM/SEC) Figure 3. Measured Wave Steed vs. Soli-ary Theory Tcrrey Pines ( + V Sever. Mile Beach ( Leadberte r ( a ) '(c) 32 PERFECT FIT BEST FIT 95% CONFIDENCE 50.0150.0 250.0 350.0 450.0 550.0 650.0 750.0 850.0 950.0 1050.0 BORE THEORY (CM/SEC) Fi igure 4. Measured Wave Speed vs. Bore Theory Torrey Pines (+) Seven Mile 3each (o) Lea abetter ( * ) 33 PERFECT FIT BEST FIT 957. CONFIDENCE 50.0150.0 250.0 350.0 450.0 550.0 550.0 750.0 850.0 950.0 1050.0 HYPERBOLIC THEORY (CM/SEC) Fraur= 5. Measured Wave Speed vs. Hyperbolic Theory Torrey Pines ( + ) Seven Mile Beach (o) Leadbe-re r ( a ) 3U TABLE I Statistical Comparison of Wave Theories LINEAR SOLITARY BORE HYPERECLIC Tonay Y=6<.3+.384X Y = 28. 6*. 855X Y=0 . 1 04+. 847 X Y=39.0>.919X Pir.es Beach Ccrr=.932 Ccrr=.951 Corr=.955 Ccrr = .920 Seven Mils Beach Y=62.5+.759X Y=36.3+.717X Y=- 3 8. 0+. 844 X Y=57.0+.b67X Corr=.749 Ccrr=.76 8 Corr=.625 Corr=.697 Lead- Y=74.5+.896X Y=30.6+.862X Y=- 1 8. 9+ . 89GX Y=51.2+.930X better Beach Corr=.897 Ccrr=.891 Corr=.383 Ccrr=.900 Total Y=5 I.9 + .932X Y=14. 5+. 836X Y=- 39. 6 +. 922X Y=20.2+.975X of ail Beachs Ccr:: = .S37 Ccrr=.936 Corr=.931 Ccrr = .93b 35 LIST CF REFERENCES Bradshaw, P., Eores and Swash on Natural Beaches, Coastal Studies Unit Technical Report No. 82/4, Umv ersity~cr" Sydney! 5y3Sey7Hre7ff7,~5us£rarra,""TU7 pp., 1982. De , S.C. , Contributions tc the Theory of Stokes* Waves, P*cc. Cambiidcj Phil. 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Guza , Energy Saturation and Phase Speeds Measured on a Natural Beach, J. Geophys. Res., 87, 9499-S508, 1932. _ 37 INITIAL EISTRIBUTION LIST 1. Eefense Technical Inf crmati cr. Center Cameron Station Alexandria, VA 22314 2. library. Code 0 1 42 Naval Postgraduate School Honterey, CA 93940 3. Professor Robert J. Eenard, Code 63Ra Eepartment cf Meteorology Naval Postgraduate School Monterey, CA 93940 No. Copies 4. Professor Christopher N. K. Eepartm^r-1: cf Oceanccraphy Naval Postgraduate School Monterey, CA 93940 Mcoers, Code 68Mr 5. Professor Edward B. Thornton, Code 58TM Department cf Oceanography Naval Postqraduate School Monterey, CA 93940 6. Er. Sassithcrn Aranuvachapun, Code 68WT Department cf Oceancaraphy Naval Postgraduate School Monterey, CA 93940 7. It. Kichael 5. Syvertsen 622 E. Iris St. Cxnard, CA 93033 8. Director Naval Oceanography Division Naval Observatory 34th and Massachusetts Avenue NW Washington, E.C. 20390 9. Commander Naval Oceancgraphy Command NSTL Station Eay ST. louis, MS 39522 10. Commanding Officer Naval Ccsancgraphic Cffice NSTL Station Eay St. louis, MS 39522 11. Commanding Officer Fleet Numerical Oceanography Centa: Monterey, CA 93940 38 12. Commanding Officer Naval Ccean Research and Development. Activity NSTL Station Eay ST. Lcuis, MS 39522 13. Comma ndincj Cfficer Naval Envir Facility Monterey, CA 93940 cnmental Prediction Research 14. Chainrar:, Cceanooraphy Department U.S. Naval Acadetry Annapclis, HD 21402 15. Chief cf Naval Research 800 N. Quincy Street Arlington, VA 22217 16. Cffice of Naval Research (Code 430) NSTL Staticn Eay SI. Lcuis, MS 39522 17. Commander Ccean Systems Pacific Eox 1 39 0 Pearl Harbor, HI 96860 39 1 Thesis S9,7 syvertsen 00861 Comparison of r« Cel«ity theories 6 Thesis 200361 S97 Syvertsen c#l Comparison of wave celerity theories with field data. C^sono.wa.celen^eo^ 3 2768 002 04943 9 ^rcw8 ' 1 . '