TRANSFORMATION OF WAVES ACROSS THE SURF ZONE Galo Padilla Teran SoP?SrGRADUArE SCHOOL MOWTERfy, CALIFORNIA 93943-5002 NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS TRANSFORMATION OF WAVES ACROSS THE SURF ZONE by Galo Padilla Teran • March 1981 Thesis Advisor: E.B . Thornton Approved for public release; distribution unlimited T200675 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS FACE (Wtton Dmtm Bnlorod) REPORT DOCUMENTATION PAGE i *eport numbTr 2. OOVT ACCESSION NO. 4. TITLE (mnd Subtitle) Transformation of Waves Across the Surf Zone 7. »uThO»(«) Galo Padilla Teran * performing organization name ano aooress Naval Postgraduate School Monterey, California 93940 READ INSTRUCTIONS BEFORE COMPLETING FORM ». REORIENTS CATALOG NUMBER '• TYPE OF REPORT ft PERIOO COVERED Master's thesis; March 1981 S. PERFORMING ORG. REPORT NUMBER • CONTRACT OR GRANT NUMBERf*) 10. PROGRAM ELEMENT. PROJECT TASK AREA * WORK UNIT NUMBERS ' 1 I. CONTROLLING OFFICE NAME ANO ADDRESS Naval Postgraduate School Monterey, California 93940 12. REPORT DATE March 19 81 II. NUMBER OF PAGES 61 14. MONITORING AGENCY NAME * AOORESSff SifeMMf Irom ControlUng Olllem) IS. SECURITY CLASS, (ol thlo riport) Unclassified IS*. OECLASSIFI CATION/ DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEMENT (ol thi* Rtpwl) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (ol <*• obotrocl onftod In Block 30, II dllloronl horn Report) IS. SUPPLEMENTARY NOTES IS. KEY WOROS (Continue on towtmo olio II nmooommrr on* tfntltf *r Woe* mmmot) Wave transformation model Shoaling Waves Surf Zone Distributions 20. ABSTRACT (Contlnuo on rowotoo »ldo II nocooomrr mnd IdmnUtf *r mlock mmmor) Goda's (1975) model, describing wave transformation from deep water to across the surf zone, is compared with a large amount of wave data obtained from experiments conducted at Torrey Pines Beach, San Diego, California. Goda's model simulates wave breaking by truncating the Rayleigh distribution in order to estimate the wave height distributions across the surf zone; wave heights are shoaled by applying nonlinear theory . DD FORM 1 JAN 73 1473 COITION OF I NOV •• IS OBSOLETE S/N 0102-014-6601 I UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAOE (Wnon Dmio Bmorod) UNCLASSIFIED ('**«■ r*tm #-«•»•* #20 - ABSTRACT - (CONTINUED) Comparisons between the empirical distributions and theoretical distributions, and between measured and theoretical rms wave heights, are made. It is found that Goda's model over-predicts the tails and under- predicts the peaks of the empirical distributions, and that the calculated rms wave heights are too large compared with measured values . The range of breaking, and the coefficients used in the breaking criteria by Goda, are modified in order to obtain a model which better fits the distribution of observed hieghts , and which matches the model and observed rms wave heights. The results are quite good, with error envelope for predicted rms wave heights less than 20%. Linear shoaling theory is applied to the model and found to be as good as applying nonlinear theory . Form 1473 0 UNCLASSIFIED Jan 73 2 ^__— 0102-014-6601 ueumw claudication or 0M« »<•«•'•*> Approved for public release; distribution unlimited Transformation of Waves Across the Surf Zone by Galo Padilla Teran Lieutenant, Ecuadorean Navy Ecuadorean Naval Academy, 1970 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL March 19 81 1 KjUi^ ABSTRACT Goda's (1975) model, describing wave transformation from deep water to across the surf zone, is compared with a large amount of wave data obtained from experiments conducted at Torrey Pines Beach, San Diego, California. Goda's model simulates wave breaking by truncating the Rayleigh distribu- tion in order to estimate the wave height distributions across the surf zone; wave heights are shoaled by applying nonlinear theory. Comparisons between the empirical distributions and theoretical distributions, and between measured and theoreti- cal rms wave heights, are made. It is found that Goda's model over-predicts the tails and under-predicts the peaks of the empirical distributions, and that the calculated rms wave heights are too large compared with measured values. The range of breaking, and the coefficients used in the breaking criteria by Goda, are modified in order to obtain a model which better fits the distribution of observed heights, and which matches the model and observed rms wave heights. The results are quite good, with error envelope for predicted rms wave heights less than 20%. Linear shoal- ing theory is applied to the model and found to be as good as applying nonlinear theory. TABLE OF CONTENTS I. INTRODUCTION 12 II. THEORETICAL BACKGROUND 14 A. RAYLEIGH DISTRIBUTION 14 B. TRUNCATED PROBABILITY DISTRIBUTIONS 17 1. Collins Distribution 18 2. Battjes Distribution 18 3. Kuo and Kuo Distribution 19 4. Goda Distribution 20 5. Summary 23 III. EXPERIMENT 24 A. INSTRUMENTS 24 B. DATA ANALYSIS 25 IV. RESULTS 29 A. TYPICAL SPECTRA 29 B. HEIGHT STATISTICS 29 C. COMPARISON OF EMPIRICAL WITH MODEL DISTRIBUTIONS 30 D. COMPARISON OF RMS WAVE HEIGHTS 33 E. WAVE HEIGHT DISTRIBUTIONS USING CURRENT METERS 35 F. COMPARISON OF MODEL AND MEASURED CUMULATIVE DISTRIBUTIONS 36 V. CONCLUSIONS 37 BIBLIOGRAPHY 56 INITIAL DISTRIBUTION LIST 59 LIST OF TABLES I. Truncated Probability Densities. The dotted lines represent the original Rayleigh dis- tributions and the heavy lines represent the modified distributions 39 II. Wave Height Statistics (W and C represent wave staffs and current meters respectively) 40 III. Measured and Calculated rms Wave Heights Obtained with Goda ' s Model and Modified Goda's Model 41 LIST OF FIGURES 1. Cross-section of surf zone showing instrument spacing and elevations relative to measured waves on 20 November 19 78 at Torrey Pines Beach, California 42 2. Definition sketch of zero-up-crossing wave heights - 43 3. Typical spectra measured during the experiments 44 4 . Empirical distribution of wave heights compared with those predicted with Goda's model, starting in deep water (P7) and going into shallow water (W21,W38) , 20 November 1978 45 5. Empirical distributions of wave heights compared with those predicted with Goda's model, starting in deep water (W21) and going into shallow water (W38,W41), 17 November 1978 46 6. Empirical distributions of wave heights compared with those predicted with the modified Goda's model, 20 November 1978 47 7. Empirical distributions of wave heights compared with those predicted with the modified Goda's model, 17 November 1978 48 8a. Range of measured and Rayleigh root-mean-square wave heights 49 8b. Range of measured and Rayleigh significant wave heights 49 9. Correlation of measured rms wave heights with calculated Goda's rms wave heights 50 10. Comparison of the changes of Hj-^ with the Goda's model applying nonlinear (heavy line) and linear shoaling (light line) . Upper figure illustrates the unmodified model; lower figure illustrates the use of the modified coefficients in the model — 51 11. Percentage error of predicted (modified model) compared with measured rms wave heights 52 12. Empirical distributions of wave heights obtained from current meters (C23, C37 and C40) compared with predicted wave heights calculated with the modified model, 17 November 1978 53 13. Comparison of measured cumulative exceedance distributions with predicted distributions (modified model) , 20 November 19 78 (W38) and 17 November 1978 (W41) 54 14 . Comparison of measured cumulative exceedance distributions with predicted distributions (modified model) , 20 November 1978 (W21) and 17 November 1978 (W38) 55 LIST OF SYMBOLS a Root-mean-square (rms) amplitude A Goda's breaking criteria coefficient C Speed of energy propagation E Energy density g Acceleration due to gravity h Local depth below still water level S Sea surface elevation H, Breaking wave height H Deep water wave height H Wave height parameterizing truncated Rayleigh distribution H Root mean square wave height rms n 3 H Transfer function that relates the velocity spectrum components to the kinetic energy H Transfer function that relates potential energy to the kinetic energy k Wave number K Goda's breaking criteria coefficient KE Kinetic energy K Shoaling coefficient L Deep water wave length m Lowest moment variance of the frequency spectrum p Probability density function of wave heights p Probability density function of unbroken waves S Horizontal velocity spectrum, x-component S Horizontal velocity spectrum, y-component T Wave period X Ratio wave height to deep water wave height X, Higher limit of breaking range X« Lower limit of breaking range z Measurement elevation m an Deep water incident angle a. Incident wave angle at breaking 6 Bottom slope 6 Delta function Y Ratio of breaking wave height to depth of water at breaking v Root mean square spread of the noise about the mean frequency p Water density 10 ACKNOWLEDGMENTS The author wishes to express his appreciation to Dr. Edward B. Thornton, Professor of Oceanography at the Naval Postgraduate School, Monterey, California, as Thesis Advisor, for his guidance, method and systematic assistance in the preparation of this study. The assistance of Ms. Donna Burych, Computer Programmer in the Oceanography Depart- ment is gratefully recognized. 11 I. INTRODUCTION The evaluation of an irregular group of shoaling waves as they approach and pass through the breaker zone is a com- plex process which requires special measurements and analy- sis considerations . The usual approach to shallow water wave transformations is to predict, from a single "represen- tative" set of deep water parameters, the wave height, the wavelength and the frequency at specific shallow water depths, using linear shoaling theory. The primary objection to this approach is that a single set of deep water wave parameters does not realistically represent the distributional charac- teristics of naturally occurring sea surface waves. A secondary objection arises from the use of linear transfor- mations which become inadequate when applied through the surf zone (Wood, 1974). Wave heights in deep water, having Gaussian surface ele- vations, are described by the Rayleigh distribution (Longuet- Higgins , 1952). Waves propagating toward shore can increase in height due to shoaling effects, refraction and wave inter- actions, and eventually reach a depth where they start break- ing. The energy dissipation due to breaking has been simu- lated (Goda, 1975) by truncating the tail of the Rayleigh distribution . Experiments were conducted at Torrey Pines Beach, San Diego, California, during November 1978. Sea surface eleva- tions, pressures and velocities were measured at closely 12 spaced locations in a line extending from 10m depth to inside the surf zone. This thesis applies Goda ' s model to the measurements in order to examine the shoaling and trans- formation of wave heights and their probability density func- tions (pdf 's) from deep water to breaking and across the surf zone to the shoreline. 13 II. THEORETICAL BACKGROUND A. RAYLEIGH DISTRIBUTION The Rayleigh distribution was shown theoretically by Longuet-Higgins (1952) to apply to deep water wave heights on the assumption that the sea waves are a narrow-banded Gaussian process. Barber (1950) had earlier presented empiri- cal evidence that the Rayleigh distribution agreed with the measured distribution of waves. On the assumption that the wave height is twice the wave amplitude, the wave height probability density is then represented by p(H) = 2H/H* exp(-H2/H^ ) CD rms rms where H is the rms wave height. rms 3 Using pressure records in the Gulf of Mexico, Longuet- Higgins (1975) observed that the Rayleigh distribution fits the observed distribution reasonably well in "fairly deep water" . He found that there is a slight excess of waves with heights near the middle of the range and a deficit at the two extremes. Since much of the high-frequency portions of the wave records were filtered out by the pressure trans- ducer, Longuet-Higgins (197 5) suggested that the narrow band approximation may not be as applicable for the unfiltered records. In shallow water with much steeper waves, the Rayleigh distribution can again be expected to be less applicable due to the non-linearities. 14 Since the Rayleigh distribution theoretically did not apply to broadband wave spectra, Goda (1970) numerically simulated wave profiles, where the amplitudes were specified by various theoretical spectra of varying bandwidth and the phase was random. He then examined the simulated records for surface elevations, crest-to-trough wave heights and zero- up-crossing wave heights. He found that, using the zero-up- crossing determination of wave heights, the Rayleigh distri- bution is a good approximation irrespective of the spectral bandwidth. Tayfun (1977) , in studying the transformation of deep water waves to shallow water waves, showed that the Rayleigh distribution for wave amplitude was generally applicable to all bandwidths . The Rayleigh distribution is applied correctly only to low waves in deep water (Longuet-Higgins, 1975), since it is assumed that the contributions from different parts of the generating area are linearly superposable . Under this assumpt- tion, the distribution clearly should not hold for waves approaching maximum height, i.e., close to breaking, as in the surf region or even in the open sea with whitecaps. It has been found by several authors (Chakrabarti and Cooley, 1977; Forristal, 1978) that the theoretical Rayleigh dis- tribution over-predicts the maximum wave in the tail compared with large wave observations. Forristal (1978) attributed the differences to the non-linear, non-Gaussian and skewed nature of the free surface. 15 Tayfun (19 80) examined non-linear effects by consider- ing an amplitude-modulated Stokian wave process with the restriction that the underlaying first order spectrum is narrow band. The surface displacements were found to be non-Gaussian and skewed, and wave heights distributed ac- cording to the Rayleigh probability law, particularly for low and medium wave height ranges. On the basis of the results obtained, Tayfun (1980) concludes that the non-Gaussian characteristics of the free surface do not directly result in reducing maximum wave heights in a manner consistent with field observations and that a more plausible mechanism is wave breaking, which is a non-linear effect not directly accounted for in the analytical wave models currently available Longuet-Higgins (1980) analyzed the effects of non- linearity and finite bandwidth on the distribution of wave heights to explain the differences with observations found by Forristal. He found that the reason for the discrepancy could be accounted for by the presence of free background "noise" in the spectrum, outside the dominant peak, and that it was not due to a finite-amplitude effect. Longuet-Higgins concludes that the distribution of wave heights even in a storm is well described by the Rayleigh distribution, pro- vided the rms amplitude, a, is estimated from the original — 1/2 record and not from the frequency spectrum as a = (2m_) The effect of finite bandwidth is estimated from a model assuming low background noise linearly superposed on a very narrow (delta function) spectrum. For narrow bandwidths , 16 he obtains the formula a2/2m0 = 1 - 0.734 v2 (2) where v is the rms spread of the noise about the mean fre- 2 quency. Values of v corresponding to Pierson-Moskowitz (broad-band) spectra also give results in close agreement with observation. Therefore, the Rayleigh distributions calculated in this paper are parameterized using the rms wave height. B. TRUNCATED PROBABILITY DISTRIBUTIONS In concept, waves are described by the joint distribu- tion of height, period (or equivalent wavelength) and direc- tion. To simplify the analysis, all authors assume a very narrow band frequency spectrum and a narrow directional spec- trum, so that all the wave heights of the distribution are associated with a single mean frequency and mean direction. Therefore, starting in deep water, the waves are described by the unaltered single-parameter Rayleigh distribution, (with the implied assumptions. The deep water wave heights are transformed into shallow water waves using shoaling theory in which frictional dissipation is neglected. Even- tually the waves reach such shallow water that they start to break, with the largest waves breaking furthest offshore first. Wave breaking is simulated by truncating the tail of the Rayleigh distribution. 17 1 . Collins Distribution Collins (1970) was the first to apply the technique of a truncated distribution to describe the effects of wave breaking, using a sharp cut-off with all broken waves equal to H, which results in a delta function at H, . Collins does b b not give an explicit formula for his distribution, but it would be the same as the described by Battjes (1974) (see below) . He used linear shoaling theory and the breaking criterion after Le Mehaute and Koh (1967 Hb/HQ = 0.76 tan1/76(H0/L0)"1/4 (3) where tan 3 is the bottom slope and Hn and L_ are the deep water wave height and length, respectively. For waves break- ing at an angle a, the bottom slope is actually tan 3 cos a. , 1/2 and H. should be replaced by H~ cos / a,.. The various dis- tributions, and how they are truncated are showed schemati- cally in Table I . 2 . Battjes Distribution Battjes (1974, 1978) again used a sharp cut-off of waves and applied the breaking criterion based on Miche ' s formula for the maximum height of periodic waves of constant form, Hb z 0.88/k tanh (y/0.88 kh) (4) where y is an adjustable coefficient. In shallow water (4) reduces to 18 Hb = y h . (5) He uses linear theory to shoal the waves. The probability density for breaking wave heights is given by: p(H) = H/2H2 exp[-l/2(H2/H2) ] , for 0 <_ H <_ Hb (6) p(H) = exp[-l/2(H2/H2) ]5(H -Hb) , for H>Hb (7) where H is the wave height parameterizing the truncated Rayleigh distribution by Battjes. All waves that have broken, or are breaking, assume the height prescribed by (7); this results in a delta function at the truncation height, H, , of the distribution (see Table I). 3 . Kuo and Kuo Distribution Kuo and Kuo (19 74) investigated the effect of break- ing on wave statistics using a conditional Rayleigh distri- bution sharply truncated, specified by the breaking wave height simply proportional to local water depth, equation (5) . The conditional probability density function of wave heights, p, (H), is calculated using the following equation, p (H) = p(H/0 ^H < Hb) = g(H) , for 0 < H < Hb b jj / p(H)dH 0 = 0 , for H > Hfa . (8) 19 Describing the conditional probability in this manner re- sults in the proportional redistribution of probability- density associated with the broken or breaking waves over the range of H. This removes the delta function at the breaking wave height, H, , previously described by Collins and Battjes. Table I shows the original Rayleigh distribution with dotted lines, and the modified Rayleigh distributions in heavy lines after applying the cut-off to the tails using the breaking criterion. The distribution by Kuo and Kuo is more realis- tic but still results in a sharp cut-off of the distribution at the breaking heights. 5 . Goda Distribution Goda (1975) derived a more realistically truncated distribution, qualitatively anyway, by requiring a gradual cut-off of the distribution. He uses a shoaled Rayleigh distribution to describe the unbroken wave heights at shore- ward locations prior to applying his cut-off which is given by p (H) = 4H/K2H2 exp(-2H2/K2H2) (9) o so so where K is the shoaling coefficient. Goda (1975) calculated s 3 the wave shoaling using the nonlinear theory of Shuto (1974) : which dictates the following: 0 < gHT2/h2 < 30: Small Amplitude Theory (10) 30 < gHT2/h2 < 50: H h2/7 = constant (11) 20 50 < gHT2/h2 <_ °°: Hh5/2 [ VgHT2/h2 -2/3] = constant. (12) Goda assumes that wave breaking occurs in a range of wave heights between H2 and H.., with varying probability. The probability density function of unbroken waves only is expressed as: Pr(X) = pQ(X) ; for X < X2 (13) X — X Pr(X) = Pq(X) -x _x2 P0(\) ; for X2 < X < Xx (13) Pr(X) = 0 ; for X± < X (15) where, normalized wave heights are defined by X = H/H , X, = H,/H and X_ = H2/H . The artifice of spreading breakers over a range partly represents the inherent variability of breaker heights, and partly compensates for the simplifica- tion of using a single wave period in the estimation of breaker height. Broken waves generally have some height smaller than X,. Since no theory is available for describing waves after they have broken, the heights of broken waves are assumed to be redistributed across the range and to be proportional to the unbroken waves as was done by Kuo and Kuo (1975) . Therefore, the conditional probability density function for all heights is calculated by Pr(X) p(X) = — ■=-*■ (16) Xl / p (X)dx 0 r 21 where : X. 2V2 1 9 -a X, / p (X)dX = 1 - (1+a X (X -X )}e L , (17) 0 r . ± z is a constant of proportionality applied to p (X) to normal- ize the pdf. In equation (17), the constant, a, is equal to /2/Ks. The breaker height is estimated using the following formula, which is an approximate expression for Goda's breaker index (1970) based on laboratory data, H L H g£ = A^{l-exp[-1.5 ^^(1+K tanpS)]} , (18) o o o o where tan 3 denotes the bottom slope and the coefficients are assigned the following values for best-fitting to the index curves, A = 0.17, K = 15, and p = 4/3. The range of breaker height, X, - X_ , is calculated by assigning the following values for A: A = 0.18 for X , A = -j A = 0.12 for X2 The upper limit of A., was selected by considering the varia- bility of breaker heights, whereas the lower limit of A2 was chosen simply as two-thirds of A, . The coefficients used 22 are based on matching laboratory data taken on a 1/10 and 1/50 beach slope and several field experiments. The Goda model is applied here to the experimental data described below. 5. Summary The common idea of these studies is to cut-off the portion of wave height distribution beyond the breaker height, which is controlled by the water depth and other factors. The methods differ in the techniques of cut-off and the formulae used to define breaker heights. 23 III. EXPERIMENT Experiments were conducted at Torrey Pines Beach, San Diego, California, during November 1978, as part of the Nearshore Sediment Transport Study. At this site there is a gentle sloping, moderately sorted, fine-grained sandy beach, The beach profile shows no well-developed bar structure and is remarkably free from longshore topographic inhomogenei- ties. Winds during the experiments were light, and variable in direction. Shadowing by offshore islands and offshore refraction, limits the angles of wave incidence in 10m depth to less than 15°. During the experiments, significant off- shore wave heights varied between 60 and 160 cm. The condi- tion of nearly normally incident, spilling (or mixed plung- ing-spilling) waves, breaking in a continuous way across the surf zone, prevailed during most of the experiments. A. INSTRUMENTS An extensive array of instruments was deployed to study nearshore wave dynamics . Measurements described here are from sensors located on an offshore transect from 10 m depth to across the surf zone (Fig. 1) . The sensors were of three types: pressure (P) , current (C) and surface-piercing-staff (W) . The pressure sensors were Stathem temperature-compensated 2 transducers with dynamic range of either 912-2316 g/cm or 2 912-3720 g/cm . They were statically precalibrated and 24 postcalibrated by being lowered into a salt-water tank and were found quite linear; the gains differed by less than 2% between calibrations. Current meters were two-axis , Marsh-McBirney electro- magnetic, spherical (4 cm diameter) probes, with a three- pole output filter at 4 Hz. Precalibration and postcalibra- tion of current meters showed little change in replicate runs with steady or oscillating velocity fields. The uncer- tainty associated with using a single gain factor for all frequencies is roughly estimated at ±5% in amplitudes (10% in variances) . The wave staffs were dual resistance wires with low noise, high resolution, and good electronic stability. The accuracy of the wave staffs was about ±3% based on repeata- bility of gain calibrations measured in the laboratory and in situ. B. DATA ANALYSIS Sea surface elevation and wave velocity components were retrieved from sensors by telemetering to shore and there recorded on a special receiver/ tape recorder, described in detail by Lowe et a_l. , (1972) . The sampling rate was 64 samples/s which was reduced to 2 samples/s by digital low- pass filtering. Record lengths of approximately 68 minutes from each data set were analyzed. It was desired to examine only the sea-swell band of frequencies between 0.05 to 1.0 Hz (20 to 1 s periods). The 25 data were first linearly detrended to exclude effects of the rising and falling tides. The da-.a were then high pass filtered with a cut-off frequency of 0.05 Hz (20 s period). The high pass filter used a Fast Fourier Transform algorithm to obtain the amplitude spectrum of the entire 68 minute record. The Fourier coefficients corresponding to 0 to 0.05 Hz were used to synthesize a low frequency time series which was subtracted from the wave record. The limiting high frequency (Nyquist) was 1.0 Hz. The energy density spectra were calculated in a similar manner for wave and velocity measurements using the Fast Fourier Transform (FFT) algorithm. A cosine-squared taper data window was applied to the time series to minimize leakage The highest maximum and lowest minimum of the surface elevation within a period interval defined, respectively, the crest and trough of a wave. A wave height H is defined as the total range of £ (t) in that interval, the time between two consecutive zero-up-crossings of c(t) (see Fig. 2). Since the average wave period was about 14 sec, the total number of waves in the 68 min. record was about 300, which gives reasonable wave statistics. The height statistics of mean wave height H, root mean square wave height H , significant wave height H, /3 (average of the heights of the 1/3 highest waves) , and H, ,.- (average of the heights of the 1/10 highest waves) , are calculated from the ordered set of wave heights. 26 The pdf and cumulative distributions of wave heights were calculated. The heights were normalized using the deep water root mean square value. Theoretical probability distributions were calculated using the Goda model and com- pared with measured distributions. A deep water reference wave height was calculated by- measuring the energy using current meter C9 located at about 4 m depth and backing the energy out to deep water. Kinetic energy spectra were calculated from the measured horizontal velocity spectra, S (f) and S (f ) ; linear theory transfer functions were used to integrate the spectra over the water column, so that the average kinetic energy is = |H„_(f) |2[S (f) +S (f)] (19) J\Ij u v 2 where |H (f) |~ is the transfer function that relates the JS.il velocity spectrum components to the kinetic energy, i it t*\ i2 1 sinh 2kh , orn lHKE(f)l = 4k p — zrrj— ; (20) cosh k(h + z ) m where z is the measurement elevation. Guza and Thornton m (1980) showed, for these same experiments, that using linear theory transfer functions to calculate average kinetic energy gave reasonable results. To first order in energy, the average potential energy equals the average kinetic energy. The potential energy in deep water is obtained by applying linear shoaling transformation 27 -0 ^ wa. ^ = |H(f) |2. , (21) deep water ' s ' 4m where H (f) = VC /C , is the linear shoaling coefficient. s g0 g The rms wave height is related to the PE and KE by = ~ p g H2 = = / df , (22) o rms _ from which the deep water rms wave height, denoted hence- forth by H , can be found. 2 o To take advantage of the large number of current meters, current data were used to infer wave heights . The velocity signals were convolved using linear wave theory to obtain surface elevations. The complex Fourier spectra of the horizontal velocity components U(f), V(f) were first calcu- lated and vectorially added. The complex surface elevation spectrum, X(f), was calculated applying the linear wave theory transfer function, H(f) X(f) = 5(f) -V.(f) (23) where H(f) = Si£h *£ - r (24) oj cosh k(h + z ) m The complex surface elevation spectrum was then inverse transformed to obtain the surface elevation time series from which the wave height distribution is calculated. The entire 68 min . record was convolved at one time in order to minimize the end effects which result in spectral leakage. 28 IV. RESULTS A. TYPICAL SPECTRA A broad range of wave and weather conditions were en- countered during the experiments. Typical velocity spectra for two days (Fig. 3, lower panel) include an example of very narrow band spectra calculated for November 20 for the current meters C22x and C23x which straddled the mean breaker line. In shallow water depths, the waves generally become more "peaky", resulting in increased energy at the harmonics. The presence of strong harmonics in the spectra indicates the importance of nonlinearities of the waves in shallow water. The spectral energy level decreases at all frequen- cies except at the very lowest from the deeper instrument C22x (heavy line) to the shallower instrument C23x due to breaking. The other typical spectra (Fig. 3, upper panel) are an example of combined sea and swell with a narrow band of energy at swell frequencies, but with broad band energy at higher frequencies. B. HEIGHT STATISTICS The wave height statistics for six days, inferred from wave staff and current meter data, are presented in Table II. The statistical parameters listed are: root-mean-square heights, H , significant wave heights, H..,-, average of the heights of the 1/10 highest waves, H1/1Q, and the maximum height, H . These parameters were obtained from 68 min. 29 records. The reference wave height H , obtained by backing the energy measured at current meter C9 out to deep water, the frequency peak of the spectrum and the depth for each instrument, are also presented. Most of the wave staff measurements were made inside the surf zone, after the waves have started breaking. Shoaling effects are observed in the data of Table II. The H , H, ,-,, and H, ,,Af increase as depth decreases, rms I/-J l/±u until they reach the breaking point after which the wave heights decrease as the depth decreases, due to breaking. The range of depths of the instruments is from about 570 cm to 40 cm. The average peak frequency (peak frequency) varied little during the experiments and was about 0.07 sec (Table II) . The relative depth for the waves at the peak frequency is h/L < 1/25 in all cases so that they can be considered shallow water waves. C. COMPARISON OF EMPIRICAL WITH MODEL DISTRIBUTIONS The Goda model is compared with measured data. The model is first run using Goda ' s original coefficients. This model results in over-prediction of the H . The coefficients r rms are changed to optimize the model's description of the wave height distribution qualitatively, and H quantitatively. The H parameter was calculated from the second moment of rms c the wave height distribution and is, therefore, a more sensi- tive parameter to describe the shape of the distribution (particularly the tail) than, say, first moments such as the mean H, H^, *1/1Q. 30 A comparison (Fig. 4) was made between the empirical wave height distribution normalized with the reference deep water wave height and Goda ' s model distribution for November 20. Starting in deep water, it is observed that the model fits quite well the empirical distribution obtained from the pressure sensor P7. It is clearly seen that the wave heights in deep water are essentially Rayleigh distributed. In shallow water the model over-predicts the tails and under- predicts the peaks. The smaller the depth the greater the errors (Figs. 4 and 5). Due to this over-prediction at the tails of the distribution the rms wave heights obtained by the model are larger than the measured ones. To obtain a better agreement between the measured and the calculated rms wave heights, the higher limit of breaking and the range of breaking of the model are changed. Goda (1975) calculated the shoaling using the nonlinear theory of Shuto (1974) (see equations 10, 11 and 12) . For all the ranges of wave heights, frequencies and depths used here, 2 2 the values of gHT /h are calculated and in all cases analyzed, fall in the third category, equation (12) . This law was used to calculate the nonlinear shoaling coefficient K . Goda assumed a range of breaking between HI and H2 that takes into account the variability of breaker heights and the use of a single frequency. The variable breaker height H, is a function of the frequency, depth and bottom slope (see equation 18) . The values of the coefficients used by Goda in his breaking criterion are the following: Al = 0.18, A2 = 2/3 Al , 31 K = 15 and p = 4/3, empirically assigned to give the best- fit with observed wave heights. The coefficients used here to give a qualitatively better fit with the measured dis- tributions and quantitatively fit with H are: Al = 0.136. ^ ■* rms A2 = 1/2 Al , K = 20; p was left equal to the previous value. The most sensitive coefficient is Al which was determined first. Values of Al were calculated for various distributions by first obtaining the higher breaker limit (XI) from the measured empirical distributions as indicated by the maximum value of the distribution, and then calculating Al using equa- tion 18 . The values of Al were then averaged to give a single representative value for all distributions. The coefficient A2 was chosen as 1/2A1, which results in a more symmetrical distribution as indicated by the results. The coefficient K, which weights the slope of the beach, was determined by trial-and-error testing of Goda's model for a variety of values looking for the best fit of all the distri- butions. The model is not very sensitive to changes in K. The comparison (Fig. 6) of the empirical distribution in deep water for November 20 with the distribution obtained applying the model used the new coefficients. In deep water, there is no notable difference from the original model's results (Fig. 4). In shallow water, the predicted values obtained with the modified model fit much better the empirical distributions than those obtained applying the original Gcda model. 32 As stated earlier, the coefficients Goda originally speci- fied were based on matching laboratory data for a two beach slopes of 1/10 and 1/50. The model was then applied to some field data, but unfortunately the beach geometry (slopes) were not given. The Torrey Pines Beach is approximately 1/50 at the beach face and the wave climate is characterized by long period (-14 sec) swell. The reason for differences in the model coefficients needed to fit this data set com- pared with Goda's suggested coefficients is not known. D. COMPARISON OF RMS WAVE HEIGHTS The model calculated H values and measured values of rms H calculated directly from the wave heights are used to test how well the model works. Table III shows the measured, Rayleigh, and the calculated Goda and modified Goda root- mean-square wave heights corresponding to the wave staffs and current meters for six different days. Goda's model-predicted H values are. in general, r rms ' 3 larger than the measured ones (Fig. 9) . In an attempt to explain the differences, H values obtained from the r rms model were plotted against depth, deep water wave height and Ursell number; but no meaningful correlation was obtained with any of these variables. Measured H and calculated values assuming the wave rms a heights are Rayleigh distributed so that H = /8m were 3 ■* ' rms o compared (Fig. 8a). Measured and assumed Rayleigh signifi- cant wave heights were also compared (Fig. 8b) . The comparisons 33 show that the agreement between measured and Rayleigh sta- tistics are good for both small and large wave heights, inside and outside the surf zone. The good comparisons with Rayleigh statistics suggests that the wave heights, although decreased by breaking, are still more nearly Rayleigh-dis- tributed than cut-off Rayleigh, as suggested by Goda. The change of the rms wave heights with depth, between the measured H and the calculated ones obtained by the rms x model first applying nonlinear (heavy line) and then linear shoaling (light line) were compared (Fig. 10) . The model with nonlinear shoaling clearly over-predicts the values, while the model with linear shoaling gives reaonable values, com- pared with measurements. The nonlinear shoaling "blows up" in very shallow water (<30 cm) and should be ignored. Measured and the calculated rms wave heights with depth obtained by applying linear and nonlinear shoaling using the modified coefficients were also compared (Fig. 10, lower panel) . This figure illustrates that the modified Goda model fits better the data than the original does. The majority of the measured H fall between the two curves obtained with rms nonlinear and linear shoaling. Based on the choice of coeffi- cients, applying linear shoaling to Goda's model can give as good, or better, results as applying nonlinear shoaling. The rms wave height values, obtained using the modified model coefficients, were plotted against depth, deep water wave height and Ursell number; no obvious correlation could 34 be noted. It is found that the new predicted H values rms compared with measured H have an error of less than +20% r rms at all depths (with the exception of one anomalous point) (Fig. 11) . E. WAVE HEIGHT DISTRIBUTIONS USING CURRENT METERS As described earlier, the surface elevations were derived by linearly convolving the velocity records and wave height distributions calculated. The basis for applying this analy- sis is the earlier work of Guza and Thornton (19 80) where they showed, for this same data set, that linear theory spec- tral transformations could be used to calculate surface ele- vation standard deviations either from pressure meters or current meters with less than a 20% error, and typically less than 10%. Examples of the derived wave height distributions are considered for November 17 (Fig. 12) . These measurements were made just outside the surf zone, at about the breaker point and inside the surf zone (current meters C23, C37 and C40 respectively). In general, the model overestimates the velocity derived wave height distributions more than the direct measurements . The reason for the discrepancy is that linear wave theory underestimates the surface elevations in convolving the velocities, particularly in the crest region of the waves . In other words , linear theory does not account for the finite amplitude of these highly nonlinear waves. 35 F. COMPARISON OF MODEL AND MEASURED CUMULATIVE DISTRIBUTIONS The cumulative exceedance of wave height distributions normalized with rms deep water wave height were calculated applying the modified Goda model (using nonlinear shoaling) and plotted with the measured cumulative distribution for comparison. The cumulative exceedance distribution empha- sizes information in the tail of the distribution. For an example in shallow water, inside the surf zone (116 cm depth) , there is a good agreement between the measured and predicted distributions with a slight underprediction by the model in the tail (Fig. 13) . With sensors located just outside the surf" zone, under-prediction of the tail is larger and over- prediction in the middle range occurs (Fig. 14). In general, there is a better agreement between the two distributions well inside the surf zone, e.g., wave staffs W38 and W41, than those (W21 and W29) which were generally either at breaking or just outside the surf zone, the most nonlinear wave region. 36 V. CONCLUSIONS It is confirmed that the wave heights in deeper water (7 m) are Rayleigh distributed. Goda's model, using the empirical coefficients originally suggested on the basis of laboratory and poorly specified field measurements, over-predicts the tail of the distribution and under-predicts the peaks. As a consequence, the predicted rms wave heights are larger than the measured. As the depth decreases, the errors of the predicted distributions increase. To get a better fit with the measured wave height distribu- tions, the coefficients in the breaking criterion used in Goda's model were modified. The values for Goda's breaking criterion giving the best fit to the measured data of this s tudy are : Al = 0.136, A2 = 1/2 Al and K = 20 . The percentage of error between the measured and the pre- dicted H values from the model with these coefficients is rms les than ±20%. Linear shoaling was found to be as good as nonlinear shoaling in applying Goda's model across the surf zone. Good comparisons were obtained between empirical and Rayleigh-derived statistics. This indicates that the wave heights, although decreased by breaking, are still more Rayleigh-distributed than the cut-off Rayleigh-distributed as suggested by Goda. 37 Goda's model was tested here with a large amount of field data for a variety of wave conditions on a 1/50 beach slope. Further comparisons should be made for a variety of beach slopes and wave climates in order to test the general applicability of the model. 38 J < 2 H n « O Q W W H EC Bn Eh H Q Eh O 3 2 W co pa a EC « EH cu W En 3 z w co co w w z « h a. Js Q W CO En W Eh 2 O H Q J W » EC > Eh H 2 Eh O H H J Eh H D ffl CQ < H a a; • O Eh CO EC CO 2 CU H O Q H Q Eh ta EC D Eh O CQ < H H u ta ps 2 ^ Eh D >H cn « cn «*-^ ^ p— . r— + ♦ m r— a» •— .e -C olo U 5 AC ■ ^I-1 jz\o ae —i -a CO ^ Ix LU ^ c >» u uo ^^ 1— O to o • LO ae X - — QJ jC ac ex '— <_> C a* ce >— 0} *J +-> CO x . "2- -= f*> z: si -0 CI 0 olo « O _| dh ■a QJ 0 CO II < 01 II s- ii >*- -a o *j J3 "O -Q X 11 c^ -olo S , XIX >*- X O <: 21 ~~" XIX ^^ t r^ S- r— a» ■ •■^ s_ s_ i- c 0 1 Ifl ra oo _i _1 —J z —• / > » 1 - L J >k i / 3 3 3 r 3: 1 / / 1 / / t / / r / / / / / / / / / / / / / -C / / 3 / -C / * 3 / ^r c .a X j '' ' /7 // // / ' ^T / /f J^S / / o / // __^i^c^ 3 / If I 1 / / .a CO J 1 ( 1 ►— t \ \\ ae. \ \\ i— \> oo \. a ^\ ^^^^ — * . . ^_^ ^_^ X X X 31 QL a. Q. a. O 3 i£ OS l/l T3 o c - — . ai *— ■» C — > ^m^ x •— o •"-> ^»- fO «3- LT> h— ■ — i — +j r» r^ ng r^ •— en *-> cn 0 cn T3 CTl O i — fO ^— 3 r— 0 f— o • — a ^ ■— 0 ^ ta CQ < Eh 39 TABLE II. WAVE HEIGHT STATISTICS (all values in cm) Date Inst h H o f H rms Hl/3 Hl/10 H max Nov 4 W41 82 42.5 .0703 33.2 44.4 53.2 74.3 W38 125 42.5 .0703 44.8 61.5 73.4 98.1 W21 177 42.5 .0703 50.6 71.8 90.9 131.3 W29 225 42.5 .0703 56.7 81.3 113.8 188.3 Nov 10 W21 170 66.1 .0632 61.0 85.5 107.4 140.4 Nov 17 W41 92 44.8 .0729 37.8 50.7 59.4 79.5 W38 141 44.8 .0729 52.0 72.1 86.9 111.5 W21 197 44.8 .0729 54.1 76.2 94.1 139.2 W29 209 44.8 .0729 52.0 72.8 93.7 154.0 Nov 20 W3 8 116 52.4 .0666 35.9 49.3 57.9 83.8 W21 153 52.4 .0666 48.6 68.1 84.2 120.3 W29 182 52.4 .0666 73.1 110.2 143.0 202.7 Nov 20 C42 39 52.4 .0703 18.0 27.0 33.8 49.1 C39 102 52.4 .0703 35.6 50.5 59.8 82.6 C37 116 52.4 .0703 36.5 53.6 62.1 81.8 C36 142 52.4 .0703 40.1 60.4 69.8 89.5 C23 147 52.4 .0703 51.6 80.2 98.0 118.9 C22 188 52.4 .0703 60.8 96.1 127.7 159.3 C19 250 52.4 .0703 68.9 105.1 149.0 210.4 C15 355 52.4 .0703 58.7 88.5 125.5 217.9 C09 571 52.4 .0703 55.8 84.0 115.2 192.4 Nov 24 W3 8 65 36.4 .0639 10.1 14.5 18.4 26.3 W21 86 36.4 .0715 26.7 36.6 43.5 55.8 Nov 18 W41 84 55.4 .0757 33.8 45.8 55.4 109.1 W21 195 55.4 .1552 61.4 87.2 103.9 131.7 W29 198 55.4 .0756 63.9 91.3 117.2 166.3 40 TABLE III. MEASURED AND CALCULATED RMS WAVE HEIGHTS OBTAINED WITH GODA'S MODEL AND MODIFIED GODA'S MODEL (all values in cm) RMS WAVE HEIGHTS SIG. HEIGHTS Date Inst. Me as . Ray. Goda Mdf . G . Meas . Ray Nov 4 W41 33.2 33.7 34.6 30.9 44.4 46.9 W38 44.8 46.2 55.9 47.5 61.5 63.4 W21 50.6 52.3 67.3 51.1 71.8 71.6 W29 56.7 58.2 65.3 58.5 81.3 80.2 Nov 10 W21 61.0 64.9 85.3 64.5 85.5 86.3 Nov 17 W41 37.8 37.9 48.6 34.7 50.7 53.5 W38 52.0 52.2 66.1 53.4 72.1 73.5 W21 54.1 55.6 70.3 56.0 76.2 76.5 W29 52.0 53.8 69.8 58.1 72.8 73.5 Nov 20 W38 35.9 36.2 50.3 43.9 49.3 50.8 W21 48.6 50.3 68.3 44.9 68.1 68.7 W29 73.1 67.5 77.6 53.3 110.2 103.4 Nov 20 C42 18.0 18.2 21.4 18.0 27.0 25.4 C39 35.6 36.1 54.0 38.5 50.5 50.3 C37 36.5 36.3 60.3 43.9 53.6 51.6 C36 40.1 40.0 70.1 53.8 60.4 56.7 C23 51.6 50.0 71.5 55.7 80.2 73.0 C22 60.8 57.5 79.1 54.9 96.1 86.0 C19 68.9 65.0 78.3 67.5 105.1 97.4 C15 58.7 60.9 72.0 64.0 88.5 83.0 C09 55.8 60.7 64.0 55.6 84.0 79.0 Nov 24 W38 10.1 10.7 26.7 24.4 14.5 14.3 W21 26.7 27.5 44.6 32.7 36.6 37.8 Nov 18 W41 33.8 35.5 33.4 31.4 45.8 47.8 W21 61.4 59.7 59.6 50.8 87.2 86.8 W29 63.9 61.7 80.7 57.5 91.3 90.4 41 pq t^J r"H w £j S m £r« pa ^P§ UT4 ^ pi P-« '-H £3 00 (J) CVJ 3 CG O EH o •'#/?* fee 7— CM* 3 ^ [ha. * V •■??•©' O 0 03 z a 03 o -P g G a) 0 £ 3 03 5-i Q) -P > • 03 rfl CQ m ■H to +1 03 G id i G c cu 0 0 5-( •H •H 5-1 -P -P 0 U id E-" Q) > 03 CD • _ NOV 20 FILE 1 U) C22X C23X o • o • CO D ru o • i i ■ 1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 FREQUENCY (HZ) Figure 3 . Typical spectra measured during the experiments 44 EMPIRICAL DISTRIDUTIO-i OP WOVE HEIGHTS N0V50 T07 1.5 1.0 0.5 0.0 1.5 1.0 o.s 0.0 1.5 1.0 0.5 0.0 / 1 / / A n / n i. M/HO Figure 4 . Empirical distribution of wave heights compared with those predicted with Goda's model, starting in deep water (P7) and going into shallow water (W21, W38) , 20 November 1978. 45 EMPlRICflL DISTRIBUTION OF u?)VE HEIGHTS N0V17 H2! I.S 1.0 0.5 0.0 1.5 1.0 0.5 0.0 1.5 1.0 O.S 0.0 A r/ 4 / 0. 1. 2. M/HO Figure 5. Empirical distribution of wave heights compared with those predicted with Goda's model, starting in deep water (W21) and going into shallow water (W38, W41) , 17 November 1978. 46 EMPIRICAL DISTRIBUTION OF ufivE hFICmTS NOV20 P07 Figure 6 1.5 t.o 0.5 0.0 t.s 1.0 o.s 0.0 1.5 1.0 0.5 0.0 FksJl 0. 2. M/hO N0V20 W?l i o. 2. H/HO Empirical distribution of wave heights compared with those predicted with the modified Goda s model, 20 November 1978. 47 EMPIRICAL D15TMCUU0M C? wfivE hCICmTS N0VI7 H21 1.5 1.0 0.S 0.0 t.S 1.0 o.s 0.0 1.5 1.0 O.S 0.0 t / 1. 2. H/HO N0V17 Mm / 71 I I \ h- i. 2. H/HO Figure 7. Empirical distribution of wave heights compared with those predicted with the modified Goda's model, 17 November 1978. 48 MEASURED HRMS AT HAVE STAFFS V5 RAUEICH HRMS 100. 80. ? 60. t HO. 20. 0. o«c IM4T otnxicm K ■l»04 M4I « «}• i« Ml ITT Ml J/l ■ »0» 10 ■21 no e HOT IT u«l « "it i*i Ml l»T w.'9 !?1 ♦ »o».-a me IK un IS J mi 1*1 X M'rt Ml a Ml 86 • Mill HI ti Ml IM «*• Iff 0. 20. 40. 60. 80. HEASURED HRMS 100. Figure 8a. Range of measured and Rayleigh root-mean- square wave heights. MEASCREO Hl/3 AT HAVE STAFFS VS RATLEICH Ml/3 100. 80. - SO. t »o. 20. OftM MSI ocrimcm ■ • 010* Dm K n* IJI Ml ITT Ml til ■ HOT 10 Ml no a M«IT U<4I « Ml iii Ml 117 Ml .'01 • MMO Ml IIS Ml IS] Mi \tl z «•»« Ml II Ml II • ■Oil* HMI II Ml III Hit III 20. MO. 60. 80. MEASURED HI/3 100. Figure 8b. Range of measured and Rayleigh significant wave heights . 49 MEASURED HRMS AT WAVE STAFFS VS GODA'S HRMS en az 100. 80. 60.