KINEMATICS OF SURF ZONE BREAKING WAVES MEASUREMENT AND ANALYSIS James Joseph Galvin SSSss n - 1 lonterey, California T KINEMATICS OF SURF ZONE BREAKING WAVES MEASUREMENT AND ANALYSIS by James Joseph Galvin September 19 75 Thesis Advisor E. B. Thornton jj Approved for public release; distribution unlimited. T Itoq b*° UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Date Fnl»t«d) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM \. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4. Tl TL E (and Subtitle) Kinematics of Surf Zone Breaking Waves Measurement and Analysis 5. TYPE OF REPORT A PERIOD COVERED Master ' s Thesis ; September 1975 6. PERFORMING ORG. REPORT NUMBER 7. AUTHORfs; G. CONTRACT OR GRANT NUMBERfs; James Joseph Galvin PERFORMING ORGANIZATION NAME AND ADDRESS Naval Postgraduate School Monterey, California 93940 10. PROGRAM ELEMENT, PROJECT, TASK AREA 6 WORK UNIT NUMBERS II. CONTROLLING OFFICE NAME AND ADDRESS Naval Postgraduate School Monterey, California 93940 12. REPORT DATE September 1975 13. NUMBER OF PAGES 96 14. MONITORING AGENCY NAME 4 ADDRESSf// dltlotent from Controlling Oltlce) Naval Postgraduate School Monterey, California 93940 15. SECURITY CLASS, (of thte report) Unclassified I5«. DECLASSIFI CATION/ DOWN GRADING SCHEDULE 16. DISTRIBUTION ST ATEMEN T (of thli Report) Approved for public release; distribution unlimited 17. DISTRIBUTION STATEMENT (ol the abetted entered In Block 20, II different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reveree side II neceaeery and Identify by block number) Breaking Waves Surf Zone Waves 20. ABSTRACT (Continue on reveree elde If neceaeery end Identity by block number) Simultaneous measurements of water surface fluctuations and horizontal water particle velocities in a line perpendicular to the direction of wave approach extending across the surf zone were taken in varying surf conditions at two loca- tions. The spectral velocities calculated using linear theory as a transfer function underestimated measured values by 79-861 at the peak of the spectrum. The coherence DD t jan*7J 1473 EDITION OF 1 NOV 65 IS OBSOLETE (Page 1) S/N 0102-014-6601 | UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Dele Entered) UNCI .ASS TRIED. $LCUWTTY CLASSIFICATION OF THIS P A O T < >» '. en l>rlm Enl»r»d values were generally low indicating non-linear and turbulent conditions. Strong harmonics in the spectra of the waves and water particle velocities further suggest a non- linear system. The theoretical phases computed using linear theory did not accurately predict the observed phases. In general breaking waves can be characterized as a strongly non- linear wave phenomenon. Measured frequency distributions were compared with both Gaussian and Gram-Charlier distributions by using the chi -square goodness-of-f it test. Qualitatively, the Gram-Charlier distribution gave the better fit to the flow velocity data. DD 1 Ja^TG I4?3 UNCLASSIFIED S/N 0102-014-6601 security classification of this paoe(T^»" rxtit Emtrmd) Kinematics of Surf Zee Breaking Waves Measurement and. Analysis by James Joseph Galvin Lieutenant Commander, United States Navy B.S., United States Naval Academy, 1966 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL September 1975 DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL So»Se^.CAUFORN»A 93940 ABSTRACT Simultaneous measurements of water surface fluctuations and horizontal water particle velocities in a line perpen- dicular to the direction of wave approach extending across the surf zone were taken in varying surf conditions at two locations. The spectral velocities calculated using linear theory as a transfer function underestimated measured values by 79-86% at the peak of the spectrum. The coherence values were generally low indicating non-linear and turbulent conditions. Strong harmonics in the spectra of the waves and water particle velocities further suggest a non-linear system. The theoretical phases computed using linear theory did not accurately predict the observed phases. In general breaking waves can be characterized as a strongly non-linear wave phenomenon. Measured frequency distributions were compared with both Gaussian and Gram-Charlier distribu- tions by using the chi-square goodness-of - f it test. Quali- tatively, the Gram-Charlier distribution gave the better fit to the flow velocity data. I ACKNOWLEDGEMENTS I am deeply indebted to Dr. Edward B. Thornton for his knowledge and assistance. I also received much aid and rapid response to requests for assistance from Mr. Robert Smith of the Research Administration Department. Petty Officer First Class Donald Antonacci, USN, of the Machine Facility, Petty Officer Second Class, John Fanning, USN, of the Oceanography Department, Ms. Sharon Raney of the W. R. Church Computer Center, and the Computer Center "night crew": Mr. Edwin Donnellan, Mrs. Kristina Butler and Mr. Mannas Anderson. I am also very deeply thankful to my wife, Barbara, for her encouragement and understanding, and my sons, Michael and J. J., for being patient while waiting for Daddy to come home from school to play. For my Mother and Father 5a TABLE OF CONTENTS I. INTRODUCTION ------------------ -10 II. MEASUREMENTS ------------------- 13 A. EXPERIMENT SITES --------------- 13 B. INSTRUMENTATION- - - - - _ _ _ _ _17 III. ANALYSIS OF DATA ----------------- 22 IV. RESULTS- -------------------- -24 A. QUALITATIVE DESCRIPTION- ----------- 24 B. PROBABILITY DENSITY FUNCTIONS- ------- -24 1. Gaussian and Gram-Charlier Frequency Distributions- -------- -24 2. Skewness and Kurtosis- --------- -31 C. COMPARISONS OF THEORETICAL AND MEASURED POWER SPECTRA ----- - -33 D. EXAMINATION OF SPECTRAL HARMONICS- ----- -37 E. PHASE AND COHERENCE- ------------ -40 F. THEORETICAL AND CALCULATED PHASE SPECTRA - - -43 V. CONCLUSIONS- ------------------ -47 APPENDIX A - BEACH PROFILES AT DEL MONTE BEACH 4-10 MARCH 1975 ------------- -49 APPENDIX B - CAPACITANCE WAVE GAUGE CALIBRATIONS - - - -50 APPENDIX C - FLOW METER CALIBRATIONS --------- -51 APPENDIX D - ANALYSIS DETAILS AND FLOW CHART ----- -52 APPENDIX E - CALIBRATION FACTORS ----------- -54 APPENDIX F - PROBABILITY DENSITY FUNCTIONS VERSUS GRAM-CHARLIER FREQUENCY DISTRIBUTIONS - - -55 APPENDIX G - POWER, COHERENCE AND PHASE SPECTRA- - - - -77 BIBLIOGRAPHY ---------------------- g3 INITIAL DISTRIBUTION LIST- - - - - - - -95 LIST OF TABLES I. Beach and Wave Characteristics --------- -14 II. Computed Statistics- -------------- -28 III. Frequencies and Amplitudes at Spectral Peaks - - -38 IV. Harmonic Frequencies and Spectral Amplitude Ratios --------------------- -39 V. Phase Relationships- -------------- -41 VI. Parameters Calculated for Phase Comparisons- - - -46 LIST OF DRAWINGS 1. Typical Beach Profile and Instrument Location at Del Monte Beach, 4-10 March 1975- ------- -15 2. Beach Profile and Instrument Location at Carmel River Beach, 29 May 1975- --------- -16 3. Schematic of Typical Instrument Tower with Sensors ------------------- -20 4. Typical Analog Record of Waves and Horizontal Velocities beneath the Waves from Del Monte Beach- ---------------------- -25 5. Typical Analog Record of Waves and Horizontal Velocities beneath the Waves from Carmel River Beach- ------------------- -26 6. Frequency Distribution of the Sea Surface Elevation at Wave Gauge #2 on 4 March 1975 - - - - -29 7. Frequency Distribution of the Horizontal Velocity at Flow Meter #2 on 4 March 1975- - - - - -30 8. Power, Coherence and Phase Spectra for Flow Meter #1 and Wave Gauge #1 on 4 March 1975- ----------------- -35 9. Power, Coherence and Phase Spectra for Flow Meter #1 and Wave Gauge #1 on 29 May 1975 ------------------ -36 10. Examples of Theoretical and Measured Phases from Del Monte Beach (top) and Carmel River Beach- ------------ -- -44 I. INTRODUCTION Wave theories developed for deep water waves can be applied with some degree of certainty, and can be tested in laboratory and field situations. These theories general ly hold until such time that the shoaling process begins, and with somewhat lesser accuracy, throughout the shoaling process up to the point of near breaking. At the breaker point, however, there is a transition from ordered to turbulent motion and the description of wave kinematics becomes more difficult. The most forthright approach to the problem of describing the kinematics of surf zone breaking waves is through direct measurements. Advances in instrument design have led to simple, sturdy, measuring devices with rapid response time which can sense the small scale as well as the large scale features. The' study of the kinematics of breaking waves in the surf zone has progressed slowly due both to the problems encountered in making direct field measurements and the difficulty in modeling the surf zone in the laboratory. Breaking waves were measured in the laboratory in early work by Iverson (1953) using photographic techniques. The length of the channel limited the wave type to plunging and surging breakers. Adeyemo (1970) made similar labora- tory measurements using hydrogen bubble and photographic methods. Gaughan and Komar (1975) applied the theory of 10 wave propagation in water of gradually varying depth, as developed by Biesel, to determine the dependence of breaker type on the beach slope tangent and the deep water wave steepness . Inman and Nasu (1956) made field measurements of water particle velocity by measuring the drag force under the wave in order to infer particle velocities. Miller and Ziegler (1964) used both acoustic and electromagnetic current meters to determine the particle motion in the surf zone. Walker (1969) made studies using propeller- type flow meters. Wood (1973) measured waves and currents in the surf zone using movies of dye movement and capacitance wave gauges. Fiihrboter and Biisching (1974) utilized a two- component current meter and two pressure- type wave meters to measure simultaneous orbital velocities and water levels Thornton (1968), Steer (1972), Thornton and Richardson (1973) and Bub (1974) used pressure meters, capacitance wave gauges and electromagnetic current meters to measure surface profiles and particle velocities; the work presented here is an extension of these studies. The objective of this research is to study the kine- matics of water particles in breaking waves within the surf zone. Simultaneous measurements were made of the instantaneous wave profile and the horizontal water particle velocities at fixed locations in the surf zone. Estimates of the probability density functions and power 11 spectra of the wave heights and particle velocities are made. The applicability of using linear wave theory as a spectral transfer function in computing velocity spectral components from the power spectrum of the waves is measured. The computed velocity spectra are compared with actually measured velocity spectra. Theoretical phase spectra between wave gauges are calculated and compared to measured phase spectra. 12 II. MEASUREMENTS A. EXPERIMENT SITES The experimental sites are in the vicinity of Monterey, California. The beaches here are some of the first inten- sively studied to gain an understanding of amphibious war- fare techniques and are described by Bascom (1964) . It was desired to measure the various types of breaking waves in the study including spilling, plunging and surging breakers. The manner in which waves break depends very much on the characteristics of the beach and near-shore bottom slope (Table I) . Plunging and spilling breakers occur most frequently at the Del Monte Beach site within Monterey Bay. The waves at this location are generally severely direc- tionally filtered and refracted by the geometry of the bay and impinge almost perpendicular to the shore with a resulting simplification in the wave description. A second experimental site was Carmel River Beach, five miles to the south, where the beach is very steep and very often has surging type breakers. Again here, the beach is within an embayment and the waves impinge almost perpendicular to the shore. A typical beach profile and instrument loca- tion for Del Monte Beach is shown in Figure 1. Del Monte Beach profiles for 4-10 March 1975 are given in Appendix A. Figure 2 shows the beach profile and instrument location at Carmel River Beach on 29 May 1975. 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A Brush 8-channel strip chart recorder was used to monitor the instrumentation performance during recording and as a means to select the appropriate data sections to be analyzed. 21 III. ANALYSIS OF DATA Record lengths of 30 minutes from each data set were analyzed. The details of the analysis system including calibration factors are given in Appendices D and E. A mean value was calculated for all data sets and the data was linearly detrended to exclude the rise and fall of the tide. The variance, standard deviation and average period were calculated. The average period was determined by calculating the time between zero upcrossings. However, perturbations caused by secondary gravity waves "riding" the primary gravity waves, capillary waves and minor instru- ment noises increased zero upcrossing occurrences and caused the calculated average periods to be lower than visually observed. Probability density functions for each data set were calculated and graphically compared with Gaussian and Gram-Charlier distributions. For each signal an auto-covariance function was calcu- lated and smoothed with a Parzen window. A Fourier trans- form was then applied to the smoothed auto-covariance function and the power spectrum determined. A cross- covariance function between data sets was computed, smoothed with a Parzen window, a Fourier transform applied, and the cross-spectrum computed. The coherence and phase were then calculated. To compare waves and flow velocity, the wave profile spectrum was converted to a theoretical velocity 22 spectrum for comparison with the measured flow velocity spectrum. The two power spectra, the coherence and the phase were then plotted. In comparing two wave profiles, theoretical celerities were calculated using linear theory and its shallow water approximation which were then con- verted to phases and plotted for comparison with the measured phase difference. Aliasing testing was performed on all spectra. Nyquist frequencies of 1.25 and 0.98 Hz were determined optimal for Del Monte Beach and Carmel River Beach data sets, respectively. The frequency bands of 0.0 to 1.25 and 0.98 Hz include the region of gravity waves (0.033 to 1.0 Hz) . The maximum lag time in calculating the covariance functions was taken as five percent of the record giving a spectral bandwidth resolution of 0.00556 Hz and results in 40 degrees of freedom for each spectral estimate. The 80 percent confidence limits for 40 degrees of freedom using a chi-square distribution are found to be between 0.73 and 1.30 of the measured power spectral estimates. 23 IV. RESULTS A. QUALITATIVE DESCRIPTION A number of similarities of wave form can be observed for various types of breakers occurring on different beaches around the world. Figures 4 and 5 are typical analog records of waves and horizontal velocities beneath the waves obtained from Del Monte and Carmel River Beaches, respectively. In general, there is a quick drawdown of water just before the breaking wave arrives, followed by a steep, almost vertical leading edge, and a sloping pro- file toward the trailing edge. On the trailing edge of the wave secondary waves are often noted. These are harmonics of the primary wave frequency and are indicative of very non- linear waves. At Del Monte Beach the waves break as plunging and spilling breakers giving a generally saw-toothed shape; the plunging or spilling occurs rapidly at the crest and moves down the wave. The surging breakers occurring at Carmel River Beach rise and fall off more gradually. B. PROBABILITY DENSITY FUNCTIONS 1. 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This is to be expected since the flow meters do not experience the surface irregularities to the same degree as do the wave gauges. In seven of thirteen fit parameter calculations for wave components, the Gaussian distribution provided the better fit. This is a result of using a truncated form of the Gram-Charlier in calculating the probability density function, Pfr , given by m, m. m_ pGCco - pGcmi.o ♦ ^Hj ♦2|h4 ♦ jf-Hg ♦ ] c where P„ = Gaussian probability density function, m~ = skewness of the data record, m. = the kurtosis minus 3, H = Hermite polynomials of degree n. Higher order moments could be included in equation (1) to possibly improve the fit, but would necessitate analyzing longer record lengths in order to maintain confidence. Stationary problems would be encountered in the longer records, thus producing other errors. Hence, the poor fit is a deficiency of the Gram-Charlier distribution in surf zone applications. 2 . Skewness and Kurtosis The physical geometry of the waves, which characteristically have narrow, steep crests and wide, shallow troughs in the surf zone produce a positive 31 skewness, indicating a greater amount of time below, rather than above, the mean water level. This is true except at Wave Gauge #3 on 6 March. (See Table II and Appendix F.) Data from this wave gauge was analyzed on this date only, because visual observations noted that most waves were actually breaking on this gauge, rather than on Wave Gauges #1 and #2 as was otherwise the case. It should follow that the wave induced particle velocities would have probability density functions similar to the waves. However, of the eleven horizontal velocity data records, four did not have positive skewness. These deviations occurred at Flow Meter #1 on 4 March and 29 May and Flow Meter #2 on 5 and 6 March. Reflected waves were originally thought to be a feasible explanation for the negative skewness. If this were so, Flow Meter #2 on A March and 29 May would also expect to have negative values. This was not the case. The reason for the large skewness and kurtosis values for Wave Gauge #1 on 5, 6 and 8 March is not known. Similar anomalous results have been found in deeper water outside the surf zone by Thornton and Krapohl (1974) . Kurtosis indicates the peakedness of each parameter. Just prior to breaking the waves achieve the greatest degree of peakedness and should have the highest kurtosis value at this time. Visual observations noted that most waves were breaking at Wave Gauge HI on 4 March and 29 May, at Wave Gauge #2 on 5 March and on 32 Wave Gauge #3 on 6 March. It should be expected that the kurtosis values would be greatest at these given gauges for the respective dates. With the exception of 29 May, this did not turn out to be true. The wide surf zone caused by varying breakers heights on 8 and 9 March make evaluation of kurtosis values difficult on these dates. C. COMPARISONS OF THEORETICAL AND MEASURED POWER SPECTRA The theoretical velocity spectra generated from the wave spectra and the measured velocity spectra were examined in order to determine the applicability of linear (Airy) wave theory in the breaker zone. Thornton and Richardson (1974) measured waves with maximum heights of 1 m and showed that the application of a linear theory transfer function to the surface profile spectrum resulted in a calculation of the horizontal water particle velocity that underpredicted measured wave-induced horizontal velocity spectral components by about 50 percent. The coherence values between waves and horizontal velocity were high, ranging above 0.75 for an approximate frequency range of 0.075 to 0.60 Hz. The phase angle computations showed the calculated velocity components leading the measured velocity components by an average of 20 degrees. Bub (1974) took measurements in mild surf resulting in theoretical horizontal velocity values 13 percent lower than measured velocity. Coherence values varied from 0.5 to nearly 1.0 over a frequency range of 0.10 to 0.65 Hz An approximate zero degree phase was observed out to 1.0 Hz 33 For this study wave heights of approximately 1 m were typically observed at Del Monte Beach and up to 2 m were recorded at Carmel River Beach. The theoretical horizontal velocity spectra were computed from the wave spectra using the linear theory transfer function S m - \° cosh(k(h+z))-.2 rr, r?. ucl±J - l sinh(kh) ] Vf) (2) where S (f) = calculated horizontal velocity power uc } l 2 spectrum (m /sec) , 2 S (f) = measured wave height power spectrum (m -sec), a = angular frequency = 2TTf (sec ) , k = wave number (m ) , h = mean water depth (m) , z = depth of flow meter below the mean water depth (m) . Figures 8 and 9 are graphs of the measured and calculated spectral velocity values at Tower #1 for 4 March and 29 May. Additional spectra are contained in Appendix G. On the average at Del Monte Beach, the calculated hori- zontal velocity values underestimated the measured veloci- ties by 7 9%. The same theoretical calculations for the data collected at Carmel River Beach underestimated the measured values by 86%. A possible explanation for the low calculated values is that the wave profiles change during the shoaling and 34 O CO CI. H CT z: UJ CD cr. cr h- UJ cn o MEASURED HORIZONTAL VELOCITY THEORETICAL HORIZONTAL VELOCITY 0.0 0.2 0-4 l 0.5 0.3 ~1 1.0 UJ o 21 UJ cr UJ n CD UJ CO CE 31 Q_ FREQUENCY (HZ) FIGURE 8. Power, Coherence and Phase Spectra for Flow Meter #1 and Wave Gauge #1 on 4 March 197 5 35 Q O". >- CD LU CD <*> CC cr h- LJ LU Cl CO CD O MEASURED HORIZONTAL VELOCITY THEORETICAL HORIZONTAL VELOCITY 0.2 0.4 0.6 0.3 1.0 LU O ~ZL LU CC LU H O o 1.0 LU CO cr. Q. FREQUENCY (HZ) FIGURE 9. Power, Coherence and Phase Spectra for Flow Meter #1 and Wave Gauge #1 on 2 9 May 19 75 36 breaking process, becoming more peaked at the crest, thus lessening the applicability of linear (Airy) theory which assumes sinusoidal wave shapes. Additionally, the wider separation between the measured and theoretical spectral values above 0 . 5 Hz may be explained by turbu- lence of the breaking waves in the higher frequency areas . D. EXAMINATION OF SPECTRAL HARMONICS An examination of the low frequency spectral components for each data set indicates that pronounced approximate harmonics are present in the data analyzed from Del Monte Beach. Such harmonics are not evident in the Carmel River Beach spectra. The frequencies and amplitudes of each analyzed spectral peak is listed in Table III. The fre- quency differences between spectral peaks and the ratios of the second, third and fourth peaks versus the primary peak are listed in Table IV. Spectral analysis shows that definite second, third and fourth harmonics are present for 4 March and 5 March with few exceptions . Evidence of second and third har- monics is present in the analyzed data from Wage Gauges #1 and #2 on 6, 8 and 9 March. An examination of the spectral amplitudes shows that there is no consistent relationship between various components either for a given data set or when comparing like components of different data sets. The maximum wave energies were concentrated at frequen- cies ranging from 0.061 Hz to 0.092 Hz corresponding to 3 7 to cm to rt rt a m rt m t— I LO r— I OO o cm o cm O CD o o o o o o LO tO LO «=* HtOHO CD rH CD CM O O O CD CD CD CD CD t— I i— I MD MD LO W^J- ^ cm cm cm cm l^- t^ rH rH LO LO LO LO CM CM CM CM rt CO OO CM to LO H H O LO CD LO o o o o CD CD CD CD rH LO rt LO CM ^J- CM CTi O CM CD rH O CD CD CD CD CD O O tO "3" OO CM f-» O'tOHO O CM O i— I O O CD CD O O CD O CD CD O tOOtOW i— I tO rH CM OHON o o o o CD CD CD O CT> CM i— i r-» O i—( o o CD O CT) t— I CM CM o to o o o o O CTl co r» CM cm r^~ t^ md oo md O CM rt t— I O O CM Or)- O o o o o o CD O CD CD O O CD rt LO CTi CTl OO CTl CD LO CD MD CTi CTi CTi CD rH rH rH CM rt rH rt l^-- rt tO LO tO LO to CM CM CM CM CM rH rH tO rH LO LO rt LO CM CM CM CM NNOOO rH OO CTl N N H vO ONN CM CM tO CM CM CM CM HK1LOO rH OO O rt O rH O rH CD CD CD CD MD LO LO rt ONON CD O O O o o o o rH O CM tO rH CD vO CD CM O CD O O CD O CD O O O O tO rt tO rt O tO O LO O O O CD CD O O CD rt rt CM O 00 tO rH O tO CD LO HNO O CD CD O O rH CD O O O O O O O OO co to to CM CM CM CM rt CO rt OO tO CM tO CM MD rH MD MD f- lO LO LO LO MD CO CO CTl CTl MD MD LO LO NNHN CT) CTi O vO rH rH CM rH CM CO tO CM CM CM rH rH CM MD LO MD OO vO CM tO rH CD rt O CTi O rH CD CD CTi rt MD vO HNMtO CD rt CD MD O O O O rt rt OO rt 00 CD CD rH CM rH CD tO CD CM O CD O O O CD r~- rt LO rH rH tO CM CM CD to CD rt oooo MD r-- CTi rH rH OO O t^ O rH CD rH O O O O MD CTi LO rH tO CD O tO O o o o ». -H rH rH MD MD MD MD CD CD CD O r>- r-- t^ r>~ MD MD MD MD O O CD CD CO OO CO CO CO r-- r-~ t>- t>- r- o o o o o rt rt rt LO 0OO0O0 N O O O O CM CM CM CM CTi CTi CTi CTl CD O CD O CO OO OO r^ r-» cm O O rH rH rH CM CM =*}= =tfc =tfc =tfc USOS «; Ph ;s u* rH rH CM CM =&= 34= ^fc: =tfe U s u s ;2 PL, IS PL, rH rH CM CM tO H^Z H^ H^fc H^t H^fc ^ LL, ^ PL, ^: rCj rC X U u u rH rH S-i CD rt rt rt +-> 2 *=H rt Q rt LO MD rH rH CM CM =8= =*te =tfe =tfe u S o s ^ PL, ^ LL, u u rt CO rH rH CM CM =te =tfe =tte =rte o s u S ^ PL, ^ PL, u rH rt CTi rH rH CM =tfc =t»: =tfc ID S O & PL, & CTi CM o -p CD #» X H rt CO TS r* C rt O CD U Ph Ph rH u CD rH *\ Ph rt X rt -M U 6 rt CD ■H Ph U i — \ LO Ph N ZZ 4-> rt II o X CO 3 •H 4-> cr rH Ph CD Ph •H rH e 5h <4H rt U CO II II 3 m rt w 38 co o w Q t— i Ph < < H CJ W a, CO Q CO w CJ W w PQ < o o OJ Cn O MD O CNI rg o LO CD ** O en OO Ol M Tj- en CO OD tO r^ tO O crj LO tO LO LO rH LO CTi tO r^ co ^t- lo tO CO tO LO LO Hen N M CO {Nt (Nl (N< o ooo rH O O O o o o o CD O O CD O rH O O rH Cx] r-| rH crj o crj IO ■H 4-1 rH rH O lONOOH M H NrO O O CD O Ol MOON N NLO K1 CD O CD CD rH LO 00 en 1 rH O I t^ CO en CT> LO CO (Nl vQ O r>- rH (Nl vO vO rH O LO r-» 0 CO (Nl rH <—t rH rH to rH rH rH (Nl rH rH rH O rH rH rH rH O (Nl (Nl (Ni CD O O LO CO {Nl CI (Nl rH O {N> rH (Nl (Nl rH LO C-J H H(MH vO vO vO lo OOOO N N H LO vO vO vO "3" CD CD O O Cn rH CO LO CD O CJ (Nl r- 1 CO vO r^ vO CO CO 0 CO rH r-«. to rO "3- 01 LO LO CTi rH en CD O vO vO vO r^ LO VO LO C- l>~ 0 r^ 0 vo 00 00 CO en CO CO CD O r~- 0 LO CD O O O O 0 0 0 0 rH O rH O O 0 0 0 0 OHH O rH rH rH t~- (Nl Ol OOOO r-- rH t-~. rH vO vO vO vO O O CD O co to co co en r~- r~- r-- r~- 00 00000 <=d" «5t LO ^h 00 CO t^ CO O O O CD o o en lo o o o in- rH rH rH O ■^t C LO ^j- lo en 000 t--. rH rH rH vO vO vO VO OOOO r- t>» r-~ r- OOOO 00 00 CO CO CO r-~ c-~ r^ r- r^ 00000 "5*- «3- "3" LO 00 CO 00 h- OOOO N (M N CM en en en en 0000 CO CO CO t-- [-- CN1 O O rH rH rH (Nl (Nl =tt= hh^ ^tfc ^tfc ^ LL, ^ LL, rH rH (Nl (Nl 3fc 3fc =tfc 3fc CJ 2 CJ S ^ LL, *: LL, rH rH (Nl CNJ to 3fc =tfc =tfc =tfc =tfc cj S 00 ^S •H +-> cr rH Ph 0 PL, •H u s r-< m a} u 00 n 11 m aj co 39 periods of 16.4 to 10.9 seconds, respectively, which agreed with visual observations. Additionally, prominent sub- harmonic peaks between 0.011 and 0.022 Hz are noted in the measured velocity spectra computed from Del Monte Beach data. This agrees with the work of Bub (1974), who observed a sub-harmonic at 0.011 Hz. Guzza and Davis (1974) state that in a theoretical analysis of edge waves excited by incoming waves, the prominent edge wave mode is the first sub-harmonic of the primary frequency. It is plausi- ble, therefore, that the observed low frequency sub -harmonic peaks were caused by edge waves in the surf zone. Such sub-harmonics are not observed in the measured velocity spectra generated from Carmel River Beach data. E. PHASE AND COHERENCE The curling crests of the unstable breaking waves tend to lead the horizontal particle velocities in the body of the waves as can be seen in the phase measurements between waves and velocities. (See Table V.) The maximum phase difference would be expected at breaking. Data from Tower #1 on 4 March indicates that the theoretical hori- zontal velocity leads the measured horizontal velocity by about 20 degrees. Since most waves observed were breaking on Tower #1, a possible explanation for the 30 degree phase angle at Tower 2 is that the waves reorganized between the towers and broke a second time at Tower #2. Since this author's observations concentrated primarily on the initial 40 CO a. h— I EC CO ^: o i—i H < W Oil W co < > •-a t/) CD o3 •H o3 CD P cq P X 10 4-> P P to CD P o3 Q U P oj =*fe u p u P oj 2 vO u p a oo 3fc P P CD CM o cm £ *= £ =**= O o H P H P CD cu > CD T3 o T3 o •H H •H H to V) P p P P 3 rt P 03 O to o to W p to p P 0 p a> a> ^ CD r^ r* Ctj r^ 03 rH CM to a3 0 OJ CD t-H 3fc 4fc =tfc o p 0 P * P x> p. X3 P P P J3 rQ P CD 0) o p p CD £ £ £ P CD P CD £ o o o CD I— 1 0) rH O E- H H W) rH bJ) r— 1 H P 03 P 03 +-> P P 03 6 o3 e P < < < _} CO -3 CO < ts) N EC N3 N EC N EC EC EC EC N EC EC o CO LO LO rH (Nl CM to LO to to C7> to rH to OO to CO LO o CD O CD CD o o o O CD o o p o p o p O P O P o p o p o p o p o p o p o o CM o o to o LO o CD CM o O o CD o LO o o rH o LO o LO rH 0 o rH rH CM rH CM rH CM rH CM rH CM rH =tfc =«= =tt= *: =tt= ^^fc ^fc =*fc * =tt= =te PL, PL, PL, s P-, PL, PL, PL, s PL, Ph P-, rH CM rH CM rH CM rH CM rH rH rH 3fc *= *= =*fc =«5 ^fc 4fc *: %: =tfc =*te CJ o ^ L9 o ^ C3 ^ o ^ u ^ C3 CJ ^ o p o3 CTi crj S CT> CM 41 breaking position, the possible reorganization and second breaking were not observed. The phase angles on 5 March are 5 and 20 degrees at Towers #1 and #2, respectively. This agrees with observations and indicates that the breakers started to curl at Tower #1 and broke at Tower #2. The zero degree phase relationships on 6 March show that the waves had not started to break until shoreward of Tower #2. This agrees with the observation of waves breaking on Tower #3. The phase angles on 8 and 9 March indicate that the smaller breakers began to curl at Tower #1 and were near breaking at Tower #2. Again, this agrees with observations. In contrast to the plunging- type breakers at Del Monte Beach, surging- spilling breakers were observed at Carmel River Beach. This was to be expected considering the beach slopes. A spilling breaker is characterized by turbulent water forming at the wave crest and eventually flowing down and covering the leading edge of the wave. Spilling begins at the crest when a small tongue of water moves faster than the wave form as a whole (Galvin, 1972). The measured phase angle at Carmel River Beach shows that the cascading crests led the horizontal particle velocities in the body of the wave by 10 degrees. The values of coherence show varying degrees of linearity in the transfer functions between different spectra. Co- herence between wave and horizontal velocity spectra at Tower #1 on 29 May is above 0.53 out to 0.S8 Hz (See 4 2 Figure 9), above 0.68 out to 0.53 Hz on 8 March and above 0.60 out to 0.41 Hz on 9 March. (See Appendix G.) Other coherence calculations show a fair to poor degree of linearity, which is consistent with the observed strong non- linearities . F. THEORETICAL AND CALCULATED PHASE SPECTRA A theoretical phase spectrum was calculated for com- parison with the measured wave spectrum from two wave gauges. The theoretical phase was computed from the relationship * = P ' (3) Ll where 4> = phase, a = angular frequency = 2irf (sec ) , x = distance between wave gauges (m) , 1 /? C, = celerity = (gh) ' (m/sec) , 2 g = gravity (m/sec ) , h = mean water depth (m) . Figure 10 shows typical examples of the theoretical and measured phases. The theoretical phases computed from Del Monte Beach data underpredict the actual phase relationships whereas an overprediction is noted in the Carmel River Beach phases. The author then determined what value of celerity (C?) would cause the theoretical and actual phases to agree out to the first zero crossing (where m •-J PQ 10 < •M H C 10 rH CD +J CD 6 u CD tsi U 1/5 Jh O +-> CD CO 5-i CD 4-> CD e CD £ ?H ■P CD ct} Q LO O vO to CD O to o CD CO LO r~- vO LO CTi tO CO CT> to LO «* "3" rH O O vO CO "5t LO ■«* >* LO o> *t f-. (NJ CO CO to CO 00 i>. vO CO CO -5* CO o «* to o u Sh CCJ to o vO f-l rH u u LO to o vO u OS vO to u •^, vO to o vO u h a} CO to LO 1— 1 t— 1 LO LO to i>- On) «* «>f ^J- vO o o vO l>- r^ o r>- vO rH LO VO vO o vO l>. CX> I— 1 rH i— 1 rH i—l rH <3- o vO LO to Oi LO CO (NJ CD •* rH to (VI vO o VO CO CO o o CT> VO LO CO LO O LO O O ** t"^ CO "!* *tf" rH en CO vO rH rH rH rH rH rH rH • • • • • • • • • • • • • • to to CO (Nl r^ LO (NJ • • • • • • • «* t-~ LO i>- to o t^- rH rH (NJ to (NJ to (NJ (NJ (NJ to (NJ (NJ CN) =8= =te =tfe *= =«= 3fc =tte C3 CD CD CD CD CD CD ^ •==- •g •=- S: & rH rH rH (NJ rH rH rH =tt= =tt= =tt= =8= 3fc =tt= =*fc CD o CD CD CD CD CD U rH 2 (NJ 46 V. CONCLUSIONS Breaking waves in the surf zone can be characterized as being highly non-linear. Qualitative observations of wave and velocity profiles show secondary waves indicative of non-linear waves. The probability density functions calculated for the flow velocity records compared better with the Gram-Charlier distribution than with the Gaussian distribution when tested using the chi-square goodness-of- fit test. However, seven of the thirteen wave records more closely approximated the Gaussian distribution. This is a result of the truncated form of the Gram-Charlier probability density function used in calculations, and points out the importance of including higher order moments in describing wave phenomena in very shallow water. The values of the horizontal power spectral components calculated from wave spectra using linear theory indicate a qualitative, but not a quantitative, relationship. Com- paring the results of Bub (1974) , Thornton and Richardson (1974) and this author, it is evident that increased breaker heights result in larger estimation errors. Additional investigation is warranted to determine if specific breaker heights and/or types result in specific underestimation values. If so, linear theory may be applied to achieve a reasonably accurate determination of flow velocities. The coherence values between waves and 47 horizontal velocity decrease as breaker height increases, indicating that the wave motion becomes more non-linear and more turbulent under this condition. The theoretical phase calculations based on the linear theory approximation for celerity did not accurately predict the observed phases in this research. The values of a, determined so as to force the theoretical phases to agree with the observed values at the first zero crossing, show no consistency. Use of electromagnetic flow meters and the improved capacitance wave gauges and instrument towers permits the gathering of accurate continuous data with sturdy and reliable equipment. 48 APPENDIX A BEACH PROFILES AT DEL MONTE BEACH 4-10 MARCH 1975 o o tn + o o + o o -J -3 2 sHiuaraiNHD ni moilvahth 49 0.0 APPENDIX B CAPACITANCE WAVE GAUGE CALIBRATIONS x r wave gauge 1 a z wave gauge 2 o : wave gauge 3 2.5 0.5 1.0 1.5 DEPTH OF IMMERSION (METERS) 2.0 2.5 50 0.40-r APPENDIX C FLOW METER CALIBRATIONS 0.00 .00 0.25 0.50 0.75 1.00 1.25 DEPTH OF IMMERSION (METERS) 51 1.50 1.50 APPENDIX D ANALYSIS DETAILS AND FLOW CHART A complete flow chart of data acquisition and analysis is presented on the following page. The parameters used during spectral analysis were chosen to balance the longest record that the computer could reasonably analyze with the best resolution over the frequency range of interest. Additionally, consideration was given to the computer run time . In order to determine the optimum Nyquist frequency for data analysis the following equations were used: At = (VDT) (NSKIP) (NCHAN) (1) and fN = TfJtJ (2) where At = sampling interval, VDT = sampling rate, NSKIP = number of samples skipped in initial data array, NCHAN = number of channels, f = Nyquist frequency. By varying the parameters of equation (1) various Nyquist frequencies were obtained and used during preliminary IBM 360 computer analysis. 52 WAVE HEIGHTS FLOW VELOCITIES DIGITAL RECORDER 1. "READ DATA I CHOOSE TWO DATA SETS X READ ALTERNATE POINTS" T CONVERT VOLTAGES TO WAVE HEIGHTS AND VETOCITIES I DETREND BOTH DATA SETS CALCULATE MEAN, SLOPE AND INTERCEPT I "COMPUTE VARIANCE, STANDARD DEVIATION AND AVERAGE PERIOD OF BOTH DATA SETS CALCULATE PDF AND COMPARE WTTTT GAUSSIAN AND GRAM-CHARLIER CALCULATE AUTO-COVARIANCE SMOOTH WITH P70TZEN WINDOW "COMPUTE POWER SPECTRUM CALCULATE CROSS- CO VART7UTCE SMOOTH WITH CALCULATE CROSS -SPECTRUM IN FORM OF CO- SPECTRUM AND QUAD- SPECTRUM I OPTTOTl 1. CONVERT HEIGHT SPECTRUM TO VELOCITY SPECTRUM 2. CALCULATE THEORETICAL PHASES PLOT GRAPHS OF: 1. GAUSSIAN AND GRAM-CHARLIER CURVES VS. PDFs 2. TWO POWER SPECTRA, PHASE AND COHERENCE VS. FREQUENCY 3. INDIVIDUAL LOG- LOG PLOTS OF SPECTRA VS. FREQUENCY 4. TWO THEORETICAL PHASE PLOTS VS. FREQUENCY (OPTION 2) 53 APPENDIX E CALIBRATION FACTORS GALA* CALM** (meters) (meters/volt) Date Instrument 4 March WG #1 and WG #2 5 March WG #3 6 March WG #1 WG #2 WG #3 8 March WG #1 WG #2 WG #3 9 March WG #1 WG #2 WG #3 29 May WG #1 WG #2 All dates FM #1 FM #2 -1.312 1.23 -1.870 4.19 -1.800 1.14 -1.422 1.23 -1.620 4.19 -1.800 1.14 -1.332 1.23 -1.570 4.19 -1.810 1.14 -1.492 1.23 -1.510 4.19 -1.830 1.14 -0.028 0.69 -0.370 0.66 0.0 3.05 0.0 3.21 * Calibration Additive Factor ** Calibration Multiplicative Factor 54 APPENDIX F PROBABILITY DENSITY FUNCTIONS VS. GRAM-CHARLIER FREQUENCY DISTRIBUTIONS =*te LO t^- w CTt o r-l E < E CO U Pi > gj < vOC 4- 4- 4- 55 $ I =«te lo t^ & Ch m 1—1 H W IE S CJ pc5 Is < o S hJ Uh •=* 56 ■J 6 •V -V * LO r-. w en CD i-i S < ac o CJ Pi 25 < ^ LO + 4- 4- 4- :-oo fCO ?x TC^i LTC 4" 57 .__! rH 4fc LO t>- & cn w r-! E-h W ac 2 u OS o g! ►J P-. LO 58 m en U < DC w < > S < C-J =te LO f«. Pi cr> W i— ( H W X S U Pi o §> H-3 ^ i-n .7 60 =ifc LO t^» W cr, C3 r-i t3 < K LD u (^ > gj W i-H E- W X S U Pi ^ tf u ^ 64 65 " to =tt= LO r- w CTi o rH D < 5C o u oi pq 3 ->■ -13 CO + +: + + + + + '.€0 ZCG ccc + + 66 =4fc LO t~- pi cr> w i— i H W ffi §■ U PS 5= o §j H-5 Ph oo 67 CM =tfc LO r^ w cr> e? i— i :=> < K u U Pi > < on -t- CV) * LO t^ ctf CD w i— 1 H fi] u: p? CJ Pt o 3 ►-4 Ph CO 70 *= LO r^ e* o> w i— i H W ac S CJ pi o gj vJ Ph CT> 71 + (V) 4b LO r^- w Ol u rH tD < ac CJ3 u Pi > gj < 72 (Nl =Jfc LO r-- cm cr> w t— i H W X S u C* •2 o J •-I tin cr> 73 LO w r^ o cr> ZD rH < o >-> w g> > < en •s CM 74 LO Pi r-- w cr> H i-H W S >H ^ o H-3 CD tL, CVJ 75 1/3 CO O 4 CV] =tfc LO W t^- CD cr> E> i-H < O >< w g> > < CT) 12 (VI APPENDIX G POWER, COHERENCE AND PHASE SPECTRA FLOW METER #2 - WAVE GAUGE #2 4 MARCH 1975 VVa 0.4 0.6 0.3 1.0 en Q. J.O FRLQULNCV [HZ) 77 WAVE GAUGE #1 - WAVE GAUGE #2 4 MARCH 19 7 5 1-0 RL'QUENCV (HZ) 78 FLOW METER #1 - WAVE GAUGE #1 5 MARCH 1975 FREQUENCY (Hi 79 FLOW METER #2 - WAVE GAUGE #2 5 MARCH 19 75 UJ O z: UJ C£ UJ X D O 8 r~ FRElQUENCV (H2) 80 f-J on CL u: on z: Lu CD a; cc h- LU Q_ on o WAVE G - WAVE GAUGE #2 0.2 1-C FRtQUENCY ( HZ 81 FLOW METER #1 - WAVE GAUGE #1 6 MARCH 19 7 5 LU O UJ cr x: o l-Q FREQUENCE fH2) 82 CD cr. > h- to LU O 7 _J cr ex h- o UJ CI- ,0 if) ' o o FLOW METER #2 - WAVE GAUGE #2 6 MARCH 19 75 o.o FREQUENCY (HZ) 83 Q CD CI Q. cr WAVE GAUGE #1 - WAVE GAUGE #2 6 MARCH 1975 >- CD "i cr. cr h- CL on D.S o.o 0.2 0.4 0.6 1.0 z: UJ GC UJ JZ D O 0.0 0-2 '^VVVsA 0.4 0.6 0-9 1.0 1.0 FRLQULNGV (HZ 84 O ' o 0"> • _J Q. 27. cr. i > i— i — i CO z: UJ CD f _J CT a: i— o UJ 1— »— o (P ' CD D WAVE GAUGE #2 - WAVE GAUGE #3 6 MARCH 1975 CO 0.2 0.4 0.6 0.3 1.0 FREQUENCY (HZ) CD * « __J Q. a. FLOW METER #1 - WAVE GAUGE #1 8 MARCH 19 7 5 > en 21 a: GC I— o UJ a. en D 0-Q 0-2 Q.a 0-5 Q.3 1.0 £. r- FRF:QUENCV MZ) 86 FLOW METER #2 - WAVE GAUGE #2 8 MARCH 19 7 5 § r- FREQUENCV (HZ) 87 L.) C7) CJ. a. y in 'ZL LU O cr. LU in ID O WAVE GAUGE #1 - WAVE GAUGE #2 8 MARCH 1975 cr. a. FRr.QUE.NGV (h 88 G- CJ- FLOW METER #1 - WAVE GAUGE #1 9 MARCH 1975 > en UJ ex cc h- o UJ en T 0-2 r\ r. G a i-Q S UJ en o en FRL'QUENCV fH2) 39 FLOW METER #2 - WAVE GAUGE #2 9 MARCH 1975 frf:quencv mz 90 WAVE GAUGE #1 - WAVE GAUGE #2 9 MARCH 1975 FRe.'QULMCV (H2) 91 WAVE GAUGE #1 - WAVE GAUGE #2 29 MAY 1975 1.0 FREQUENCV f.H2) 92 BIBLIOGRAPHY Adeyemo, M. D., "Velocity Fields in the Wave Breaker Zone," Proceedings of the Twelfth Conference on Coastal Engineering, v. 1, p. 435-460, ASCE , 1970. Bascom, W. , Waves and Beaches, Doubleday and Co., Inc., 1964. Bub, F. L., Surf Zone Wave Kinematics, M. S. Thesis, Naval Postgraduate School, Monterey, California, 1974. Fiihrboter, A., and Biisching, F. , "Wave Measuring Instrumenta- tion for Field Investigations on Breakers," Proceedings of the International Symposium on Ocean Wave Measurement and Analysis, ASCE, 1974. Galvin, C. J., "Wave Breaking in Shallow Water," Waves on Beaches and Resulting Sediment Transport, p. -413-456 , Academic Press , Inc . , 1972 . Gaughan, M. K. , and Komar , P. D., "The Theory of Wave Propagation in Water of Gradually Varying Depth and the Prediction of Breaker Type and Height," Journal of Geophysical Research, v. 80, p. 2991-2996, 20 July 1975. Inman, D. L., and Nasu, N., Orbital Velocity Associated with Wave Action near the Breaker Zone, Technical Memorandum Nc~ 79 , U. S. Army Corps of Engineers, Beach Erosion Board, March 195 6. Iverson, H. W. , "Waves and Breakers in Shoaling Water," Proceedings of the Third Conference on Coastal Engineering, Council on Wave Research, p. 1-12, ASCE, 1952. McGoldrick, L. F., A System for the Generation and Measure- ment of Capillary-Gravity Waves, Technical Report No. 3 , University of Chicago, Department of the Geophysical Sciences, August 1969. Miller, R. L., and Ziegler, J. M. , "The Internal Velocity Field in Breaking Waves," Proceedings of the Ninth Conference on Coastal Engineering, p~! 103-122, ASCE, 1964. ~~ Steer, R. , Kinematics of Water Particle Motion within the Surf Zone, M. S. Thesis, Naval Postgraduate School, Monterey, California, 1972. 93 Thornton, E. B., "A Field Investigation of Sand Transport in the Surf Zone," Proceedings of the Eleventh Conference on Coastal Engineering, p. 335-351, ASCE~ 1968 . Thornton, E. B., and Krapohl , R. F., "Water Particle Velocities Measured under Ocean Waves," Journal of Geophysical Research, v. 79, p. 847-852, 20 February 1974 Thornton, E. B., and Richardson, D. P., The Kinematics of Water Particle Velocities of Breaking Waves within th~e Surf Zone, Technical Report NPS- 58TM74011A, Naval Postgraduate School, Monterey, California, January 1974. Walker, J. R. , Estimation of Ocean Wave- Induced Particle Velocities from the Time History of a Bottom Mounted Pressure Transducer, M. S~. Thesis, University of Hawaii, 1969. Wood, W. L., A Wave and Current Investigation in the Nearshore Zone , Department of Natural Science, Michigan State University, E. Lansing, 1973. 94 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center 2 Cameron Station Alexandria, Virginia 22314 2. Library, Code 0212 2 Naval Postgraduate School Monterey, California 93940 3. Department Chairman, Code 58 3 Department of Oceanography Naval Postgraduate School Monterey, California 93940 4. Assoc Professor E. B. Thornton, Code 58 Tm 5 Department of Oceanography Naval Postgraduate School Monterey, California 93940 5. LCDR James J. Galvin, USN 3 7 7 Highland Road Somerville, Massachusetts 02144 6. Oceanographer of the Navy 1 Hoffman II 200 Stovall Street Alexandria, Virginia 22332 7. Office of Naval Research 1 Code 480 Arlington, Virginia 22217 8. Dr. Robert E. Stevenson 1 Scientific Liaison Office, ONR Scripps Institution of Oceanography La Jolla, California 92037 9. Library, Code 3330 1 Naval Oceanographic Office Washington, D. C. 20373 10. SIO Library 1 University of California, San Diego P. 0. Box 2367 La Jolla, California 92037 95 11. Department of Oceanography Library- University of Washington Seattle, Washington 98105 12. Department of Oceanography Library Oregon State University Corvallis, Oregon 97331 13. Commanding Officer Fleet Numerical Weather Central Monterey, California 93940 14. Commanding Officer Environmental Prediction Research Facility Monterey, California 93940 15. Department of the Navy Commander Oceanographic System, Pacific Box 1390 FPO San Francisco 96610 16. Commander, Naval Weather Service Command Naval Weather Service Headquarters Washington Navy Yard Washington, D. C. 20390 17. Coastal Studies Institute Louisiana State University Baton Rouge, Louisiana 70803 18. School of Marine and Atmospheric Science University of Miami Miami, Florida 33149 19. Mr. John F. Borchardt Gilbert Associates, Inc. P. 0. Box 1498 Reading, Pennsylvania 19603 20. LT Frank L. Bub , USN Fleet Weather Central COMNAVMARIANIS, Box 2 FPO San Francisco 96630 96 Thes i s G142 CI Galvin • .'2 Kinematic* ** 2one break?n SUrf ysis, ' 2 Thesi s G142 Galvin c.l Kinematics of surf. zon<=> break inn waves: measurement and analysis. thesG142 Kinematics of surf zone breaking waves i mi i minimi m mum 3 2768 002 01018 3 DUDLEY KNOX LIBRARY