ChaSs ad a CoastEng. eS. CETA 80-5 WHO! | pe AENT F 7 ECTION Hi uietation of Wave Energy Spectra by Edward F. Thompson COASTAL ENGINEERING TECHNICAL AID NO. 80-5 JULY 1980 & NC gt Stains Approved for public release; distribution unlimited. U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER Kingman Building Fort Belvoir, Va. 22060 OoN70 Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22161 The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. MBL/WHOI OA DB 8301 Goer 2 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER CETA 80-5 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Coastal Engineering Technical Aid PERFORMING ORG. REPORT NUMBER INTERPRETATION OF WAVE ENERGY SPECTRA 6. 7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) Edward F. Thompson 10. PROGRAM ELEM AREA & WORK U A31463 NT, PROJECT, TASK IT NUMBERS 9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of the Army Coastal Engineering Research Center (CERRE-CO) Kingman Building, Fort Belvoir, Virginia 22060 E N 12. REPORT DATE July 1980 13. NUMBER OF PAGES 21 15. SECURITY CLASS. (of thi report) UNCLASSIFIED 11. CONTROLLING OFFICE NAME AND ADDRESS Department of the Army Coastal Engineering Research Center Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) DECL ASSIFICATION/ DOWNGRADING SCHEDULE 15a, 16. DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the abetract entered in Block 20, if different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse side if necessary and identify by block number) Spectral analysis Wave gages Wave hindcast Wave energy spectra Wave height Wave period ABSTRACT (Continue em reverse side if mecessary and identify by block number) Guidelines for interpreting nondirectional wave energy spectra are presented. A simple method is given for using the spectrum to estimate a significant height and period for each major wave train in most sea states. The method allows a more detailed and accurate description of ocean surface waves than that given by a single significant height and period, yet it eliminates much of the formidable detail of a full spectrum. An example problem illustrating application of the method is presented. Spectral analysis and display techniques, and the natural variation of spectra in space and time, are discussed. 20. FORM DD 1 JAN 73 1473 EDITION OF ? NOV 65 IS OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) eben aN mil ee en a peyote Prager eee ean cy 7 i ' ; | ' ame SA ey it Bees SAMY TAY Tar It he wet a Oe nt ey a ae lege byt wn iia ey pmo Adt . : feral Mens) MR ine Aaikk. Leto tq" Ors : . Bieta spo Pe inwtdt. s -oS ike ay] | F Sige is GORA HTT { i Tween Poy hats. AY. aon nge , : eh tarry ad 7O. Spine Ue seal lite 7 fy ; - Bil Aw Goltx 2h ina getzev tags} oh { feunen sit Gan , 29) ee Uo if 2 (Ferenc ih. ay i A gM mae i erm sedeoteaae Ti eeAL WG. i erties! wales ued op MAY Hiroe SRM ae i Ree ¢ ! I a ee, PREFACE This report presents guidelines for interpreting nondirectional wave energy spectra. The guidelines apply to spectra derived from both wave gage measure- ments and from numerical wave hindcasting models. A method is provided for using the spectrum to estimate a significant height and period for each major wave train in a sea state, except major wave trains with nearly the same period and different directions cannot be distinguished. The method has undergone limited testing and has been applied to 7 station-years of gage data, but fur- ther testing in well-documented field situations is needed. The guidelines and method are consistent with but are more practical and explicit than the material in the Shore Protection Manual (SPM). The work was done under the wave measure- ment program of the U.S. Army Coastal Engineering Research Center (CERC). This report was prepared by Edward F. Thompson, Hydraulic Engineer, under the general supervision of Dr. C.L. Vincent, Chief, Coastal Oceanography Branch. Helpful reviews by Dr. C.L. Vincent, Dr. D.L. Harris, P. Knutson, and P. Vitale are acknowledged. Dr. D. Esteva provided the data for Figure 6. Comments on this publication are invited. Approved for publication in accordance with Public Law 1966, 79th Congress, approved 31 July 1945, as supplemented by Public Law 1972, 88th Congress, approved 7 November 1963. Colonel, Corps of Engineers Commander and Director IV VI CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) SYMBOLS AND DEFINITIONS . INTRODUCTION. PRIMARY GOAL OF SPECTRAL ANALYSIS . PRACTICAL LIMITATIONS OF SPECTRAL ANALYSIS. 1. Calculation Procedures . 2. Display Formats. 3. Natural Variability. INTERPRETATION OF SPECTRA . 1. Highest Spectral Peak. one 2. Major Secondary Spectral Peaks . 3. Example Problem. INTERPRETATION OF SPECTRA FOR APPLICATIONS SENSITIVE TO SPECIFIC FREQUENCIES. SUMMARY . LITERATURE CITED. TABLE Spectrum for Huntington Beach, California FIGURES Spectrum for Wrightsville Beach, North Carolina . Energy spectra for wave record at 0400 e.s.t., 17 March 1968. Energy spectra for record composed of three superimposed sinusoidal waves . Five formats frequently used in displaying wave energy spectra. Wave energy spectra from bottom-mounted pressure gages at Channel Islands Harbor, California . A Wave energy spectra from pier-mounted continuous wire gages at the CERC Field Research Facility. Directional spectrum obtained in the Atlantic Ocean 68 miles east of Jacksonville, Florida. Spectrum for Huntington Beach, California . Page 18 13 15 18 CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: Multiply by inches 25.4 2.54 square inches 6.452 cubic inches 16.39 feet 30.48 0.3048 square feet 0.0929 cubic feet 0.0283 yards 0.9144 square yards 0.836 cubic yards 0.7646 miles 1.6093 square miles 259.0 knots 1.852 acres 0.4047 foot-pounds 1.3558 millibars TONS Texans ounces 28.35 pounds 453.6 0.4536 ton, long 1.0160 ton, short 0.9072 degrees (angle) 0.01745 Fahrenheit degrees 5/9 eo To obtain ee millimeters centimeters square centimeters cubic centimeters centimeters meters square meters cubic meters meters square meters cubic meters kilometers hectares kilometers per hour hectares newton meters kilograms per square centimeter grams grams kilograms metric tons metric tons radians Celsius degrees or Kelvins! TTT eee lTo obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: To obtain Kelvin (K) readings, use formula: (5/9) (F -32). K = (5/9) (F -32) + 273.15. Af, (Af); SYMBOLS AND DEFINITIONS wave component amplitude water depth sum of all E; spectral energy density values energy density for the highest spectral peak frequency of spectral component in hertz frequency corresponding to the highest spectral peak frequency corresponding to the ith highest spectral peak acceleration due to gravity significant wave height corresponding to the full spectrum significant wave height corresponding to the ith highest spectral peak indices representing the lower and upper bounds of the highest spectral peak indices representing the lower and upper bounds of the ith highest spectral peak total number of spectral frequency components spectral energy values energy contained in ith highest spectral peak time period corresponding to the highest spectral peak period corresponding to the ith highest spectral peak frequency bandwidth represented by each spectral energy density (in hertz) sea-surface elevation referenced to local mean water level phase of spectral component frequency of spectral component (in radians per second) INTERPRETATION OF WAVE ENERGY SPECTRA by Edward F. Thompson I INTRODUCTION The ocean usually has more than one independent train of waves propagating along its surface in U.S. coastal areas. The common practice of using a single significant height and period for a sea state can be misleading because no indi- cation is given to the existence or characteristics of other trains. On the other hand, an estimate of the wave energy spectrum provides more information than is generally used in coastal engineering. The spectrum can be reduced to estimates of significant wave height and period for all major wave trains pres- ent. A knowledge of these characteristics for major wave trains is often im- portant to coastal engineers. Spectra are becoming widely available through various field wave measurement programs, laboratory tests with programable wave generators, and numerical wave hindeasting projects. Because of the availability and applications of spectra, practicing coastal engineers should become familiar with spectra and their interpretation. II. PRIMARY GOAL OF SPECTRAL ANALYSIS A fundamental parameter for characterizing a wave field is some measure of the periodicity of the waves. For many years a significant period, which could be subjectively estimated in various ways, was used. However, the ocean surface often has waves characterized by several distinct periods occurring simultane- ously. A record of the variation of sea-surface elevation with time, commonly called a time series, frequently appears confusing and is difficult to interpret. Developments in computer technology and in mathematical analysis of time series have provided a practical approach to an objective, more comprehensive analysis of periodicity in wave records. The approach is to express the time series as a sum of periodic functions with different frequency, amplitude, and phase. The simplest functions to apply are the trigonometric sine and cosine functions. Thus, the time series of sea-surface deviations from the mean sur- face, n(t), is expressed by equation (3-11) in the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977) as my Gt) a aj Cos (wjt-o3) (1) J where a; = amplitude w7 = frequency in radians per second $j = phase t=) ‘tame Frequency is often expressed in terms of hertz units where one hertz is equal to one cycle per second. One hertz is also equivalent to 21 radians per second. If the symbol f; denotes frequency in hertz, then anf = Ws. The amplitudes, aj, computed for a time series, give an indication of im- portance of each frequency, f;. The sum of the squared amplitudes is related to the variance of sea-surface elevations in the original time series (eq. 3-12 in the SPM) and hence to the potential energy contained in the wavy sea surface. Because of this relationship, the distribution of squared amplitudes as a func- tion of frequency can be used to estimate the distribution of wave energy as a function of frequency. This distribution is called the energy spectrum and is often expressed as ; ag (Ej) (Af) 3 = $= S3 (2) where Ej = E(fj) is the energy density in jth component of energy spectrum, (Af)j the frequency bandwidth in hertz (difference between successive fj), and Sj = S(fj) energy in jth component of energy spectrum. Equation (2) is similar to equation (3-15) in the SPM. An energy spectrum computed from an ocean wave record is plotted in Fig- ure 1. Frequencies associated with large values of energy density (or large values of a%/[2(Af)-]; see eq. 2) represent dominant periodicities in the orig- inal time series. Frequencies associated with small values of a4/[2 (Af) 4] are ~ usually unimportant. It is common for ocean wave spectra to show two or more dominant periodicities (Fig. 1). 25,000 [__] Region used to Compute Spy 20,000 ZA Region used to Compute Spe = Py = ‘S 15,000 = e [=< @ © 10,000 ~ mo o (=< ty 5,000 0 y 0.2 0.3 I 0.5 fi i Frequency (Hz) Figure 1. Spectrum for Wrightsville Beach, North Carolina, 0700 e.s.t., 12 February 1972; Hg = 4.2 feet (128 centimeters), Af = 0.01074 hertz, and depth = 17.7 feet (5.4 meters). The primary goal of spectral analysts ts to objectively tdenttfy all tm- portant frequenctes in a wave record. Since wave period in seconds is equal to the reciprocal of frequency in hertz, important wave periods are also identified. III. PRACTICAL LIMITATIONS OF SPECTRAL ANALYSIS 1. Calculation Procedures. The appearance of a spectrum can be noticeably influenced by the methods used for calculation and display, neither of which is standardized in coastal and ocean engineering activities. Spectra are computed from both digital wave records and analog records for which an assortment of analog spectral analysis devices exists. A spectrum is computed from a digital wave record by either (a) computing the fourier trans- form of the autocovariance function of the record, or (b) computing the fourier transform of the record directly from the record using the fast fourier trans- form (FFT) approach (see Harris, 1974, for further detail). Both of these algorithms are used in conjunction with wave records subjected to various fil- ters and smoothing functions before analysis. Further, some form of smoothing, averaging, or summing is often applied to the computed spectral components. Different methods for calculating a spectrum will produce slightly different estimates of the spectrum when applied to a particular wave record. Major dif- ferences in the height of the spectral peak were shown by Wilson, Chakrabarti, and Snider (1974) when different approximations to the autocovariance function were used and different smoothing functions were applied to the spectrum of a field wave record (Fig. 2). Major differences in the height of the spectral peak and energy levels between peaks were noted by Harris (1974) when a time series composed of three superimposed sinusoidal waves was analyzed by several accepted methods (Fig. 3). Hindcast wave energy spectra are computed by estimating atmospheric input of energy to the sea surface and redistribution of energy within a spectrum. The estimates are based on a series of equations derived from the physics of air-sea interaction and waves. The quality and characteristics of hindcast spectra are a function of the model used to perform the calculations (Resio and Vincent, 1979) as well as the accuracy of the input wind field. Spectra obtained from either measurements or hindcasts are also limited by the resolution of the computation technique. The energy density and frequency at a spectral peak can be noticeably distorted if the frequency bandwidths, (Af);, are not small enough to permit clear definition of major peaks. 2. Display Formats. The appearance of a spectrum can be strongly affected by the display format used. Harris (1972) showed five often used formats for plotting spectra (Fig. 4). Each format alters the appearance of the spectrum. Format E shifts the relative magnitudes of spectral peaks enough that the second highest peak in formats A, B, C, and D becomes the highest peak in format E. A and C are the two most frequently used formats. Figure 2. Figure 3. 900 800 600 400 Energy Density (m*/Hz) 200 0 0.05 0.10 0.15 0.20 Frequency ( Hz ) Energy spectra for wave record at 0400 e.s.t., 17 March 1968, from a weather ship in the North Atlantic, computed with different approximations to the autocovariance func- tion and different spectral smoothing functions (after Wilson, Chakrabarti, and Snider, 1974). soos FET used with |.024-second time yf series smoothed by cosine bell ---- Avg. of spectral values computed with FFT for eight 128-second time series —-— Autocovariance procedure me Se Relative Energy Density (arbitrary units) 1/64 1/8 1/4 3/8 1/2 5/8 3/4 7/8 Frequency (Hz) Energy spectra for record composed of three superimposed sinusoidal waves. Simulated frequencies indicated by vertical lines from top of graph (after Harris 1974). 10 wD ~ 3.0 Go Ibs) 10,0 0 n > a Ss 20 2 ho = re C — wo ~— Cc a uw E> al). = = ® 10 Ops ® Cc > Cc uJ rs) w oO! Cc wo 0.0 Ss 0.0 oe 00 0.0 0.5 10 @ 001 0.1 1.0 0.01 0.1 1.0 Frequency (Hz) re Frequency (Hz) Frequency (Hz) 1Q0 0 3.0 a 100 a Se = 2.0 E = 1/0 a ® ® 1.0 Cc OM Ww 00 0.0 0.0 0.5 1.0 0.0 10.0 20.0 30.0 Frequency (Hz) Period (s) Figure 4. Five formats frequently used in displaying wave energy spectra. The actual spectrum is identical in all five graphs. The frequency band- width, (Af)3, is constant for all j so that energy values and energy density values differ by a constant factor (see eq. 2). The plots would look the same if energy were replaced by energy density, but the vertical scale would change (from Harris, 1972). 3. Natural Variability. Wave energy spectra are naturally variable simply because they are based on a finite length record of a wave field which varies in time and space. Spectra computed for successive records of a relatively stationary wave field are never identical and often differ noticeably. The magnitude of spectral variation in time is illustrated by spectra derived at 2-hour intervals from two pressure gages along the southern California coast (Fig. 5). The significant wave height is nearly constant in the figure. Spatial variation of the spectrum over short alongshore distances in shallow water is also shown in Figure 5. Each spectrum in the top row of the figure can be compared to the spectrum immediately below it to see variations between spec- tra from two gages 80 feet (24 meters) apart. In this figure, spatial variations are smaller than temporal variations. Spatial variations would be expected to be greater if the gages were farther apart or the water depth varied between measurement points. Variations between spectra from gages situated along a line perpendicular to shore are shown in Figure 6. The spatial variations are more prominent in this figure than in Figure 5. "98e3 oy. JO adUasLOUGnS 03 OeNp UOTIeNUS}3e DTWeUApoIpAY TOF peresuedwos useq eAey ei390edS *(Satej0M ['9) 1900} OZ FO yAdep 1907eM ur j2ede (SitoJOW pz) 190F 0B OtEM sade ‘7 O3e3 WOLF MOT WOZI0q ut §[ 03e3 worz. moz doz ut e4iqdedS -etTULOFITeD) ‘LOqIey SpueTSsT Touuey) 32 sese3 aanssoerd paqunouw-wo.0q worz eIjZIeds ASIOUS SAM (ZH) Aouanbas4 20 0 70 ZO ce} 44 ObSO 414 Obed ¥VS3 IYObIO 261 5ny 81 4392 =SH sog=91 AUObvE? 44 ObI2 44 Ov6l 46? $38 448°? $89 Wile Spl WB8e se9 We?’ sos aan] ¥vs® 1YOvIO 14 Ovle 14 Ov6! 26! 5ny 91 42°2=5H 49°2 G2 soe=% $89 sos Ge S Ol G2 S Ol G2 S Ol g2 S Ol G2 S Ol G2 s)=01 Ge (S$) pOluad 44OvZl 492 sZgl *g oan3sty Ol rsa 14 QvSl p26) ny 21 43 2°2 = SH sp2=4) 182 JY QbGI p26] Soy ZI 482 =5H d Spiel oO 02 Ov (49d) Ajisuag Absau3 12 13 Jon. 1979 1258 hr es.t. 13 Jon.1979 1058 hr e.s.t. Hg = 5.5 ft (168 cm) Hg = 5.5 ft (168cm) Hg= 4.6 ft (139 cm) Hs = 4.5 ft (136 cm) Energy Density Energy Density (0) 0.1 0.2 0.3 0.4 (0) 0.1 0.2 0.3 0.4 Frequency (Hz) Frequency (Hz) 13 Sept. 1978 0000 hr e.s.t. 11 Sept. 1978 1100 hr e.s.t. Hs= 3.5 ft (108 cm) Hg = 3.3 ft (100 cm) Hg = 2.2 ft (67cm) Energy Density Energy Density Hg = 2.2 ft (67 cm) Hg = 1.8 ft (54cm) Hg= 1.8 ft (54cm) (0) (om) 0.2 0.3 0.4 (0) 0.1 0.2 0.3 a4 Frequency (Hz) Frequency (Hz) Figure 6. Wave energy spectra from pier-mounted continuous wire gages at the CERC Field Research Facility (FRF) near Duck, North Carolina, showing variation along a line perpendicular to shore. Solid lines represent a gage at the seaward pier end (depth 29 feet or 8.8 meters); dashlines represent a gage 480 feet (146 meters) from the seaward pier end (depth 22 feet or 6.6 meters); dot-dash lines represent a gage 840 feet (256 meters) from the pier end (depth 17 feet or 5.1 meters). IV. INTERPRETATION OF SPECTRA Because of natural vartabtlity tn the spectrum and artifictal variability tnduced by analysts and display techniques, the spectrum should never be in- terpreted as an exact representation of energy density versus frequency for a wave fteld. However, certain major features of the spectrum are consistent and meaningful. 1. Highest Spectral Peak. a. Frequency and Period. Frequency corresponding to the highest spectral peak, f,, is usually a reliable measure of the dominant wave frequency; fp is shown in Figure 1. Period corresponding to the highest spectral peak, Tp, is equal to the reciprocal of fp and is usually a good estimate of the domi- nant wave period. b. Energy and Significant Wave Height. Energy contained in the highest peak, Spl> is defined as the total energy in the vicinity of the highest peak. Ko Lae Ej; (Af); (3) ; oe : : S where K, and Ky are indices representing the lower and upper bounds of the main peak. The upper and lower bounds sometimes represent a broad range of frequencies (see Fig. 1). Spl is relatively consistent, and is less influenced than the magnitude of the highest peak by data collection procedures, by analy- sis and summarization procedures, and by temporal and spatial variation. Some spectral analysis procedures are designed so that (Af); = Af isa constant for all j, which leads to Lp 3a Sol = (Af) ) By oe j=K +1 Significant wave height corresponding to highest spectral peak Hg), is an estimate of the significant height for the wave train represented by the highest spectral peak. It is computed by the relationship Hs1 = 4VSp1 (4) Energy density at the highest spectral peak, Emgxz, can be an indicator of how well focused the wave energy is in frequency. Although this parameter is variable, major differences in Emg, (on the order of 50 percent) between spectra analyzed by the same method can be meaningful. Emgqr, is shown in Eaooremle 2. Major Secondary Spectral Peaks. a. Identification of Major Secondary Peaks. Major secondary spectral peaks are often indicative of independent secondary wave trains characterized by dif- ferent heights, periods, and directions than the train represented by the main peak (examples are given in McClenan and Harris, 1975). Identification of major secondary spectral peaks involves some subjective judgment, but an objective 14 test for major secondary peaks has been developed and used at CERC. The test is applied to the difference in energy density between a spectral peak and the preceding and following spectral valleys. If that difference exceeds 3 percent of the total of all spectral energy density values, E, then the peak is con- sidered major. Details of the procedure with a computerized version are given in Thompson (1980). The procedure was applied to the spectrum in Figure 1, com- puted from an ocean wave record, and two major peaks were identified. The pro- cedure has been applied to 7 station-years of shallow-water spectra by Thompson (1980) to show that two-thirds of the ocean and gulf coast spectra have more than one major peak. If two independent trains have nearly the same frequency but different di- rections, they cannot be identified by the method presented in this report. However, an analogous method could be developed for use with directional wave spectra to identify all wave trains which have distinct frequencies or direc- tions. Directional spectra which give estimates of the distribution of wave energy density as a function of both frequency and direction are becoming in- creasingly available, mainly from spectral hindcasting models but also from improved measurement devices. A directional spectrum obtained by the Data Buoy Office of the National Oceanic and Atmospheric Administration (NOAA) from an experimental large discus buoy (Burdette, Steele, and Trampus, 1978) is shown in Figure 7. The figure indicates a concentration of low-frequency wave energy coming from the north-northeast and a concentration of high-frequency energy coming from the quadrant between east and south. Figure 7. Directional spectrum obtained in the Atlantic Ocean 68 miles (110 kilometers) east of Jacksonville, Florida, at 1900 e.s.t., 30 March 1977. Contour values are in units of 0.001 meter squared per hertz per degree. The radial coordinate is frequency in hertz; the azimuthal coordinate is direction from which energy is coming (from Burdette, Steele, and Trampus, 1978). Identification of major spectral peaks can also be a useful step toward approximating water velocities and accelerations from a complex record of sea- surface elevations, although procedures for doing this are not well established. Velocities and accelerations could be estimated for each peak separately, using equations in the SPM (Sec. 2.234); then they could be added together to give estimates of resultant velocity and acceleration. If direction estimates are available, the velocity and acceleration components should’ be added vectorially. An alternative approach is to use the nondirectional spectrum to estimate a velocity spectrum (Harris, 1972). Occasionally, spectra have major secondary peaks which do not represent independent wave trains. In particular, when waves are very steep or in very shallow water the spectrum will often have secondary peaks at frequencies which are integral multiples of the dominant frequency. The secondary peaks indicate that additional frequencies are needed to represent the nonsinusoidal wave pro- files (the peaks do not necessarily represent independent wave trains). When a major secondary peak appears at twice the dominant frequency, its source can usually be identified by referring to a plot of the time series from which the spectrum was computed. The time series is expected to show wave profiles which are clearly nonsinusoidal if the major secondary peak is nonindependent. Coarse guidelines for when nonindependent secondary peaks may occur are steepness: 2 5 0.008 (5) slp or : d relative depth: —» < 0.01 (6) B slp In cases where both steepness and relative depth approach the above guidelines, nonindependent peaks may also be evident. b. Frequency and Period. Frequency and period corresponding to major secondary spectral peaks, f,; and T,;, usually indicate secondary frequen- cies and periods at which major amounts of energy are present. fp2, corre- sponding to the second highest spectral peak, is shown in Figure 1. Equations (5) and (6) can be helpful in identifying cases where major secondary peaks do not represent independent energy concentrations. c. Energy and Significant Wave Height. Energy and significant wave height corresponding to major secondary spectral peaks, Spz, and Hg7z, indicate the relative importance of secondary peaks. Since major secondary peaks often rep- resent independent wave trains, Sp; and Hs; can be estimates of the energy and significant height of secondary trains. A method used at CERC to estimate Spi is to partition the spectrum at its lowest point between every pair of ad- jacent major peaks (Fig. 1). The energy between partitions is then totaled by t Spits aide aeolA)g (7) where Kiz is the index representing the lower bound of the peak, and Kz the index representing the upper bound of the peak. If (Af)j = Af = constant, then 16 Koz SoG = (Af) ) Ej (7a) j=Ki¢t1 Significant height is estimated from the total energy assigned to each peak by an equation similar to equation (4). 3. Example Problem. This problem illustrates a method for estimating peak frequencies, periods, and significant heights for a spectrum with two major peaks. GIVEN: Wave spectrum from Huntington Beach, California, for which the signifi- cant height is Hy = 5.7 feet (175 centimeters) (Fig. 8 and Table). Energy density is expressed as a percent of the sum of energy densities for all f; listed in Table. H, = 5.7 feet is based on 100 percent of the energy in the spectrum. FIND: Estimate a separate significant wave height and peak period for each wave train indicated by the spectrum. SOLUTION: To identify major spectral peaks, the difference in energy density between peak 2 and the valley between peaks 1 and 2 is estimated along the vertical axis in Figure 8 or from Table (about 4.5 percent). Since this is greater than 3 percent, peaks 1 and 2 are accepted as major peaks. Several other peaks appear at frequencies higher than 0.15 hertz, but there is no other combination of peak and valley for which the difference in energy den- sity exceeds 3 percent. Therefore, peaks 1 and 2 are the only major peaks. Frequencies for peaks 1 and 2 are estimated along horizontal axis or from Table: fp = fg = 0.081 Hz fp2 = £13 = 0.135 Hz Since fp2 is not integral multiple of fp, assume that peaks 1 and 2 repre- sent independent wave trains. The reciprocal of each peak frequency gives peak period. 1 Tn, = ~—=12.3 5s Pp fp 1 T = S p2 fo Compute total energy density, E, in the full spectrum by combining equations (3a) and (4) and rearranging to get Se yee ey a te J Na Af 52 _ (Sof 23% i saan Gel 0.01074 Hz 192 ft2/Hz 17 Peak 1 [__] Region used to Compute Spi VA Region used to Compute Sp2 Energy Density (pct) 0.3 Oa fp2 Frequency (Hz) Figure 8. Spectrum for Huntington Beach, California, 0640 e.s.t., 24 August 1972; Hg = 5.7 feet (175 centimeters), Af = 0.01074 hertz, and water depth = 26 feet (7.8 meters). Table. Spectrum for Huntington Beach, California, 0640 e.s.t., 24 August 1972.! .005 -016 .027 - 038 . 049 -059 .070 .081 .092 -102 .113 -124 7355 .145 156 - 167 .178 - 188 .199 -210 221 Zou 46 -242 47 .253 48 to 100 Sa) Leet (175 centimeters) ; AE = 26 27 28 29 30 31 32 nS3 34 35 36 37 38 39 40 41 42 43 44 45 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0. 0 0 0 0 0 0 0 0 0 FPAWFPUNAINWAWAODORFPODTOWRFRYENOOCSO CORPO OCOFRRPNSNUWAMNNK OUNF OC POR RF NF NWEEPNOWWAE NYP RWeENH PPWW 18 Estimate the energy in peak 1 using equation (3a), the tabulated energy den- sity (Table), and the above value of E to convert the sum of energy density from percent to foot squared. 5 So1 = (Af) Es P dn 3 (0.01074 Hz) (FsG- FEE) co £t2/Hz) Splice Similarly estimate the energy in peak 2 using equation (7a) 100 S Q & (Af) ) Ej , j=12 36.4 pet 2 (0.01074 Hz) ( 100 pet (192 £t“/Hz) On7S se" Estimate significant height for peak 1 using equation (4). Hy1 = 45,1 ayf1.31 ft? = 4.6 ft Note that H,, is 1.1 feet lower than H, based on the full spectrum. Simi- larly, significant height for peak 2 is estimated from equation (8) Hao = 3.5 ft Hs, Hs1 and Hg2 are related in this example by Hg =f Hg1 ae H30 V. INTERPRETATION OF SPECTRA FOR APPLICATIONS SENSITIVE TO SPECIFIC FREQUENCIES The interpretation of spectra in terms of wave trains is appropriate for most coastal engineering work. However, certain engineering applications, es- pecially applications in which resonance can occur, are highly sensitive to one frequency or a small range of frequencies. Estimates of how much energy can be expected at that frequency or range of frequencies are required. The estimates are obtained directly from the spectrum. Since the estimates are sensitive to data collection and analysis procedures, it is especially important that the procedures be optimum when such applications are intended cr anticipated. For frequency-sensitive applications, it is generally assumed that each frequency with nonzero energy represents an independent wave component, regard- less of whether it is a spectral peak. Thus, in a resonance problem sensitive to frequency fp, spectral energy at fp is treated as an independent wave. If fp corresponds to a major spectral peak, it would be appropriate (and conservative) to estimate the energy from equation (7). Ina floating break- water problem where energy at frequencies lower than some cutoff frequency, 19 fe, will be transmitted through the breakwater, the transmitted energy is esti- mated from an incident spectrum by summing all energy at frequencies lower than fe, regardless of where the spectral peaks are located. The assumption of independent spectral components seems adequate most of the time for such applications. However, for very steep waves or waves in very shal- low water the assumption is often incorrect. Cases have been documented in which all major spectral components at frequencies higher than the main peak frequency are closely tied to the main peak and do not represent independent energy con- centrations. The steepness and relative depth criteria in equations (5) and (6) can be used to indicate cases where the assumption of independent spectral com- ponents may be poor. VI. SUMMARY This report has presented guidelines for interpreting nondirectional wave energy spectra, including a simple method for identifying major spectral peaks and for estimating significant wave height, period, and energy for each major peak. Each major spectral peak is generally assumed to represent an independent wave train. Coarse guidelines are presented for identifying cases where major peaks do not represent independent trains. Spectral analysis and display tech- niques, and the natural variation of spectra in space and time, are discussed to show that the above method for interpreting spectra provides a relatively consistent description of general spectral characteristics. The method allows a more detailed and accurate description of ocean surface waves than that given by a single significant height and period, yet it eliminates much of the formi- dable detail of a full spectrum. 20 LITERATURE CITED BURDETTE, E.L., STEELE, K.E., and TRAMPUS, A., 'Directional Wave Spectral Data from a Large Discus Buoy," Oceans '78 Conference Record, Marine Technology Society, 1978, pp. 622-628. HARRIS, D.L., 'Wave Estimates for Coastal Regions," Shelf Sediment Transport, Dowden, Hutchinson, and Ross, Stroudsburg, Pa., 1972, pp. 99-125. HARRIS, D.L., "Finite Spectrum Analyses of Wave Records," Proceedings of the International Sympostum on Ocean Wave Measurement and Analysts, American Society of Civil Engineers, Vol. I, 1974, pp. 107-124 (also Reprint 6-74, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., NTIS A002 113). McCLENAN, C.M., and HARRIS, D.L., "The Use of Aerial Photography in the Study of Wave Characteristics in the Coastal Zone,'' TM-48, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Jan. 1975. RESIO, D.T., and VINCENT, C.L., "A Comparison of Various Numerical Wave Pre- diction Techniques," Proceedings of the Offshore Technology Conference, American Society of Civil Engineers, 1979. THOMPSON, E.F., ''Energy Spectra in Shallow U.S. Coastal Waters,'' TP 80-2, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Feb: 1980. : U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protection Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp. WILSON, B.W., CHAKRABARTI, S.K., and SNIDER, R.H., "Spectrum Analysis of Ocean Wave Records," Proceedings of the Internattonal Sympostum on Ocean Wave Measurement and Analysts, American Society of Civil Engineers, Vol. I, 1974, pp. 87-106. 2| ‘aa Fytl : oul fehl ni ae or cd uz i Le: 1 damn? 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