y OSA Oe Eu, Bs STP TP 80-2 Energy Spectra in Shallow U. S. Coastal Waters Gen. by me Se. ae \ Edward F. Thompson \ 2, epi N : \ TECHNICAL PAPER NO. 80-2 \ FEBRUARY 1980 7 Approved for public release; distribution unlimited. U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING oa “RESEARCH CENTER oe . Kingman Building BL Fort Belvoir, Va. 22060 Vu DO 2) 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 Contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. 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. HOi MBL/W. INIA IAIN q991 0 I Wi oO 0301 0086 UNCLASSTFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS 1. REPORT NUMBER 2. GOVT ACCESSION NO,| 3. RECIPIENT'S CATALOG NUMBER TP 80-2 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Technical Paper 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(a) ENERGY SPECTRA IN SHALLOW U.S. COASTAL WATERS - AUTHOR(as) Edward F. Thompson 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS A31463 12. REPORT DATE 13. NUMBER OF PAGES 15. SECURITY CLASS. (of thia report) UNCLASSIFIED 1Sa. DECLASSIFICATION/ DOWNGRADING 1 SCHEDULE Approved for public release; distribution unlimited. - PERFORMING ORGANIZATION NAME AND ADDRESS Department of the Army Coastal Engineering Research Center (CERRE-CO) Kingman Building, Fort Belvoir, Virginia 22060 - 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) 16. DISTRIBUTION STATEMENT (of this Report) DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES - KEY WORDS (Continue on reverse side if necessary and identify by block number) Coastal engineering Wave gages Spectral parameters Waves Wave energy spectra 20. ABSTRACT (Continue em reverses aide if meceasary and identify by block number) Digital wave analyses for 3 to 12 months of data from each of 11 U.S. coastal gages are summarized and discussed. Water depths at the gage sites were typically between 5 and 9 meters. The gage designs included step resist- ance, continuous wire, pressure, and accelerometer buoy. The analysis for each record included computation of the energy (or variance) spectrum and the distribution function of sea-surface elevations. Parameters of the spectrum and distribution function of sea-surface elevations were also computed. continued) DD anys 1473 EDITION OF 1 Nov 65 1S OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Wren Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) Spectra and parameters for the 24 highest energy cases from each gage location are presented individually. Spectra for all cases are grouped according to significant wave height and peak spectral period. Mean spectra and standard deviations about the mean are presented for most height-period groups. Parameters for all cases are summarized, including number of major spectral peaks, spectral-peakedness parameter, and third and fourth moments of the normalized distribution function of sea-surface elevations. Multipeaked spectra are common at all locations. Mean spectra show evi- dence of systematic changes in shape as a function of significant height and peak spectral period. Values of the spectral-peakedness parameter range from about one to eight. The distribution functions of sea-surface elevations indicate a tendency for more extreme high values than low and for a more narrow distribution function than the Gaussian distribution. Observed char- acteristics are related to physical wave behavior and illustrated with three cases from cnoidal wave theory. Evidence is presented that the ratio of Significant wave height to water depth does not exceed 0.55 in depths of 5 to 9 meters. 2 SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE This report provides coastal engineers and researchers with wave energy spectra and spectral parameters for nine shallow-water gage locations along the U.S. Atlantic, Pacific, gulf, and Great Lakes coasts. Insight is also provided on the physical meaning of shallow- water spectra, which are becoming increasingly important in coastal engineering work. The work was carried out under the waves and coastal flooding research program of the U.S. Army Coastal Engineering Research Center (CERC). This report was prepared by Edward F. Thompson, Hydraulic Engineer, under the supervision of Dr. C.L. Vincent, Chief, Coastal Oceanography Branch, and his predecessor, Dr. D.L. Harris. The guidance and constructive comments provided by Drs. Vincent and Harris are gratefully acknowledged. The computer program in Appendix D was developed by Dr. Harris. The assistance of D.G. Dumm in preparing some of the tables and figures is appreciated. Special appreciation is due the CERC Automatic Data Processing Office for outstanding support in producing the voluminous computer printouts and plots generated for this report. Comments on this publication are invited. Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. Colonel, Corps of Engineers Commander and Director CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI). SYMBOLS AND DEFINITIONS. 2°. 2 500 3 we st 3 8 I INTRO DUGIVUIONMEy yin eure relancrm oakcun teres Tel PHYSICAL CHARACTERISTICS OF SHALLOW-WATER OCEAN WAVES. iver WAVE GAGE DATA COLLECTION AND ANALYSIS . Ie Colle ctrloms teens cieem ver CepiteL acetic wets De INDEMIVYASIESS go 6 io 00 6.00 0 0 OD IV SPECTRA, SPECTRAL SUMMARIES, AND SEA-SURFACE ELEVATION DISTRIBUTION FUNCTION PARAMETERS . . 1. High-Energy Wave mea 0 0.0 2,eMean\ (Spectrasy ya. ue 3. Spectral and Sea- Surfaces Elevation. Distribution Function Parameters. ..... e V COMGHUSIIONSS 66.5 lo o 100066 6'10..6.0 06661690 VI SUM MINS CS, Gi Gg oo, 6 oO 01015 160 6 O00 © 6a) 6 EITERATURES GRE Dir iienetierice ct Micali sireiits APPENDIX A INDIVIDUAL HIGH-ENERGY WAVE SPECTRAL PLOTS . B MEAN SPECTRA GROUPED BY HEIGHT AND PERIOD. ... . C MEAN SPECTRA GROUPED BY HEIGHT, PERIOD AND WATER LEVEL . D DESCRIPTION OF COMPUTER ROUTINE SMOOTH FOR IDENTIFYING MAJOR PEAKS AND VALLEYS IN AN IRREGULAR SIGNAL . TABLES Information on wave gage locations .... . Peak period grouping intervals . Percentage of cases in each significant height group corresponding to number of major spectral peaks . Linear correlation between number of major spectral peaks, significant height and period corresponding to the highest PEAK a lics oy ety tar aN votre es aco ceuibeyiateten iyo o Mele! fi oled telnet om LiMo etc Mmion El Percentage of cases in each peak period group corresponding to number of major spectral peaks . .. +--+ +2 + & 4 Page 140 145 20 25 42 43 43 10 11 12 13 CONTENTS TABLES--Continued Correlation between spectral-peakedness parameter and other WAVEmDARAMe COT:SmesmeNh s: (eMh Milas dictioue) (oie oO = s 1 3° FRF Mid Pier (Depth 2m) & a 0 : A 1 16 20 00 04 08 12 16 20 19 Jan 78 Time (Eastern Standard Time) 20 Jan 78 Figure 3. Time history of significant wave height for three North Carolina gages during a large winter storm. For some of the gage sites included in this report, and perhaps for all except the accelerometer buoy sites, a sandbar was often seaward of the gage. The bar is a somewhat transient feature which can have a major influence on the waves by causing high waves to break before they arrive at the gages and by acting to reflect energy seaward or trap it between the bar and shore. Evidence presented by Wang and Yang (1976) indicates the bar acts mainly as an energy dissipator at low tide. At high tide for nonbreaking waves, the bar tends to reflect wave energy seaward during onshore winds and to trap wave energy between the bar and beach during offshore winds. Since ocean waves in a sea state are never strictly uniform in either height or frequency, a sea state is conveniently characterized by the distribution of wave energy as a function of frequency (the spectrum). During active wave growth, most energy from the atmosphere is added at frequencies slightly higher than the dominant frequency. The energy is then redistributed by nonlinear transfers within the wave field to fre- quencies slightly lower than the dominant frequency and to very high frequencies where it is lost to breaking processes. These phenomena result in a gradual shift of the spectral peak to lower frequencies. 16 For waves propagating outside the wave-generation area the nonlinear energy transfer to low and very high frequency continues without the accompanying replenishment of energy at midfrequencies. Thus, swell waves are characterized by narrow spectra with peaks at low frequency. There is evidence that the intensity of nonlinear energy transfer increases rapidly with decreasing water depth. The nonlinear transfer may also narrow the directional spread of energy in shallow-water waves (Herterich and Hasselmann, in preparation, 1980). Aerial photos of nearshore ocean waters indicate remarkable organi- zation in the surface waves (Fig. 4); the photos also indicate that sea states dominated by a single visible wave train are relatively unusual. Photos typically show two or three apparently independent wave trains with different, well-defined directions and wavelengths. The relative prominence of the trains often varies with proximity to shore due to frequency-dependent shoaling and refraction, and in some cases due to redistribution of energy with frequency by nonlinear interactions between trains. For example, although a long swell wave train often dominates the breaker zone in photos taken along the southern California coast, the train is invisible two or three wavelengths seaward of the breaker zone. The swell waves are preferentially amplified nearshore. McClenan and Harris (1975) show several examples of this process occurring. The swell waves are also refracted more completely than higher fre- quency waves so that at breaking they are often nearly parallel to the shore. The water particle velocity associated with low-frequency swell waves attenuates slowly with depth. High-frequency waves which have not been substantially refracted can travel into the swell-created breaker zone with a large component of longshore motion; yet, their particle velocities attenuate rapidly with depth. Since longshore sediment move- ment requires both a mechanism for initiating sediment movement and a longshore drift to transport it, two concurrent wave trains may move sediment alongshore more effectively than either train acting alone. The presence of a current near a gage site can alter the measured characteristics of waves. The longshore current in and near the breaker zone affects some of the gage measurements in this report. Tidal cur- rents and wind-generated currents may also affect the measurements. The Gulf Stream may affect deepwater waves approaching the U.S. Atlantic coast by steepening waves enough to cause breaking or, in special cases, by refracting the waves so that they never reach the shore (Kenyon, 1971). The rate of wave growth in the vicinity of the current may also be affected. III. WAVE GAGE DATA COLLECTION AND ANALYSIS 1. Collection. The wave gage data used in this report were collected as part of the CERC field wave data collection program from 11 gages at’ the 9 locations 1%, ee atiote ee oe long the ams) al ilometers south of Port le wave trai Hy) 1t © & Ix ing mu 1a coas 4. Aerial photo show southern Californ igure Fy 1 istration). minis Hueneme on 29 March 1977 (courtesy of Nationa Aeronautics and Space Ad 18 shown in Figure 1. Table 1 gives some details of the gage locations, gage types, and water depth at the gages. The gage designs and charac- teristics are described in detail by Thompson (1977), and for the Great Lakes accelerometer buoy installations in Thompson (1978). The continuous-wire staff gage (manufactured by the Baylor Company, Houston, Texas) is the more accurate sensor because the gage directly senses surface waves and it interferes very little with waves being measured. The continuous-wire gage and the pressure gage (fabricated at CERC) give comparable surface wave height estimates when the pressure- gage record is compensated for the depth-dependent attenuation of the dynamic component of pressure due to surface waves. Suitably compen- sated pressure-gage spectra are also comparable to continuous-wire gage spectra except at high frequencies (Esteva and Harris, 1970). The accelerometer-buoy gage or Waverider (manufactured by Datawell, Haarlem, the Netherlands) gives wave heights and spectra which are reasonably comparable to those given by the continuous-wire gage for frequencies between 0.065 and 0.5 hertz, including virtually all impor- tant frequencies in the Great Lakes (Pitt, Driver, and Ewing, 1978). The step-resistance staff gage (fabricated at CERC) has been shown to have a consistent bias toward high wave heights (Esteva and Harris, 1970). Significant height estimates from the step-resistance gage may be 30 centimeters too high during low wave conditions and 20 percent too high during high wave conditions. Aside from the tendency for over- estimating wave height and energy, Esteva and Harris show evidence that spectra for step-resistance gage records are reasonably consistent with continuous-wire gage spectra. Time series of sea-surface elevation from staff gages, subsurface pressure from pressure gages, and double-integrated sea-surface vertical acceleration from buoy gages were transmitted by telephone line to the CERC laboratory and used to generate pen-and-ink, strip-chart records and digital records on computer-compatible, seven-track magnetic tape. Digital records are 20 minutes long. A digital record from a single station contains data points at 0.25-second intervals giving a total of 4,800 data points per location per 20-minute record. Further details of CERC field wave data collection procedures are given in Thompson (1974, 1977) Zana lySasts a. Spectral Analysis. The digital field wave gage records were analyzed by the CERC routine analysis system (Thompson, 1977). One record per 6 hours is routinely analyzed; however, one record per 2 hours was analyzed for some of the Great Lakes data. The routine anal- ysis times are chosen to approximately coincide with synoptic times (0100, 0700, 1300, and 1900 e.s.t.). The procedure includes editing each 20-minute time series to eliminate data points which are obviously bad, computing and testing the distribution function of the edited data points, and computing the variance or energy spectrum for the time series 19 ZL6T *20d ainsseld "M 160.6TT ‘N iLO0o¢S "FITTED ‘n3ny Iq ZL6T *29C eITM snonuTqUOD *M 10008TT ‘N 16£0S8 ‘yorag uowutquny SZ6T °*AON saInssetd 1T°L0008 2 OT och "eq ‘aTs] onbsorg SL6T “AON Aong 10°20.08 SST och "eq “eTs] enbsergq SL6I “AON eINssoa.td 1$°6S098 10°Zbolb spur ‘AIT uestyotW SL6I *°AON Aong 10°65.98 "N iv Sholt ‘pur ‘AID uestysty 696T °390 02 896T soueqstsez daas 16PoT8 “N 180.92 ‘ely ‘sotden 696T ~390 02 896T aouejstsar daa 120008 VLE 097 "WTA “YOM ayeq 696I °290 02 896T aouejstsez dais 195 oSL 1SS,S¢ ‘O'N “peasy sden 696 ~390 02 896T aoueystser daqs 18S0SL 1TS09E “BA “YyoReg BTUTSITA 696T aoueystser doag SZobL TZ o6E "CoN ‘AITO OTIUeTIy pues aAoqe o1rez a8e3 Jo aystoy yadep ra.eK soqeUurpi1oo) uoTe907 *SUOTIVIOT Oded JAEM UO UOTJRUIOFUT “fT 9TqQRL 20 smoothed with a cosine bell data window. The spectrum is computed with a fast Fourier transform (FFT) algorithm applied to 4,096 data points (17-minute and 4-second record). The spectral analysis procedure assigns a fraction of the total var- iance of the record to each of 1,024 frequencies, or frequency lines. It is difficult to deal with such finely resolved spectra. To gain sta- tistical stability, groups of 11 successive lines in each spectrum are combined into bands. The resultant band width is 0.01074 hertz over the full range of frequencies considered (0.03 to 1.00 hertz). The uniform spectral resolution over the frequency range gives a nonuniform resolu- tion in wave period. Significant height is estimated as four times the Square root of the total variance assigned to wave frequencies of inter- est, usually 0.03 to 1.00 hertz. The frequency-dependent responses of some of the gage types required some special treatment of the spectra. Accelerometer-buoy gage spectra were terminated at the high-frequency end at 0.5 hertz. A special low- frequency bound of about 0.065 hertz on the spectrum would have been appropriate but was not used because energy at frequencies between that and the low-frequency cutoff for other gage types (0.03 hertz) was in- Significant in the Great Lakes buoy records. Pressure-gage records were compensated for hydrodynamic attenuation of the pressure signal with depth by multiplying the energy in each spectral band by a frequency-dependent factor. The compensation factor becomes very large at high frequencies where the small pressure signal is often obscured by noise. Hence, it is necessary to terminate the high-frequency end of pressure-gage spectra at a frequency below the standard 1l-hertz cutoff for staff gages. The high-frequency cutoff used for the Michigan City and Presque Isle pressure gages is 0.33 hertz. The cutoff for the Pt. Mugu pressure gage is 0.31 hertz. All spectral energy assigned to frequencies above the cutoffs was omitted from both the spectrum and the significant height estimate. The neg- lected energy is unimportant during high wave conditions, but it can increase significant height by as much as 0.5 meter during low recorded wave conditions. The latter effect is often partially balanced by slight overcompensation of the high-frequency energy retained in the spectrum. b. Spectral Summarization. (1) Averaging. Individual shallow-water ocean wave energy spectra are quite irregular and can change substantially and somewhat erratically with time (see Fig. 5). Because of these problems, it is difficult to identify typical spectra for each wave gage site. However, it is desirable to condense the masses of individual spectra into a more concise form. At this time, no completely satisfactory technique is in use for isolating and summarizing general characteristics of field spectra. One ail Figure 5. Energy Density (cm?/Hz) 00 Of 0.2 0.3 0.4 0.5 Frequency (Hz) Sequence of spectra computed at 6-hour intervals for Nags Head, North Carolina, beginning at 0642 e.s.t., 20 April 1969. H, is 190 centimeters for the top spectrum decreasing to 97 centimeters for the bottom spectrum. Major spectral peaks are marked with asterisks. 22 common approach is to compute spectral statistics which are often based on moments of the spectrum, where the nth spectral moment, My LS defined as By = fs f df (1) 0 where f is the frequency and S(f) the spectral energy at frequency f. This approach is used in many cases because it is the only tractable way to deal with large numbers of spectra. However, some of these spectral Statistical parameters have disadvantages which will be discussed later. A relatively new and promising approach to spectral parameterization involves the use of eigenfunctions (Vincent and Resio, 1977). Another approach to the problem of how to characterize and summarize ocean wave spectra has been used in conjunction with ship-response pre- diction. Hoffman (1974) organized deepwater spectra into 10 groups from a site in the North Atlantic according to significant height. For each group, the mean and standard deviation of spectral energy in each band were computed. The average energy of the one-third highest and one-third lowest values in each band was also computed. The importance of accounting for variability of spectral shape when predicting ship- bending moment responses was demonstrated. An averaging procedure for summarizing deepwater spectra from a site in the North Atlantic was also used by Gospodnetic and Miles (1974). Spectra were grouped according to both significant wave height and average wave period. The average spectra were made dimensionless to facilitate comparison of spectral shapes. Although most of the varia- tions were removed by using the dimensionless format, it was concluded that even the dimensionless spectra vary systemmatically with both height and period. In very shallow water, spectral characteristics would be expected to have a pronounced systematic relationship to both significant height and dominant spectral period, although the relationship would be somewhat obscured when major secondary wave trains exist. Wave height is impor- tant because high waves lead to wave breaking in shallow water. Thus, any reasonable attempt to compute average spectra at a shallow-water location must include stratification of the spectra according to wave height or energy. The extent of wave shoaling is specified by the relative water depth, d/L,, which is directly related to the dominant wave period. Waves in very shallow water assume a nonsinusoidal profile which gives rise to spectra with peaks at multiples of the dominant frequency. An example of the profile and spectrum for several cnoidal wave cases is shown in Figure 2; another example is an aerial photo spectrum pair in McClenan and Harris (1975, p. 63-64). Because of such systematic effects on the shallow-water spectrum, spectra should also be stratified by d/L, or by a characteristic wave period. 23 Deviations of the wave profile from sinusoidal occur even in deep water for very steep waves as discussed previously. Spectra for such waves can also be expected to show energy at multiples of the dominant frequency. In such cases the spectral form is related to both wave height and period. Because of expected systematic influences of wave height and period on the shallow-water spectrum, spectra in this study were grouped into sets by both significant height and period corresponding to the highest spectral peak. Significant height and peak period are convenient because they are comparable to height and period parameters used extensively in past nonspectral wave analyses. Despite the grouping, major differences between individual spectral shapes in each set can still occur. Differences between spectra at one location for several cases in which the significant height and peak period were near the annual means are shown in Figure 6. The differences in spectral shape between individual spectra in a set were greater than the differences in the mean spectrum between sets in this study and in Hoffman's (1974) study. However, any systematic characteristics of each set should emerge in the mean spectrum and standard deviation if the number of samples is sufficient. This does not imply that the mean spectra necessarily represent characteristics. 25,000 eee OATIESRE 26 Nov. 1968 15 yon. 1969 17 Jon. 1969 24 May 1969 25 May 1969 27 Moy 1969 20,000 15,000 10,000 Energy Density (cm2/Hz) 5,000 Frequency (Hz) Figure 6. Several individual spectra in the same H, - Tp group (Hg = 91 to 122 centimeters; Tp = 8.5 to 9.3 seconds), Nags Head, North Carolina. ; 24 Spectra were grouped by 30-centimeter intervals of significant height and by variable intervals of peak period (Table 2). Variable period intervals were needed because energy is computed for nonuniform period bands (but uniform frequency bands) in the CERC spectral analysis program. Selection of longer period intervals was necessary so that each interval encompasses only one spectral band. If the intervals had included more than one spectral band, the averaging procedure would tend to produce an unrepresentatively low and broad peak in the average spectrun. Table 2. Peak period grouping intervals. Frequency bands Frequency bands Period interval | in interval | in interval (No. ) (No. ) Period interval 1 47 -8 to 8.5 2 -5 to 9.3 3 -3 to 10.3 4 -3 to 11.6 5 6 to 13.3 6 3 to 15.5 6.6 -5 to 18.6 7.2 6 to 23.3 fe) Period intervals of 1 second were used for periods shorter than 6 seconds (Table 2). Individual spectra with relatively short peak pe- riods are generally more poorly focused in frequency than spectra with long peak periods. Thus, period intervals encompassing more than one spectral band could be used for short periods without creating much artificial broadening of the peak in the average spectrum. This treat- ment is further defensible on the grounds that (a) short-period waves are in relatively deeper water than long-period waves at the measure- ment site and hence exhibit less depth-induced modification, and (b) spectra with peak periods shorter than 6 seconds are generally not im- portant for coastal engineering design applications at ocean sites. After the spectra were grouped into height-period intervals, average spectra were computed. Intervals were selected for averaging to include all intervals containing a reasonable number of cases and all high wave intervals regardless of the number of cases. The minimum number of cases considered for low and moderate waves was 10 at most locations. The wave height above which all intervals were considered varies between locations. The average spectrum for a height-period interval was obtained by aver- aging energy values in each band. The standard deviation of spectral energy values about the mean for each band was also computed. Averages and standard deviations of spectra for the locations listed in Table 1 are discussed in the next section. It is important to note that the spectrum for any individual record never duplicates the mean spectrum. Any application which is sensitive to the precise form of the spectrum should make use of individual spectra rather than the mean spectrum. One reasonable approach might be to select a small random sample of spectra from each group, apply each spectrum to Ze) the problem at hand, and combine the individual results to estimate response probabilities, as suggested by Hoffman (1974, 1975). (2) Spectral Parameters. (a) Number of Peaks. It is relatively common in the ocean for two or more independent wave trains with different frequency and ‘direction to occur simultaneously (e.g., Harris, 1972; McClenan and Harris, 1975; Ochi and Hubble, 1976; Thompson, 1977). The marine sur- face observation reporting code even includes provisions for reporting major secondary wave trains observed visually. Certainly, the concept of a simple, single-peaked spectrum is misleading in many ocean wave records. Yet, there is a lack of reliable quantitative information as to how often major secondary trains occur and how much energy they contain. To investigate the occurrence of multiple wave trains, a computer routine was adapted for use in identifying major spectral peaks. The routine (described in App. D) smoothes over spectral peaks and valleys which differ in energy density by less than 3 percent of the total energy in the spectrum. Examples of individual spectra were shown in Figure 5. The major peaks identified in each case by the computer rou- tine are marked with asterisks. A fraction of the spectral energy is assigned to each major spectral peak by a simple method. The spectrum is partitioned at the lowest point between successive major peaks (Fig. 7). All spectral energy within the partition for a peak is assigned to that peak. The energy is expressed as a dimensional quantity and as a fraction of the total spectral energy. 10,000 ; Ss Energy in Pec« 1 8,000 Energy in Peck 2 = ZZ Energy in Peck 3 E © 6,000 & 4,000 = WwW 2,000 0 0.0 0.1 0.2 0.3 0.4 0.5 Frequency (Hz) Figure 7. Technique for partitioning spectral energy and assigning the energy to major peaks. 26 Statistics on the occurrence of multiple spectral peaks identified by the above techniques (presented in thé next section) are based only on the part of the spectrum between 0.03 and 0.5 hertz for the staff and accelerometer-buoy gages. The high-frequency cutoff for the pressure gages was 0.33 hertz in the Great Lakes and 0.31 hertz at Pt. Mugu as discussed earlier. (b) Spectral Peakedness. It is highly desirable to param- eterize a spectrum. Various parameters have been considered in the lit- erature, but most are derived from moments of the spectrum as defined in equation (1). In practical application equation (1) becomes f My, = far S(£) f"d£ (2) fon where f77 is the low-frequency spectral cutoff, and fyr the high- frequency spectral cutoff used for computation. When applied to field records of finite length, the integral in equation (2) becomes a summation. Equation (2) shows that the higher the value of fyp, the greater the magnitude of m, provided that S(f) is greater than zero at high frequencies. In practice, the choice of fyp has a significant influ- ence on the values computed for m, especially for the higher moments. The choice of f;, is less ambiguous in practice because contributions to m, from the low-energy and low-frequency part of the spectrum are generally negligible. The sensitivity of m, and parameters based on m, to the choice of fyr was investigated by Rye (1977). He considered two theoretical spectral shapes: a sharply peaked JONSWAP spectrum and a flat JONSWAP spectrum equivalent to a Pierson-Moskowitz spectrum. Rye recommended three spectral parameters for general use: (a) Significant wave height, H, where Hg = 4¥m (3) (b) period of the spectral peak, Tp (c) spectral-peakedness parameter, ® where i) Ex Q = <> | f[s(#)12aF (4) 0 The spectral-peakedness parameter was originally proposed and shown to be directly related to the average number of high waves oceurring in AT succession in simulated wave records by Goda (1970). However, Goda (1976) could not find a clear relationship between Q and the extent of group- ing of high waves in field records. The value of Qp as a spectral parameter is shown in Figure 8. The two spectra shown have nearly the same significant height and peak period. The spectra also have comparable values of the frequently used spectral- width parameter, €, defined as 2 m eee loa! (5) Mo Mm, However, Q) for one spectrum is 66 percent greater than for the other, indicating substantial differences in spectral shape. Energy. Frequency Figure 8. Comparison of two measured spectra from the North Atlantic Ocean with Hg = 3.3 meters, Tp = 10.5 seconds, and « = 0.61 (after Hoffman, 1974). 28 Both Q and e are designed to fit the concept of a single-peaked spectrum, but multipeaked ocean wave spectra are commonly observed. The suitability of spectral parameters for representing multipeaked field wave spectra has received little attention in the literature, although Ochi and Hubble (1976) have developed a computationally complicated tech- nique for parameterizing field spectra with two major peaks. Since Qp seems to be a potentially important and useful parameter, it was computed for each of the spectra considered in this report. was based on the part of the spectrum between 0.03 and 0.2 hertz for staff and accelerometer-buoy gages, between 0.03 and 0.33 hertz for the Great Lakes pressure gages, and between 0.03 and 0.31 hertz for the Pt. Mugu pressure gage. c. Parameters of Distribution Function of Sea-Surface Elevation. The distribution function for instantaneous sea-surface elevations pro- vides useful insight on shallow-water spectra and spectral parameters. It also provides probabilities associated with instantaneous surface elevations above the mean. The distribution function of sea-surface elevations can conveniently be parameterized by its moments defined as: N = 3 iM? oh. 6 qd, sep ie Gg) (6) where Gn = nth moment of the distribution function of sea-surface elevations N = number of intervals in the distribution function n; = sea-surface elevation associated with the ith interval in the distribution function p(n,) = probability associated with n,; L The zeroth and first moments, Gg and q,, are equivalent to the mean and variance of the distribution function. q3 and q, are often re- ferred to as the skewness and kurtosis of the distribution function. The distribution function of sea-surface elevations is often assumed to be Gaussian. When normalized, the Gaussian distribution function has a mean of zero, a variance of one, a skewness of zero, and a kurtosis of three. Steep waves in shallow water assume a decidedly nonsinusoidal pro- file with broad, flat troughs and narrow, high crests. The distribution of sea-surface elevations measured at a point is obviously non-Gaussian. 29 The skewness and perhaps the kurtosis of surface elevations in very shallow water can be expected to differ from the values for a normal- ized Gaussian distribution. For the cnoidal wave profile in Figure 9, the skewness and kurtosis are equal to 2.0 and 5.8, respectively. The figure also shows the distribution function, skewness, and kurtosis for several idealized wave profiles. Distribution Function of Sea—Surface Elevations Wave Profile Sinusoidal Wave = =) Triangular Wove 7, St) 7 Inverted Cnoidal = <9 Wave Figure 9. Distribution function of sea- surface elevations for several idealized wave profiles. Both skewness and kurtosis for the normalized distribution function of sea-surface elevations were computed for each record and are summa- rized in the following section. For the accelerometer-buoy data, the distribution functions represent doubly integrated vertical-surface accelerations rather than directly measured sea-surface elevations. For the pressure-gage data, the distribution functions represent sub- surface pressure fluctuations. 30 IV. SPECTRA, SPECTRAL SUMMARIES, AND SEA-SURFACE ELEVATION DISTRIBUTION FUNCTION PARAMETERS 1. High-Energy Wave Spectra. Since high-energy spectra are of greatest concern in most engineering applications, plots of individual spectra containing the most energy for each station are provided in Appendix A. Spectra are grouped by gage location in the order listed in Table 1. The appendix includes 24 spec- tra for each location, arranged and numbered in descending order of sig- nificant height. Each plot in Appendix A shows the significant wave height, period corresponding to the highest spectral peak, spectral-peakedness parameter (Qo), and number of major spectral peaks identified by the computer rou- tine SMOOTH (App. D). Spectral plots for days associated with tropical storms and hurricanes are labeled accordingly. The energy density scale can vary between plots. Pressure-gage spectra in Appendix A are cut off at high frequencies above 0.33 hertz for the Great Lakes gages and 0.31 hertz for the Pt. Mugu gage. Several of the Pt. Mugu high-energy spectra were obviously overcompensated for attenuation of pressure with depth so that the high- frequency energy was much greater than the low-frequency energy. These spectra were omitted from Appendix A. This problem can arise because the high-frequency cutoff is constant while the compensation increases as the depth of water over the gage increases with the tide. Thus, the high-frequency end of the spectrum is sometimes overcompensated when water levels are high. This problem did not arise with pressure-gage spectra in the Great Lakes (where there is no tide). Energy computed for wave periods longer than 30 seconds (frequencies less than 0.043 hertz) does not directly represent wind-generated waves. In some cases it is questionable whether such very long-period energy has any physical meaning. Therefore, it was omitted from spectral plots for the Great Lakes and Pacific coast locations and should be ignored in plots for the Atlantic and gulf coast locations. Spectral plots for each location in Appendix A are followed by a table which contains auxiliary data for each plot, including sequential number, date, time, significant wave height, peak wave period, water depth, rela- tive water depth (where is the shallow-water wavelength for a wave with frequency equal to the frequency of maximum spectral energy density), ratio of significant wave height to water depth, wave steepness, ratio of depth to gTf (which is related to wavelength), spectral-peakedness pa- rameter, and skewness and kurtosis of the distribution function of sea- surface elevations. Wavelength used in the tables was computed in each case from period corresponding to the highest spectral peak and the equa- tions of linear wave theory. The largest ratio of significant wave height to depth is 0.54. None of the significant height-to-depth ratios approach the commonly used limit of 0.78. 31 Multiple major spectral peaks are evident in many of the spectra in Appendix A. However, some of the multipeaked spectra and many of the single-peaked spectra appear to approximate the form of the JONSWAP spec- trum for waves undergoing active generation (Hasselmann, et al., 1973). Only seven of the spectra selected for inclusion in Appendix A were col- lected during hurricanes. This small sample does not illustrate any clear differences between spectra generated by hurricanes and spectra generated by extratropical storms. 2. Mean Spectra. a. Grouped by Height and Period. Spectra from each gage site were grouped into sets by significant height and peak spectral period. Inter- vals of 30.5 centimeters were used for significant height; intervals of about 1 second were used for peak period (Table 2). Mean spectra were plotted for all height-period intervals containing more than one case with heights greater than a cutoff height, which varied with location. In addition, mean spectral plots were generated for height-period intervals with heights below the cutoff if there were a reasonably large number of cases in the interval. This approach pro- vided mean spectral plots for the most statistically meaningful low wave cases and for all high wave cases. Energy computed for periods longer than 30 seconds was omitted from plots for Great Lakes and Pacific coast locations as discussed earlier. The mean spectral plots grouped by location are given in Appendix B. The approximate relative water depth and ratio of significant wave height to water depth for each location are listed in Table B-1; the dates covered are in Table 1. The range of significant wave height and peak spectral period repre- sented and the number of cases are listed on each plot. Each plot repre- senting more than one case contains several dashlines in addition to the solid line showing the mean spectrum. The dashline nearest the mean and above it represents the mean plus one standard deviation. Similarly, the dashline nearest the mean and below it represents the mean minus one standard deviation. When the mean minus one standard deviation is nega- tive, it is plotted as zero. For plots representing more than two cases, another dashline above the mean shows the highest single-energy density value occurring in each band. This line forms an envelope inside which all of the individual spectra would fall. The dashlines should not be treated as a spectrum, and are included only as indicators of the observed variability of energy density in each spectral band. Some of the plots show a relatively large concentration of energy at the lowest frequency (0.005 hertz). This concentration is at a frequency which is irrelevant to wind-generated gravity waves and should be ignored. The mean spectral plots for each location in Appendix B are arranged in order of increasing significant height and peak period in most cases; i.e., all mean spectra in a colum represent the same peak-period interval 32 and all mean spectra in a row represent the same significant height interval. Thus, the effect of increasing height on spectra with similar peak periods and the effect of increasing peak period on spectra with Similar heights can be easily followed. A small number of plotted mean spectra are not included in Appendix B because the spectra (a) usually represented low or moderate wave heights with relatively few cases or high wave heights with only one case for which the spectrum is shown in Appendix A, or (b) were high-energy cases for Pt. Mugu which had obviously been overcompensated for attenuation of pressure with depth. Mean spectra for low wave heights and peak periods less than 5 seconds along the Atlantic coast show evidence of secondary peaks at low frequency in many cases. The low-frequency peaks are broad and flat, are generally located near 0.1 hertz, and are likely to represent low-energy swell in the Atlantic Ocean. Corresponding broad, low-frequency secondary peaks are also evident in mean spectra for Pt. Mugu on the Pacific coast. These peaks are not evident in mean spectra for the gulf and Great Lakes gages, although the Presque Isle pressure-gage spectra show a sharp secondary peak which is consistently centered on 0.038 hertz. This low-frequency peak at Presque Isle may not be physically meaningful. For all locations, the differences between mean spectra are smaller than the differences between the individual spectra used to compute the means. Mean spectra from the ocean gages for peak periods longer than about 10 seconds show evidence of secondary peaks at high frequency, especially for high wave heights. For example, several selected mean spectra for high wave, long-period groups are plotted in Figure 10 in dimensionless form. Energy density was made dimensionless for each mean spectrum by dividing by the total energy in the spectrum and converting to percent; frequency was made dimensionless by dividing by the peak frequency. The figure clearly shows that the main secondary peak in each mean spectrum is at twice the peak frequency. The spectrum for one cnoidal wave (see Fig. 2) is also shown to clearly resemblé the field spectra. A section of the Lake Worth pen-and-ink strip-chart record taken at nearly the same time as the Lake Worth spectrum in Figure 10 is shown in Figure 11. The profiles of the larger waves closely resemble the cnoidal wave profile. Many of the high-frequency secondary peaks in mean spectra for high, long-period waves are probably indicative of cnoidal-type wave profiles. Spectra for high, long-period waves in which secondary peaks at about two or three times the peak frequency represent independent wave trains are expected to be rare. The plots in Appendix B also show interesting variations with wave height in mean spectra with short peak periods. In some cases, the mean spectra for high waves exhibit secondary peaks at low frequency. The emergence of the low-frequency secondary peaks for Nags Head is shown by dimensionless spectra plotted in Figure 12. The figure also indicates that the largest secondary peak is at one-half the frequency of the dominant frequency. A section of pen-and-ink strip-chart record corresponding to two high wave, short-period cases is shown in Figure 13. 33 50 y | | 40 I I I He T I, Symbol Location No.Cases (cm) (s8 oe | ——— __ Nags Head 12 183-213 116-13.3 o | —»—— ___ Nags Head 8 183-213 13.3-15.5 — 30 | —-— Lake Worth 1 183-213 13.3-15.5 > ication Cnoidal Wave 286 14 _— 7) i= ao (=) a> om i @ i= uw Figure 10. Dimensionless mean spectra for several high wave, long-period groups and for the cnodial wave shown in Figure 2, case 3. of Same Record % Continuation —HA10$ K- <— Time Hs:199cm d= 5.6 m H| 10s |+— Hg=118¢m d=45m Hs/d 20.26 s:006 Tp 24.8 s 9.70.92 d/gT,'=0.020 is p 3 p op Qp=1.52 4473.8 d/Lp20.163 35 ; 3 °6.0 carer aols ——— — (b) PH: —+| 10s be H=171¢m d=52m H,/d =0.33 20,000 Tp= 6.48 9370.77 — d/gTy’=0.013 23 Q,=1.37 9473.9 — d/Lp=0.125 as —— 5 0 0.2 i (c) 152 em ¢ (Hz) H, =105¢m d=89m H,/d=0.12 8,000 Tp 28.85 a,7031 4 /aT,=0.011 = 4 : ‘ sz Q) 21.40 qg32 d/L 20.117 = 5 °o0 0.2 0.4 ¢ (Hz) Figure 18. Pen-and-ink strip-chart records and corresponding spectra for some cases with five major spectral peaks: (a) Nags Head record at 0400 e.s.t., 25 December 1968; analysis at 0643 e.s.t., 25 December 1968; (b) Nags Head record at 0800 e.s.t., 24 February 1969; analysis at 0642 e.s.t., 24 February 1969; (c) Huntington Beach record at 1900 €.s.t., 9 May 1972; analysis at 1840 e.s.t., 9 May 1972. small peak at high frequency. Small peaks at frequencies less than 0.05 hertz appeared in some of the Great Lakes spectra, especially spectra from the Presque Isle pressure gages (App. B). These peaks were consid- ered spurious and are not counted as major spectral peaks. Double-peaked spectra are shown to be common at both the gulf and Great Lakes locations. It was noted that secondary spectral peaks do not always represent independent wave trains. When waves are nonlinear, i.e., when H,/d or Hg/Lp are large, the wave profile will be nonsinusoidal. For such cases, a secondary spectral peak at twice the dominant frequency is expected even though a photo would show a single train of waves. The effect is most evident for high, long waves at low tide; i.e., cases with high values of Hsg/d and low values of d/Lp. Appendixes A and B clearly show that secondary peaks tend to occur systematically for high, long shallow-water waves. However, the joint distribution table of number of major spectral peaks versus significant height at each location shows 41 no clear evidence of a relationship (see Table 3). The correlation be- tween number of peaks and Hs is less than 0.3 at all locations (Table 4). The joint distribution table of number of major spectral peaks ver- sus period corresponding to the highest spectral peak at each location also shows little evidence of a relationship (Table 5). The correlation between number of peaks and Tp is also very low (Table 4). For most wave conditions the occurrence of secondary spectral peaks does not appear to be systematically related to Hg and Tp. The overall sta- tistics on multiple peaks as indicators of multiple wave trains should be reasonably correct. Several other attempts have been made to quantify the likelihood of having multiple wave trains. Shepard and Inman (1951) report visual wave observations to identify secondary and dominant wave trains from the Scripps pier in southern California during a 10-month period. They report secondary wave trains in 56 percent of 302 observations. This figure may be an underestimate because of the difficulty of visually identifying secondary trains, but is reasonably consistent with the Huntington Beach gage data. Table 3: Percentage of cases in each significant height group corresponding to number of major spectral peaks. Major spectral pouks a) ca a SS eS A] Nags Head, N.C. INo observed cases for combinations of H, and number of peaks. 42 Table 4. Linear correlation betwecn number of major spectral peaks, significant height and period corresponding to the highest peak. Correlation vs. Location Hg Tp vs. No. of peaks No. of peaks Atlantic City, N.J. 0.12 0.07 Virginia Beach, Va. 0.10 0.10 Nags Head, N.C. 0.16 0.06 Lake Worth, Fla. 0.02 0.05 Naples, Fla. 0.17 0.05 Michigan City, Ind. (buoy) H 0.24 0.17 Michigan City, Ind. (pressure) 0.26 0.24 Presque Isle, Pa. (buoy) 0.09 0.18 Presque Isle, Pa. (pressure) 0.19 0.32 Huntington Beach, Calif. 0.01 0.17 Pt. Mugu, Calif. 0.02 0.16 — z —— Table 5. Percentage of cases in each peak period group corresponding to number of major spectral peaks. | a A ic a al el ls Nags Head, N.C. to to to to to to to to to to to to to to WOMWNIAMHRWN eH a OonNwWRP oO NW OO to to to to to to - to 4 to 27 40 20 13 = 15 2 OBDNAMNHPWNHE ' ' a) oO N wn N un ' ES to 18 61 9 9 to 10 35 45 20 == to 11.6 40 53 7 SS = 43 to 13.3 4S 4S 10 1 = 173 e@ 1555 39 47 13 —— > 250 to 18.6 37 $1 10 1 = 147 €0) 235.5 29 50 21 == = , 4 43 Hoffman (1978) reports on the occurrence of multipeaked spectra from hindcasts generated for a grid point in the North Atlantic Ocean by the Spectral Ocean Wave Model of the U.S. Navy Fleet Numerical Weather Cen- tral, Monterey, California. The criterion for accepting a secondary peak was that its peak energy be at least 30 percent of the energy in the highest peak and that it be reasonably separate in frequency from the main peak. About 25 percent of 572 cases considered were multi- peaked. There was a tendency for multipeaked spectra to be more common during low wave conditions than during high wave conditions. It is of some engineering interest to know whether a spectrum with a certain Tp value is more likely to contain major secondary peaks than a spectrum with a different Tp value. Evidence at many locations shows that spectra associated with certain ranges of Tp do indeed ex- hibit a preference for multiple peaks. At Atlantic City, Virginia Beach, and Nags Head on the Atlantic coast and at Huntington Beach on the Pacific coast, multipeaked spectra are more common for long and short T,, cases than for intermediate T, cases (shown in Fig. 19). Multipeaked spectra are least common for T, values between about 8 and 10 seconds at the three Atlantic coast locations. At Huntington Beach multipeaked spectra are least common for Ty values of 12 to 13 seconds. At Lake Worth and Naples, there is no clear evidence of a relationship between T,, and the occurrence of secondary spectral peaks. Data on secondary peaks from Pt. Mugu and the Great Lakes locations were not considered because of | difficulties with the compensated pressure spectrum and because of the small number of Great Lakes cases. Additional insight on the occurrence of secondary spectral peaks can be obtained from a detailed look at their characteristics for a repre- sentative Atlantic coast location and the single Pacific coast location. For example, Figure 19 shows that 70 percent of the Nags Head spectra with T between 6 and 7 seconds have major secondary peaks; Figure 20(a) shows in a histogram that the periods of those secondary peaks range from 2 to 17 seconds. The figure also shows that secondary peak periods are fairly evenly distributed over the range of 3 to 17 seconds. The rela- tively low percentage of secondary peaks at 7 to 8 seconds shown in Fig- ure 20(a) is artificially low because the CERC analysis procedure gives a nonuniform resolution in wave period which tends to decrease the number of cases in the 7- to 8-second intervals and increase the number of cases in the 6- to 7- and 8- to 9-second intervals. The procedure also precludes any ‘cases wath intervals) of 11 tol2, 15) to, 14) 15) to lo. “anid pili7stomzo0 ; seconds. Although the histograms in Figures 20 and 21 show how often secondary peaks occur, no indication is given of how much energy is contained in the secondary peaks. Therefore, another curve was added to the figures to show the average energy for secondary spectral peaks in each period interval. The energy associated with each peak was estimated by the pro- cedures discussed in Section III and shown in Figure 7. Relatively high- energy secondary peaks tend to occur in Figure 20(a) at periods, slightly less than T. and at periods of 12 to 14 seconds. The long-period sec- ondary peaks generally represent swell waves. 44 Fraction of Spectra Containing at Laast No. of Cases: 0.2 a ” o @ g 1.6 BE o = oo c 2 a8 =o 0.8 a = eho Cee © 9 0.6 Kas) Paap) i=) = ae no 0.4 = os °o < c S2 iz Co So _— we No. of Cases: ° 9° @ S aS One Major Secondary Peak fo) fo) S ty Figure 19. 3 6 28 54 13273 258 142 &7 105 51 6 co) fe) (a) Nags Head 5 4 7 24 15 44 20 43 173 Fraction of spectra with at least one major secondary peak versus Tp. 45 18 Percentage of Records with Secondary Peaks at Indicated Perlods a to} (a) Nags Head Tp =6-7s 132 Cases 5) oO Use Right Scale h fo) bh fo) fo) a o ~ oO ips) fo) fo) Average Energy (cm®) Use Left Scale Peaks at Indicated Periods (o} Percentage of Records with Secondary at °o © a + a © < o =x = ° = B=) ° c @o a (0) 2 4 6 8 10 12 14 16 Period for Secondary Peak (s) 60 (b) Nags Head 50 Tp =12-13s Use Left Scale 105 Cases 40 30 Use Right Scale 20 oe ee oo 10 33 << os = ax 0) L—JQ 0 2 4 6 8 10 12 14 16 18 20 Period for Secondary Peak (s) Figure 20. Frequency of occurrence and average energy for secondary peaks as a function of secondary peak period at Nags Head, North Carolina. 46 (+S) fo} fo) 400 Average Energy (cm?) (a) Huntington Beach a (2) Tp =6-7s 24 Cases ES fo) ~ fo) Peoks at Indicated Periods w ro) Percentoge of Records with Secondary Period for Highest Peok 0 2 4 6 8 (OMI2e Sa: Pome onne 20 Period for Secondary Peak (s) (b) Huntington Beach Tp =12-13s 173 Cases a fe) oa 3 Average Energy (em?) L fo) Use Right Seale e) o L fo) fo) 20 Peaks at Indicated Periods 10 Percentage of Records with. Secondary a o © a > J @ <= o = Period for } (0) 2 4 6 8 10 12 14 16 18 20 Period for Secondary Peak (s) Figure 21. Frequency of occurrence and average energy for secondary peaks as a function of secondary peak period at Huntington Beach, California. 47 bh oO (o} 200 Average Energy (cm?) Figure 20(b) shows characteristics of secondary spectral peaks at Nags Head when period of the dominant peak is 12 to 13 seconds. Secondary peak periods of 4 to 9 seconds are common. In most cases these secondary peaks represent locally generated wave energy or swell waves generated near the Nags Head coast. The average energy associated with these peaks is relatively high. Secondary peaks at periods longer than 10 seconds are almost totally absent. At Huntington Beach, spectra with relatively short Tp have a very strong tendency for a secondary peak at long period (see Fig. 21,a). Locally generated waves at Huntington Beach are usually superimposed on long-period swell waves. Huntington Beach spectra with long Tp have a small tendency for secondary peaks with short periods (see Fig. 21,b). They also have a small tendency for secondary peaks at long period, which presumably correspond to secondary swell waves superimposed on the domi- nant swell. However, most Huntington Beach spectra with long Tp appear to be dominated by the main spectral peak. b. Spectral Peakedness. The spectral-peakedness parameter, > defined by equation (4) was computed for each spectrum. The spectral- peakedness parameter is shown on each plot in Appendix A. Cumulative distribution functions of Q) are shown in Figure 22 for the selected locations. The Q») values cover a wide range at most locations—from less than one to greater than five. values tend to be smallest for the Atlantic coast sites. The highest Qp values tend to occur for the Great Lakes gages, with the pressure gages having higher Q> values than the buoy gages. Q values for the gulf and Pacific coast gages tend to be intermediate to values for the Great Lakes and Atlantic coast gages except for the highest 5 percent of the Pt. Mugu Qp values. The cumulative distribution function for Pt. Mugu shows a rapid increase in Qp» values associated with probabilities of less than 5 percent. This feature is a result of occasional large overcompensation of the pressure spectrum which creates a high spectral peak at high frequency. Several cases in which Qp was greater than nine are omitted from Fig- ure 22. These cases all correspond to very low wave conditions except for several overcompensated Pt. Mugu cases. Goda (1976) indicated that high values of are associated with strong grouping of high waves. Pen-and-ink record traces for sev- eral cases with reasonably high waves and high values are shown in Figure 23. Some grouping of the high waves is evident in the records. Records with low Qp) values in Figures 13 and 18 show little evidence of wave grouping; yet the record in Figure 11 for high, long waves shows evidence of high waves occurring together, but the Q> value is only Sore The spectral-peakedness parameter might be expected to weakly corre- late with the number of major spectral peaks so that single-peaked spec- tra tend to have high peakedness. The data indicate a low correlation at some locations, but not at others (Table 6). Qp) also appears to be 48 Symbol * ) C) 1} Ga 4 v 0 t) x is} Location No. of Cases Atlantic City 1037 Virginia Beach 982 Nags Head 945 Lake Worth 982 Naples 866 Michigan City (Buoy) 246 Michigan City (Pressure) 127 Presque Isle (Buoy) 110 Presque Isle (Pressure) 119 Huntington Beach 746 Pt. Mugu 799 99.9 99 90 80 60 40 20 10 1 01 Cumulative Percent Figure 22. Cumulative probability distribution functions for spectral peakedness. Hy=184cem d=8.4m S| Tp=l4.1s 4320.87 2] Q)=6.7 G4 = 3-9 10s —s| —— Ho= 45cm Q) = 8.9 q3 = 0.27 s Tp =2.0s d=5.7m q% =3.1 —1 10s I— Hy=3icm = Qp=12.1 q3 =-0.08 =33 44 300cm od 10s k-y Figure 23. Pen-and-ink strip-chart records for selected cases with high values: (a) Huntington Beach record at 0100 e.s.t., 3 September 1972; analysis at 0040 e.s.t., 3 September 1972; (b) Naples record at 1220 e.s.t., 9 June 1969; analysis at 1340 e.s.t., 9 June 1969; (c) Michigan City (buoy) record at 2025 e.s.t., 14 September 1975; analysis at 2220 e.s.t., 14 September 1975. 49 Table 6. Correlation between spectral-peakedness parameter and other wave parameters. Location Cases Correlations } Cases | Correlations ® vs. abe Vs. Q | Tp vs. Q bit aun ah peo ay No. of peaks } (single-peaked cases} Atlantic City, N.J. ; 1,037 0.22 0.08 0.27 Virginia Beach, Va. 0.13 0.02 QO. Nags Head, N.C. 0.31 0.20 0. Lake Worth, Fla. 0.05 0.08 0. Naples, Fla. 0.06 0.16 0. Michigan City, Ind. (buoy) 0.09 0.27 0. Michigan City, Ind. (pressure) 0.06 0.61 0. Presque Isle, Pa. (buoy) 0.10 0.14 0. Presque Isle, Pa. (pressure) 0.06 0.36 QO. Huntington Beach, Calif. 0.34 0.24 0. Pt. Mugu, Calif. 0.22 0.25 0. weakly correlated with relative water depth as evidenced by a small cor- relation between peak period and for single-peaked spectra (Table 6). This may also be a result of the definition of Qo. p values have a tendency to increase with Hg for Tp shorter than the mean and decrease with Hs for Hg and Tp greater than the annual means. High values of Hg are often associ- ated with relatively high mean Q) for short Tp, but low mean Qp for long T,. The latter is mostly a result of the nonsinusoidal pro- file for long waves in shallow water. On the basis of evidence in this report and Goda (1976), it seems reasonable to hypothesize that high peakedness parameters are indicative of strong wave grouping, but low peakedness parameters in shallow water are not necessarily indicative of weak grouping, especially when the relative water depth is small. c. Skewness and Kurtosis of Sea-Surface Elevation Distribution Function. The distribution function for sea-surface elevation also pro- vides useful insight on shallow-water spectral statistics. Mean values of skewness and kurtosis (defined in eq. 6) for the selected locations are given in Table 8. It is evident that, except at the Great Lakes sites, the shallow-water distribution functions tend to have more high than low extremes (high skewness) and that they tend to be more sharply focused than the normalized Gaussian distribution (high kurtosis). These findings are consistent with cnoidal wave profiles shown in Figure 2. Cumulative distribution curves for skewness and kurtosis are shown in Figures 24, 25, and 26. The Atlantic coast locations have a tendency for higher skewness and kurtosis values than the gulf and Pacific coast 50 Table 7, Mean % values for each H, - Tp group.! Tp (s) Hg (cm} 2 3 1 10 to 30 | 30 to 61} 61 to 91 {91 to 122 | 122 to 152 | 183 to 213 WAWDWWUNONAANRWNH O ONAWrPowWwowonn dA RPePRPH to to to to to to to to to to to to to to 6 7 7 8 9 10 11 SOMO) [15:5 to Tannual means: Hs = 94 centimeters and T, = 8.9 seconds for Nags Head; H; = 87 centimeters and Tp = 13.2 seconds for Huntington Beach. WwW Ulan oO — acne 225 255 a 1.98 2.01 2.31 = 1.89 2.16 2) a — 2.00 2.46 WDWAWNANOWONADUNAWNHH O PR rr RPO WNNAANAPWN PH MWUAnrHOr + + «= « DNNWAW oo 2No cases in the Hg - p groups. 3parentheses indicate value is based on less than 10 cases. Table 8. Statistics on skewness and kurtosis of distribution of sea-surface elevations. Location Mean Mean Correlations sk i ewness kurtosis Hy vs. skewness Atlantic City, N.J. Virginia Beach, Va. Nags Head, N.C. Lake Worth, Fla. Naples, Fla. Michigan City, Ind. (buoy) Michigan City, Ind. (pressure) Presaue Isle, Pa. (buoy) Presque Isle, Pa. (pressure) Huntington Beach, Calif. Pt. Mugu, Calif. N° ATTEN OLR OT TD CEE EET OOO RIES Mt OLS LY SOT GAS 5 TTR Symbol Locaticn jto_of Cosas. | Penk Atlantic Ciry $037 t iss 2 C) Virginio Beach 282 = e Nags Head 945 c i) Loke Worth 982 = e a Noples 866 z- 2 8 Michigan City (Buoy) 246 ce v Michigan City (Pressure) 127 HM 0 so o Presque Isle (Buoy) - 110 oitos uae e Presque Isle (Pressure) 119 dexe B O 321 x Huntington Beach 746 i eae! . xeao Se a Pt. Mugu Uk Is ea ss & ere Bove 29 4 P e* 6x 8 @o Vv By | Mee Melacbess ot dai) aitak fat ie ace = Oj-—-— — o— — -o*%-e— w— O—M— -0— —y— — — — — —;] a BD xex eos & v0 ee e 48 +0 B29 e Ae vo e 4 vo Figure 24. 10 Cumulative Percent Cumulative probability distribution functions for skewness of sea-surface elevation distribution function. 9 z ae Symbol Location No, of Cases el * Atlantic City 1037 & | e Virginia Beach 982 ea a= [ ° Nags Head 945 = 7 B Loke Worth 982 elicies = ws ——Noples 866 28 Sp W eo 6; 9 ons oe Gk 3 G@e “<5 bees Su 5} Gc 2 72) cowed P 2 © ogt °° a” De S Spe 2 Normalized Gaussian a fer” oO Fe . . ° eos poo DSETbUTT ON od ge esi te wey ae) chalet eel eee Fats) enitkie:) mis B 9 Ox e,£ % Blk 6 pe ° 2 ny as 1 = 0 eae, Codes Laem eanuan ches $3.9 99 90 fo) S@ 3 10 1 0.1 Cumulctive Percent Figure 25. Cumulative probability distribution functions for kurtosis of sea-surface elevation distribution function at Atlantic and gulf coast locations. 52 Symbol Location No. of Cases Michigan City(Buoy) 246 Michigan City (Pressure) 127 Presque Isle (Buoy) 110 Presque Isle (Pressure) 119 Huntington Beach 746 Pt. Mugu 799 Normalized Gaussian Distribution Kurtosis of Sea-Surface Elevation Distribution Function 99.9 99 90 70 50 30 10 1 0.1 Cumulative Percent Figure 26. Cumulative probability distribution functions for kurtosis of sea-surface elevation distribution function at Great Lakes and Pacific coast locations. locations. The Great Lakes locations and Pt. Mugu tend to have the lowest skewness values. The Michigan City locations and the Presque Isle buoy gage show a tendency for occasional very high kurtosis values which are associated with very low significant wave heights. Although several kurtosis values greater than nine occurred for the Michigan City locations and Pt. Mugu, the values are omitted from Figures 25 and 26. The Presque Isle pressure gage shows a tendency for exceptionally low kurtosis values. A section of pen-and-ink record for a case with high kurtosis (equal to 8.0) was shown in Figure 11. Some of the kurtosis values in Figures 25 and 26 are questionably high. Since kurtosis is based on the fourth power of the deviation of sea-surface elevations from the mean (i.e., n = 4 in eq. 6), it is more affected by a few elevations far from the mean than any other parameters considered in this report. Noise or momentary signal losses in a record which are comparable to the size of the largest waves will pass uncor- rected through the editing process described earlier. These bad data points usually have little effect on the spectrum but they can have a Significant effect on the kurtosis of the distribution of sea-surface elevations. Another factor which leads to questionably high values of kurtosis is the occasional tendency of the step-resistance gage to become elec- trically shorted at a particular elevation, often by marine growth, and to produce wave records in which the troughs are unrealistically flat. Because kurtosis is especially sensitive to these problems (known to have occurred intermittently in the data considered in this report), 53 the percentages associated with kurtosis values in Figures 25 and 26 higher than about 4 should be considered somewhat inflated. The correlation between skewness and peak spectral period is less than 0.25 at all locations; for single-peaked cases it is similarly low. However, the correlation between skewness and significant height is moderatly high for the ocean and gulf coast locations (Table 8). There is a clear tendency for skewness to increase with significant height for the ocean and gulf coast locations as evidenced in a plot of mean skew- ness versus significant height (Fig. 27). This tendency is not evident for the Great Lakes locations (Fig. 28). It is apparent that cnoidal wave profiles (see Fig. 2) with very high skewness and kurtosis values are rarely observed at the selected gage locations. However, some of the records contain wave profiles which are more nearly cnoidal than sinusoidal. Figures 24, 25, and 26 indi- cate that cnoidal profiles appear in about 10 percent of the records or less. V. CONCLUSIONS The following conclusions are drawn from evidence and discussion presented in this report: 1. About 60 to 70 percent of the ocean and gulf coast energy spectra in shallow water have one or more major peaks in addition to the dominant peak. 2. At least 50 percent of the spectra in Lakes Michigan and Erie have one or more major secondary peaks. 3. Major secondary spectral peaks are common for both high and low energy spectra. 4. The very small sample of spectra for hurricane waves examined (seven spectra) does not reveal any special characteristics of hurricane- generated spectra. 5. Major secondary spectral peaks are usually indicative of secondary wave trains. 6. When secondary spectral peaks occur at frequencies of two or three times the dominant frequency, they may represent a single train of steep waves rather than independent wave trains, especially if the relative water depth for the dominant wave is small. 7. When secondary spectral peaks occur at frequencies of 0.5 or 1.5 times the dominant frequency and the waves are high enough for breaking to occur, the secondary peaks may not represent independent wave trains. 54 12 c o3 = s 2 10 ivy $s 5208 marr o as 33 Qé Bo 3 0 Sino af sn x symbol Location No. of Cases Pr [ ) * Atlantic City 1037 . x CY) Virginia Beach 982 e x ) Nogs Head 945 = a Loke Worth 982 rane Naples 866 8 : x Huntington Beach 746 ; a ts) Pt Mugu i 799 | 122 183 244 305 366 427 488 Hg (cm) Figure 27. Mean skewness of sea-surface elevation distribution function as a function of H, for ocean and gulf coast locations. 03 jmbol Location No. of Cases Michigan City (Buoy) 246 Michigan City (Pressure) 127 Presque Isle (Buoy) 110 0.2 Presque Isle (Pressure) 119 ) 0.1 t 2S = Distribution Function i] S n> Mean Skewness of Seo-Surface Elevation ' S w OL on Gh Ol “Aen ke Aa ae eS Hg (cm) Figure 28. Mean skewness of sea-surface elevation distribution functions as a function of Hg, for Great Lakes locations. 55 8. There is evidence that wave breaking limits the ratio H,/d at the shallow-water gage locations considered to less than 0.55 rather than the commonly used limit of 0.78. 9. Individual shallow-water ocean spectra with comparable signifi- cant heights and peak periods are highly variable in shape. 10. Mean spectra for the same significant height-peak period group show a surprising similarity between locations; however, there is a nota- ble lack of low-frequency energy in gulf and Great Lakes spectra and a lack of high-frequency energy in most southern Pacific coast spectra. 11. The distribution of individual energy values about the mean energy for each spectral band tends to be nearly Gaussian in the vicinity of the dominant peak but non-Gaussian for low-energy bands away from the peak. 12. For most exposed ocean locations, a spectrum is more likely to contain major secondary peaks when the period of the dominant peak is long or short than when it has intermediate values. 13. High values of the spectral-peakedness parameter, Q» indicate that high waves may be occurring in groups; however, high Q, values in shallow water are usually associated with low-energy wave conditions. 14. Low values of Q. generally indicate that high waves are poorly grouped. However, if the spectrum has secondary peaks which do not repre- sent independent wave trains, Q . is not a direct indicator of the extent of wave grouping. e 15. Q, values tend to be higher in the Great Lakes than along the ocean and gulf coasts. 16. The distribution function for sea-surface elevations in shallow water tends to have more high than low values far from the mean and tends to show less variability than the normalized Gaussian distribution. These features are consistent with cnoidal-type, shallow-water wave profiles. VI. SUMMARY Digital wave records were obtained from 11 gages at 9 shallow-water sites along the U.S. Atlantic, Pacific, gulf, and Great Lakes coasts. The records were analyzed with a fast Fourier transform procedure to produce energy spectra. Selected spectral parameters, including the number of major spectral peaks and Goda's (1970) spectral-peakedness parameter, and parameters of the distribution function of sea-surface elevations were also computed. The physical meaning of the shallow- water spectrum and computed parameters was discussed and illustrated with three cases of cnoidal wave profiles. 56 Spectra for the 24 highest energy cases from each gage were summa- rized individually. Many of the spectra had more than one major peak. Although many of the high-energy cases represent breaking wave condi- tions, the ratio of significant height to water depth was less than 0.55 in all cases. This value differs significantly from the value 0.78 which is usually used in conjunction with depth-limited wave heights. All spectra were grouped according to significant height and period corresponding to the highest spectral peak. A mean spectrum was com- puted for most height-period groups. The standard deviation of spectral values about the mean was also included. The mean spectra show a clear lack of low-frequency energy at the gulf and Great Lakes sites and a lack of high-frequency energy at the southern California site. They also show evidence of systematic variations in shape as a function of signifi- cant height and peak period. Multipeaked spectra are common; 60 to 70 percent of the ocean and gulf coast spectra and at least 50 percent of the Great Lakes spectra have more than one major peak. Major spectral peaks are usually indica- tive of independent wave trains. High values of the spectral-peakedness parameter appear to be related to the occurrence of high waves in groups. However, low values of the spectral-peakedness parameter are not necessarily indicative of a lack of grouping of high waves in shallow water. The distribution function of sea-surface elevations in shallow water tends to deviate from a normalized Gaussian distribution in a way which is consistent with cnoidal-type wave profiles. In the course of this investigation, a computer program was developed to identify major peaks and valleys in an irregular signal. The program has many potential applications, including identification of major peaks in a spectrum and identification of meaningful crests and troughs in a time series of sea-surface elevation. 57 LITERATURE CITED BARNETT, T.P., "Wind Waves in Shallow Water," Final Report, Westinghouse Ocean Research Laboratory, San Diego, Calif., June 1969. EDMISTEN, J.R., "Toward Fulfillment of An Urgent Need, Coastal Wave Data Acquisition and Analyses," Shore and Beach, Vol. 46, No. 3, July 1978, pp. 3-14. ESTEVA, D.C., "Evaluation of the Computation of Wave Direction With Three-Gage Arrays," TP 77-7, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., July 1977. ESTEVA, D.C., and HARRIS, D.L., "Comparison of Pressure and Staff Wave Gage Records,"' Ch. 7, Proceedings of the 12th Coastal Engineer Con- ference, American Society of Civil Engineers, 1970, pp. 101-115 (also Reprint 2-71, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Washington, D.C., NTIS 732 637). : GODA, Y., ''Numerical Experiments on Wave Statistics with Spectral Simu- lation," Report of the Port and Harbour Research Institute, Japan, Vol. 5 NOs Ss SGaes MOVO, 0 S-Sio GODA, Y., "Estimation of Wave Statistics From Spectral Information," Proceedings of the International Sympostun on Ocean Wave Measurement and. Analysts (WAVES '74), American Society of Civil Engineers, Vol. 1, 1974, pp. 320-337. GODA, Y., "Irregular Wave Deformation in the Surf Zone," Coastal Engt- neering tn Japan, Tokyo, Japan, Vol. 18, Dec. 1975, pp. 13-26. GODA, Y., "On Wave Groups," Ch. 6, Proceedings of the First Conference on Behavtour of Offshore Structures (BOSS '76), Norwegian Institute of Technology, 1976, pp. 28-41. GOSPODNETIC, D., and MILES, M.D., "Some Aspects of the Average Shape of Wave Spectra at Station 'India' (50° N., 19° W.)," Proceedings of the International Symposium on the Dynamics of Marine Vehicles and Struc- tures tn Waves (IMECHE '74), Institution of Mechanical Engineers, 1974, pp. 19-32. HARRIS, D.L., "Wave Estimates for Coastal Regions," Ch. 5, Shelf Sedt- ment Transport, Process and Pattern, D.J.P. Swift, D.B. Duane, and O.H. Pilkey, eds., Dowden, Hutchinson § Ross, Inc., Stroudsburg, Pa., 1972, pp. 99-125. HASSELMANN, K., et al., ''Measurement of Wind-Wave Growth and Swell Decay During the Joint North Sea Wave Project (JONSWAP) ,'"' Deutsches Hydro- graphisches Institut, Hamburg, Germany, 1973. HERTERICH, K., and HASSELMANN, K., "A Similarity Relation for the Non- Linear Energy Transfer in A Finite-depth Gravity--Wave Spectrum," Journal of Fluid Mechanics (in preparation, 1980). 58 HOFFMAN, D., '"'Analysis of Measured and Calculated Spectra," Proceedings of the Internattonal Sympostum on the Dynamtces of Marine Vehicles and Structures tn Waves (IMECHE '74), 1974, pp. 8-18. HOFFMAN, D., "Wave Data Application for Ship Response Predictions," Final Report, David Taylor Naval Ship Research and Development Center, Bethesda, Md., Oct. 1975. HOFFMAN, D., and CHEN, H.H., "The Use of Directional Hindcast Spectra Design Wave Data," Proceedings of the 10th Annual Offshore Technology Conference 3279, Vol. 4, 1978, pp. 1989-1996. IRIE, I., "Examination of Wave Deformation with Field Observation Data," Coastal Engineering in Japan, Tokyo, Japan, Vol. 18, Dec. 1975, pp. 27-35. KENYON, K.E., "Wave Refraction in Ocean Currents,' Deep-Sea Research, Vol. 18, No. 10, Oct. 1971, pp. 1023-1034. 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. OCI, M.K., and HUBBLE, E.N., ''Six-Parameter Wave Spectra,'' Ch. 18., Proceedings of the Coastal Engineering Conference, American Society Ora CvalEnginecers), Vol. 1.) 1976), pp S0l-3528) PITT, E.G., DRIVER, J.S., and EWING, J.A., "Some Inter-Comparisons Between Wave Recorders,'' Report No. 43, Institute of Oceanographic Sciences, Taunton, Somerset, United Kingdom, unpublished, 1978. RYE, H., "The Stability of Some Currently Used Wave Parameters," Coastal Engineering, Elsevier Publishing Co., Amsterdam, Netherlands, Vol. 1, Mar. 1977, pp. 17-30. SAWARAGI, T., and IWATA, K., "Wave Spectrum of Breaking Wave,'' Ch. 33, Proceedings of the Coastal Engineering Conference, American Society of Civil Engineers, Vol. 1, 1976, pp. 580-594. SEELIG, W.N., and AHRENS, J.P., "Estimating Nearshore Conditions For Irregular Waves," U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va. (in preparation, 1980). SHEPARD, F.P., and INMAN, D.L., ''Sand Movement on the Shallow Inter- Canyon Shelf at La Jolla, California,'' TM-26, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Nov. 1951. 59 THOMPSON, E.F., ''Results from the CERC Wave Measurement Program," Proceedings of the International Sympostun on Ocean Wave Measurement and Analysts, American Society of Civil Engineers, Vol. 1, 1974 (also Reprint 7-74, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., NTIS 002 114). THOMPSON, E.F., "Wave Climate at Selected Locations Along U.S. Coasts," TR 77-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Centex, Jane to 7: THOMPSON, E.F., ''An Evaluation of Two Great Lakes Wave Models," TR 78-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Oct. 1978. U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protectton Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., Ws) dha Aoyd Hoio VINCENT, C.L., and RESIO, D.T., "An Eigenfunction Parameterization of A Time Sequence of Wave Spectra,"' Coastal Engineering, Elsevier Scientific Publishing Co., Amsterdam, Netherlands, Vol. 1, No. 1, 1977, pp. 185-205. WANG, H., and YANG, W-C., ''Measurements and Computation of Wave Spectral Transformation at Island of Sylt, North Sea," Technical Report No. 3, Office of Naval Research, Washington, D.C., Nov. 1976. 60 APPENDIX A INDIVIDUAL HIGH-ENERGY WAVE SPECTRAL PLOTS 61 *Aostor Mon *AQT) OTAueTIV ‘szoTd TerzD0eds oAeM ABIOUS-Y3 TH (TH) ADNANOIes 3-0 v0 s"0 2-0 to oo loodatm a 3 3 2 loodozS = oe loodos = 3 8 SS = x joodor ~ ‘o0d03. (2H) AIN3NOIws Or) 50 z"0 tro o-0 T aT 0 joodolm 3 2 joodoz 9 2 S loodos = 3 8 = t =guuad" ON 5 62° 1 240 joodar — 8 11°68 *uad W290 =1H ALID Jiine die ‘oodos (2H) AIN3NOIMS a0 0 6-0 20 to oo 2 *8Nuad" On 86° 1 #40 soz Zt sydd WI 292 21H ALI) Jinetiy (ZH/ZHI)AL16M30 4903N9 (THD AININOIuS a0 vo 6°0 z-0 to oo 3B =9)b3d" ON @O°t =d5 Cate uid WW) 662 51H ALIJ QEAMB VL (ZH/ZHI1AL16N390 A043N3 (TH) AINaNOIus a0 70 6-0 20 to GO G joodot > a & joodazg z S joodos = 3 8 SS 1 =suuaa:on a 4 = S 60° *u3a oe W3_9rz 21H VORTEC CRTC) = ——— SS joodos 63H) Awan Iu + or) 60 2-0 to oo) ads ies 3 3 2 loodos & & 2 loads» = 3 $ 2 ssyuad: on a a vt agi = 8 6L°9 suid pore M2943 21H Abt JILIN TB foods. 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oo or) 00 6"0 a0 Oe o-0 9-0 "0 6:0 30 bo 0'0 }ogb I “te } iW Sedud 216 (2H/7M3:4118990 L08INI (ZW/7U9 4118830 AoaIeI CEM/2MI 4116990 AOuRRI 5 oy 3 poade yoobo = nd WOG'e 82971 WrLUs 3 © eSuead' ON nN 9892 loocbo ~ (ZW/2H9)4116"30 ACwaID (TH) Adnanoaus a0 La) a) ao to o-0 : looobs g 3 obs = s 3 sgnyae: z eer looobe ~ " nonw id ooobs (aH) AINANOAYS so vo 6-0 20 to oo o R 2 é g = s a 6 =Syy * 31° = (2H) AQMNANB AWS a0 0 60 0 to 0-0 m § <. g 4 & es a a Fy Sy = = Tetzoeds oAem ABLOUN-UZTH (2H) AINaNOT 60 20 ro o'o i g R & 3 Ss ‘ o A g & 2 rs) 3 IN 3 *S¥¥3d" On = (2H) ADNANDAYS 90 0 i) 30 10 9°09, 001 i obs R a Pt poe = locate ~ joobe "TI-V ean3ty (IH) AINaND as so Oo a) a0 ro oo (aH) ADNANDIWs et) 0 a) 20 Vo oo (v4) AININDIYS so ro fo te to a) L =ss¥qu" ON 0 CP) ENN te (ZH/7H3)AL18M20 AOUINS (2H/ENIALIGMSS Cuan, 93 61100 980°0- 0s00°0 SO" 0 TgZ0° “0 82°6 ata a' 760 °0 $£00°0 TOTO*O 99T °0 £090°0 9S°6 $9°9T 82£0°0- 8820 °0 oss0*0 8ST °O 0607°0 90°0T | 26°S 080°0 zS00°0 8T10°O 2st ‘0 0SZ0°0 97° OT aaa As 9LT 0 8£00°0 6600 °0 9S1°0 z£90°0 ce OT S9°9T Opt °O- 7870 °0 8fs0°0 p9T*O £907°0 88 °6 L6°S 89T °0 Z'00°0O bczt0°0 pLT°O T1TZ0°0O 672°6 cL * er 60T °0- 0Tz0°0O Sc£0°O poop i) vs9t*o Ov's 8£°9 c7z°0 zc00 °0 8800 °0 pst °o LLb0°0 T6°8 87°02 8S0°0- L1T£0°O SZ££0°0 L9T°0 ¢pe7z°0 8L°6 OS z00°0 ££00°0 94T0°0O S61 °O TO60°0 Lv'8 v8 °Oor pzt°O £00 °0 8410°0 961 °0 9060 °0 ss°8 bs °or 860°0 £120 °O 86c0°0 ST ‘0 OOZT*O 78 °6 Ss3°9 820 °0- £920°0 8920°0 8st °o T96T °O 0z°6 L6°S 020°0 S00 °0 TT10°O Sst °0 z090°0 Tp°6 S9°9OT 6bT “0- T¢70°0 Orso*o 061 °O S6ZT “0 vz°6 se°9 6 227° 0- 2020°0 L£9£0°0 0c72°0 S99T*O 97°8 82 °9 8 9L£°0 8860°0 OzTT*O OST *O STc9°0 8Z°OT | 92°¢ L $720 °0- $800°0 Z810°0 z6T °O ¥£60°0 vl°6 vs°or 9 T60°0O £300 °O P6T0°O T0270 9960°0 6S°6 v3 °Ot $s $60°0- $920 °0 zZt0°O biz°0 696T°0 927°6 SL 6S 4 6L0°0 6600°0 s£70°0 07z2°0 890T°0 07°6 2-3) £ 879°0 £¢00°0 9£T0°0 ez °0 ¢ss0°0 £6°8 SORT G 6770 °O crbd*O 8Tc OGST*O 96°6 829 T g0ejFins-e9s FO uorqzouns (w) (s) (wo) di8/p P/*1 dap P oh aH owt aqeq | ON UOTANGCTLYISTP POZTTCWION "eTULOZT[eD “non ‘Iq ‘erqoods ASLoua-ysTY TUNPLATPUT IOF stajyouezeg “TT-V 9TqeL 94 APPENDIX B MEAN SPECTRA GROUPED BY HEIGHT AND PERIOD 95 Table B-1. Approximate relative depth and ratio of signif- icant height to depth for mean spectra grouped by height and period. Parameters based on period | Parameters based on height i (s) (cm) Atlantic City DEPNOCHOWAUIADNAW NPWUN@WOHLOHLUWMN fad fed fed fed PNOWMBMDUDAUVAN KWoOUMPOhMHAUNUY aa DENOUWOABDNAQAM OF 26 Cieeounts NEFPUWUwWNAOhAAMN OOFOPOLOrOLrcOcOKOrOre a ey 96 Table B-1. Approximate relative depth and ratio of signif- icant height to depth for mean spectra grouped by height and period. --Continued Parameters based on period | Parameters based on height RPWoNntfuunnn SO SCY SOeyoyres) Ot) S) ocooooo0o0o0cceoo ° 2.5 3.5 4.5 5.5 6.4 6.8 7.4 8.0 8.8 Yoo oooo0ooocooco ‘jJooo0ooocoo°ec°eco DIDAH HPWH OoOfoahunNuUunn ooooooocooco 97 Table B-1. Approximate relative depth and ratio of signif- icant height to depth for mean spectra grouped by height and period. --Continued Parameters = on period | Parameters based on height (s) (a) Presque Isle (buoy ooooocoocco WNFWeONWMDOP OAM ooooooooocoeoco Pt. Mugu WORMONIDAVAW 7 Ol oO on 6 Oo 6-00 6 5. 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Ay 8 S e302 91 & Sasu2 02 x avai t a So-Ciutd loodbz ~ 1-160 e 6 nonw id bina ty " ons ~ + 4 1 nods “TZH/ZUTIALIENZO OWRD 136 (2H) AoNaNDIWs 270 to 0-0 ao Oo eo 2 a g & & 3 3 = s3su) | 3 $ €€2-9817uId Md 261-2200 8B 6 NOW 1d (1H) AaNanDaus 3:0 0 6:0 20 to 9-0, ood t & g jooobz 2 & sods = S 8384) Ol & S$ C02-9Bi sud loodby ~ Md 221-162 € non id (2H) AdmaNOgud a0 La 20 to o-o (2M/ZU21AL19MI0 AONBAD (1H) AdMano aNd "0 "0 60 270 0 0-0 loos i g oodbs & & e] o0dRe = a = 635) 1h = 3 ooobe ~ bw mun id oogwL (2H) A9Nanb aud 3-0 00 6-0 20 to o'0 o & 2 Et 2 8 s 3 = 3 826) 99 a § 901-G Siu AD 221-16 sae 6 MONW id (2H) AININDIWs 30 £ 60 a) S3su2 tt S901-¢ 20 17H) AINANOIYA 90 0 co zo UMD ALT EMIO 19NIND $3sv) 12 SOM-C Sit wIE A219-OF am © nonw id (2H) AQNaNOaed a0 Oo a) 2-0 to 0-0 (ZH/Z2M3 1411620 L04¥9N9 6360) oO SO SI-C Ct: u3d BI UGt- 22 aw & nonw id =: (TH) ADNANDAY 80 0 6:0 20 1-0 oro, ooo loca & pods = 8 Sagu2 ty a 6 CEI CEI ude ode ~ AI 2-1 LH 6 nonu id loodbs (1H) A2NaNOaNd 3°0 0 6-0 20 a) 0-0 0 008 0001S & ods = Fo = SEGI-€ jogbe - AD 16-19 1 6 nonw id Titi (1H) AQNINODeS a0 0 60 zo ro 0-0 g2evd LE SOC C Crude RI 19-Of LH 6 nonw id CZH/ZM2ALIGM2O ADNIND (2H) AININDIYd loodbi a > g loodoz & = & =) joovE = 8 NY crn) ob 4 SOCI- 9 UIE odor ~ AD 21-2200 dw 6 nonw id oocos (1H) ADNaNOIWs oo Ca) a) 20 to 0-0 2 a 3 $ ¢ a = S 3 ry $ Fs = S38) 2 a SECI-9i UI AD 221-16. 1 € non le (1H) ADNaND awd v0 "0 60 20 to oro, loos = & 3 $ C} oocb1 S & S| loots! = 3 83u) 99 = z ; loocbs ~ oooks (7H) ADNAN 8-0 Or) a) 20 10 9-0, loodz = & Y & g lobdr g B 2} ooo 5 a 137 (THD ADNANODD a0 La 0) 0 io oo AJNANDAds aol "0 20 10 oro sisud§ 8OW-CCI Odd MO 26k ah Ae (IH) AINANOIeS so H 6-0 a0 S28v2 Ol Ch DEC fH WI 221-16 218 6 non id sav) Lt BON-COlgId AD 16-198 48 6 non ad (2W/2wI ALISO ASTD (24/243) 411630 1.08383 a0 La) 0 2-0 0 o'o (7H) AIN3NDAWs RW) 2G1-22he dw # Rorw id (2H/2M2 AL 18890 AOURND a0 (1H) ADNINOIME a F°O 20 s3su) ot SG@-O4:03d WI 260-22 aw 6 nonw id (24) AINANDaUd ro a) z0 io 0-0 33 a 3 2 OT01 S 3 z = Situ = -225024 jocbz ~ Wd 261-2202 bw 6 nonw id 4 — ——— ooR7 138 (IH) ADNANDAUS so Oo 60 8-0 lo o°0 : : (ZH/7MI 4119830 AOUIND g Es Sasu2 ' SEEI- OG UI HD COi-zGheaw 6 non te 3 (2/79 14118030 165203 TEM/2UTTALIGMIO A009 SO N-COIbIe WI CI2- COV w (7H) ANAND Aus (UH/EMIALIENsG AOURND (ZH/2UIALIGHIO AONaND (Hb AsniOsed 8'0 oo Go 20 to 0-0 8abud 1 $6 9-98 ude Md boz-C1Zehe 6 none id (2H) A2NINOIUE 30 a} a) 30 to o-o (2H/2uI) AL 16820 ASyINI (2H) AdNanoawa oO v0 6-0 a0 to 0-0 6asu) iy (2H/8u9 4118820 ASUaI SoCi- 902 9d 2 €O1-2Gh9 40 6 nonu td 139 APPENDIX C MEAN SPECTRA GROUPED BY HEIGHT, PERIOD, AND WATER LEVEL Interpretation key Mean Water level 1 low 2 mid 3 high SSS 140 12H) AINIPDIws so oO 60 z0 bo 0-0 6 =Ryau $389 9 $€0I-C6+u3d po Aad OF4H Quah SObN +. 4. = ‘0000; PE b3bbd FI $CO1-€6'UId 219-08 ' 1H O¥Jh SOUn (70) 43N3ND38> vn ro FO 20 o 3 rc x lpa0¥ ooo a 2 19-0C 11H Ova SOUN godt 6 Z4/ZH9 1) LTGNIO AO8INF (ZH/ZHI A 1IGNIO A0G3NF iD ADIN 3 (7H/ZHD Aa 1G 1) ADNINDISS § SNbIW bb) 6 S€6-GO: U3 AD 19-08 s AW QU7H GOON La (7h) AJNIND IS Ct) 20 2 [mya S3Sb) AZ S€6-COruId AD 19-OF 1H Qua4 SOUN v cey3y $38v9 (2 S€6-GO'u2d Le) Of an QbaH SON ELE] A ZH/2MI)ALIGNIO A9N9NF (7hs ZHI 15 1L1GNID J 0d: (ZH/ZHIIALIGNIO J 0¥9NF y AIn3NDIY4 a0 a) 2-0 to a0, 002 in 2 3 3 ooor = 3 : 9008 5 3 8 6 =NUQY > s3sud Lk a Leuia looda ~ RIIF-OC 11H Ovit GOBN a a 4 4 20001 a7hd ADNAPDSad a9 Ue) £0 29 99, s00z = A & 0 poor & = é a looos 5 g 3S 3 onyau = 33$U) 22 5 $60-Oc9u3d jos0e A? 19-08 an QU3H SObN —e 4 7 4 0901 .7H) AInINOId so ro eo 290 to 0507 loooz = a > s ooo © z & loop = 3S 1 ewe 3d > BIS 92 = S60-0%1¥3d pode A? 19-069 ba Ob3H SOuN n — 4. 9000 i i) , oO w7H) 7 ININ03NS ae) 20 ig 6 ENYIN sash) oL $Ou-24' 3d RI19-CEhe dH Qv3H SObN 4 4 4 4 so ia (7/4) AININOIUS fo 20 2 ©NYIW 63Su3 al SOu-22:U3¢ AI 19-OC 3m Quail SOUN a 4 4 (1H) 43N3NOIWd 6-0 20 to v onvau s3sv) SOu-22:uId WI 19-06 o Aw QuzH soUN joods ooo (74/ZWIIILIGN3O A0b3hF AObINI (2H/2M 14418630 (2H/ZH/AL1GMIO A0¥INI 141 (7H) ASNINOIUS .2H) AININOJed a0 f0 0 oo, so a) 20 to 0-0 loos & S joodbi e = fr] = loons! = 3 6 ony SS 6 =wyau easua Fr a sau) 9 $6 6-94: U3a oadbz ~ $9-C 1 uid code A) 16-19: 1H AD 16-195 1H QUsH SObn QuIh SObN looduz 1 2 a: ‘0001 (HI LING dds (7h) AIN3N03b3 s0 v0 6:0 z70 bo 0-0 so 6-9 2:0 10 o'0 — t) aes locos & z Ss 2 looo01 E loot = 3 pus 3 2g Sas) 6 a 6abu2 $6 8-84=U3d oodbe ~ $9-6:U3d WD 16-19 1H WD 16-19 21H Qu3H BUbH QU3H SON oogez (7Hy JINBNDSHS € 20 70 2 AQNIND VS 6 ery bis) 21 SS O-84:03d WD 16-19: LH Qu3H BOUM omuay eisbo OL $9-GUid WD 16-19: 1H 0¥3H S0bh 1 4 4 joo08 oogv! FE (24/7H IF 118690 408 (ZM/ZHALIGAIO J0UINZ (ZH/7W35)118K30 4003962 AININDGds z0 Us, G6 ENUM sisu) 8 SECI-9 UI AD 19-OF 1H OvaH SOUN eee a TH) AINANDIUs so eo 20 : o-0 a ; C) ooo! loooz 000s 3 eM $35) 9 é Seei-9 U2 psp W) 19-0C'2 Qua S00N == 4 ze 1. loo0s (7h) AININO Pd 20 Oo O° so ro FO SeCi-91 RD 19-O0F 1H ‘QUIK DUN — may — + O08, (ZH/ZMI)ALi6R390 ,OU9NF 200545 30 (Dh/7 ALIEN 6 ZH/7H34419K30 500387 6 =NUIH $3Sv) 2 COluid #2 19-O0F 1H Ob3H SObN — =: AZ) AQNINOIEY oO 6-0 770 2 syd $36 Ol SOU-COI UI WD 19-OC 1 AN Oy3H SOUN n = 1 oRyzy Sib) orl S9N-C Old WD 19-00 84H ObI1, SOLN 4 4 \7S) AININDIS 20 OOF 0008 (Z4/Z7WIIALIGNIG A0bINF aa" (7/291) L1GN390 190: «ZH, ZI 14118630 108 142 (2H) AIMANDIUS a0 0 €0 2:0 to o-0, loci = a 3 loopoz o a loodbs = Fd fe loodor ~ OD 261-2217 Ah O¥3H SOHN rn 4 He 4 ooDs 0 9 ONO em z ° & lpoa0. 2 & naning = 3 3 ony Iu s s3su) 7 ES 89-G2UR¢d jonone MI 261-2200 Aw OH3N SOW — — a — —noons i2Md AININD AMS so v0 60 2-0 to 00 0 oot ir & ‘ joo007 2 z 3 = qs ila T CRU DW : sisu) 2 a $9-G° 02d loodby - 2 261-2208 Aw Quah SOUN (7H) AININDIb4 a) 0 2-0 ro so 30 $60I-6G2U2d maT Fd 4 m Z 3 | u g joo99t = 3 § =Hyau > sisb) Zt a | $C ON-C6sUld Phelte © Wd 221-16 1H 2 eth SObN 4 4 + 00:92 (74) 1 0N3PO3M4 vo es) 70 9 oo Sa. = a ees eae, Se ~ t none = anon , z y 2 4aocs1 5 = 2. =NBIM > $78u) 91 a SCOI-CGs ude qeneinz Wd 221-162 40 ‘OBI SObN 4 4 a + noas7 47H) AM3ND3Ud 20 6:0 z0 to 0°3 <1 t) om = fd 3 6 g g e = 8 = 5 ey > 93609 1 a (TH) AIMINOIed a0 0 6-0 zo to oo 3 joooot 2 z 3 jooast 5 De G My2u 5 2 SHSHI IL & $C O1-C6*UId ong07 ~ loo097 so 9 4 £ é nana 2 ® & z foots 1 = = 2 Dey ah 5 _. S3Sb) at ra r $6'O1-C6 urd jonnaz ~ 2 16-19: 1h UH SOBN 7 = : o90sz 1H) 49N3N03Md v0 v0 60 20 to 0-0 3 2? & 3 1 Nea S sis) Ut z {> $©00-668uad joog0z - #2 16-190 48 QbaW SUbn — 1 .- 1 — ‘90092 (1H) 49N39PB3US 60 2-0 ia = a 2 s 2 2 = = = looos = 3 x G6 omyay = s3su) EN E $06-CO> uid joooe — > 16-19 AH OvIH SNUN = 4 = + loodot (TH) A9N300Fds €0 0-0 ) ooo? = Ea 3 3 990% © z = ii n099 5 4 Es 2 ENUIH S _,§38b) St i = SO6-CO suid ljosda ~ A) 16-1929 AW Quah SObN — rn ap 50091 6) AIN3NDId> 9 70 we) 0-0 C) loooz sn a 3 S ooor © = =) looo9 = 2 = SJ = = loose ~ Q¥3H SObN ————— out 143 (UW) AdNaNOIUY so oO "0 a"0 to o°0. 6 oMesy sis) 8 oURd 2 COr-Z617 AH OU34 SOvn (Urs 2H) o418NIF A00IMmd (TH) AIN3NOId 6-0 a0 3 S Ee (ZA/ZUI)AL18M30 100383 g Ef 3 Gum suuN 2 Z (TH) AININDIUd ao 1 90 6-0 2-0 to SCGi- ON UIE Wd 2Gi-2tre am QUIN SOUN LOM SLEU-ETT) 90 0 FO 2-0 to =NUaY sibyr O8qH Soh 4 4 —e i joooot js00st on07 onda 7M/Zd2 124i GRID 1 D09N7 (ZH/7H3 14118830 A0B2m9 (7H) ASNINO Id a) 000 I = a4 s edb? F 3 loogbe iS 3 Z joooDe - AD 261-2208 bh ava! SOUN —!. —t—_____ 00009 (7H) AINIOIeE go Oo 60 z0 to 0-9 & 4 ’ 3 ooo © 2 a ‘conus = 3 8 Bi nooor ~ 2261-2202 AH OWIH SOUN 4 4 - nogoe (74) AININOIed v0 v0 FO zo to oo ZH/2d2AL1GMIO 49077 Ibi SOLN (ZH) AINANDIVs & 3 ooo & & 2 si a § oy au = 8350) t) ~ S$OU-202 UId CCo0s RD 2Gi-22r 2 aw OUIH SOUn oooyz (2H) AININOINS 30 é eo zo a ey Sisv) 8 0U-24 2 UI RD 261- 2202 Aw QH7H SOUN (eee Sees n — 2m/2a214415N90 290Im9 (TH) AQNINOIMY v0 0 eo 20 to 0-0 (TM/2M 314415970 10099 MI 261- 82h bw Ov TH SOUN re 144 APPENDIX D DESCRIPTION OF COMPUTER ROUTINE SMOOTH FOR IDENTIFYING MAJOR PEAKS AND VALLEYS IN AN IRREGULAR SIGNAL The computer routine SMOOTH is useful for deleting small, inconse- quential peaks and valleys from an irregular digital signal. Peaks and valleys that remain after application of SMOOTH represent major extrema which in many cases are more meaningful than small wiggles in the signal. SMOOTH is a very general routine which can be applied to many differ- ent Situations. It has been applied at CERC to the following three situations: (a) Computation of statistics on individual waves. SMOOTH was applied to a time series of sea-surface elevations: (b) Computation of statistics on tidal highs and lows. SMOOTH was applied to a time series of tidal elevations. (c) Identification of major spectral peaks. SMOOTH was applied to a wave energy spectrum. The operation of SMOOTH is most conveniently described in terms of its application to a time series of sea-surface elevations, although its other applications are analogous. The general scheme of operation con- sists of a check on the time difference and elevation difference between successive extrema. If either is less than the specified acceptable min- imum then one peak and one valley are deleted from the time series. The input to SMOOTH consists of several control parameters and an array (EXTIM) containing time and elevation for each extremum in the time series. Figure D-1 shows five extrema in a hypothetical time series. If the point labeled ''-1" were the first point in the time series, then the first 10 values in the EXTIM array would be t_y; N-y> to, No> ti> Ni» to, No» t3, and N35 where t; and nz; are defined as the time and elevation associated with the ith point. The control parameters which must be specified are: FURST = time associated with the first point desired in the time series. ITEMS = total number of values in EXTIM array (= twice the number of extrema). CHP = minimum acceptable time difference between successive points (critical half period). HMIN = minimum acceptable elevation difference between successive points. 145 Cia nie) / “XN YY zy [ (tgs) ” Figure D-1. Hypothetical time series. After completion of SMOOTH, the times and elevations of major peaks and valleys are stored in the first ITEMS elements of EXTIM. The value of ITEMS has been reduced in accordance with the number of small peaks and valleys eliminated. Since the smoothing algorithm cannot work prop- erly at the end of a record, the last few points are usually accepted regardless of whether or not they satisfy the acceptance criteria. Thus, the last six elements in EXTIM may not represent major peaks or valleys and should either be checked or categorically eliminated. A complete list of subroutine SMOOTH with comments is provided in Figure D-2. A flow chart for the main processing loop (identified as loop 1520 in the listing) is provided in Figure D-3. The symbols nj and t; and the point numbers referred to in the flow chart are con- sistent with definitions in Figure D-1. Tpyp and Hyry in the flow chart are equivalent to CHP and HMIN in the listing. 146 SUBROUTINE SMOOTH (FURST> ITEMS) SUBROUTINE SMNUTH TAKES A RECORD OF PEAKS AND VALLEVS AND ELIMINATES YNCONSEWUENTITAL PrakKS AND VALLEYS, THE ELIMINATION CRITERIA ARE A MINIMUM DIFFERENCE BETWEEN PEAK AND VALLEY ELEVATIONS (HIN) AND A MINIMUM HORIZONTAL SPACING (BE IT TIMEs POSITIUONe ETC) RETWEEN PEAK AND VALLEY (CHP), INPUT PARAMETERS ARE DEFINED aS FOLLOWS) FURST & STARTING TIME ITEMS 3 TOTAL NUMBER OF EXTREME VaLUES CINCLUDES TIME AND ELEVATIUN VALUES) CHP 2 CRITICAL HALF PERIOD HVMIN © MINIMUM HEIGHT TO BE CONSIDERED ExTIMc(ODD) s TIME EXTIMCEVEN) © EXTREME ELEVATION aSSOCJATED WITH EXTIM(EVENe}) VALUE OF TIME COMMON /SMOT/ EXTIM(100) ig SET VALUES OF CHP AND HMIN YO BE USED DATA CHPe HMIN / Ve000 3,0 # C INITIALIZE VARIABLES AND FIND STARTING POINT FOR PROCESSING 1503 IITEMSSITEMS Js} ITEM7TSITENSo7 DO 1502 TEyoTITEMSo2 TSKIJPsI IF CEXTIMN( I) GEeFURST) GO YO 1504 JsJe2 ITEMSSITEMSe2 1502 CONTINUE 1504 JSTARTSISKyP CHREKKEAEA TRATES HES SHER HHSH EA ESE REESE SKE K HE SAE KH KSEE SEK SKEKA RS SEEKS EFI ELH RESS MAANMARAVANANN qanaan ¢ BEGIN MAIN PROCESSING LOOP DO 1520 ISYSTARTOTTEMTe2 Ls! IFCISTARY.GT,JTEM7)GO TO 1520 TFCEXTIMCTISCHP,GToEXTIMN(1¢2))G0 TH 1595 TF CABSCEXTIMCI $1) PEXTIM(T$3)) LT HMIN)GO TO 3511 € IF NO TRANSFER, THIS EXTREME ACCEPTED EXTIM(JVSEXTIMCT) ExTIN¢@ Jel SExTImnc led) JET+2 GO TO 1520 Cc WHEN THE NEXT INSTRUCTION JS REACHEDe ONE HIGH AND ONE LOW WILL C BE DELETEN 1591 IFCEXTIMC 191) GT ,EXTIN(1+3)IGO TO 1514 G IF NEXT INSTRUCTION IS USEDe THIS 7$ A LOW IFCEXTIMC145) ,GT,EXTIM(1¢1))IGO TO 1512 TF CEXTIMCI¢3) GT EXTIN(C1=1))GO TO 45138 GO TO 1517 1542 IFCEXTIMC( 143) GT EXTIM¢1+7))GO0 TO 1519 GO 19 1516 Cc WHEN NEXT INSTRUCTION JS USED» THIS IS 4A HIGH 1544 IFCEXTIM( 145) STEXTI4(1+1))G0 Yo 1515 IFCEXTIMNC 167) GT ,ExTIN(1+3))GO TO 4519 GO TO 1516 15145 IF CEXTIM( 73) GroEXTIM(I91))GO To 1517 GO TO 1518 Cc SET THE vVaLUE OF SCaSE 1516 ICASEs{ GO TO 1524 1517 ICASEs2 Go TO 1524 Figure D-2. List of subroutine SMOOTH. 147 1518 ICASES3 GO TO 15214 1519 ICASEs4 1521 JislIeed Jesle7 c DELETE ONE HIGH AND UNE LOW ACCORDING TO THE VALUE OF ICASE GO 10(1522515235152491525) ICASE. c STORAGE PLAN A 1522 EXTI4(Je2)stxTInclea) ExTIM( Jeol )SExTIM(lej) ExTIM(J)SEXTIM(T) EXTIM(J+1)SExTIM(I¢)) EXTIM( J+e)SEXTIM( 146) EXTIM( J*#3)SEXTIMCI¢7) GU TO 1526 c STORAGE PLAN B 1523 EXTIM(Je2)sExTIM(Ie2) EXTI*(Jel)sExTIncles) EXTIM(J)SExXTIM( I 64) EXTIM(JeI)SEXTIMCIT eS) EXTIM(J+2)SExTIM( 146) EXTIM(J+3)SExTIMCI¢7) GG TO 1526 c STORAGE PLAN € 15e4 EXTIM(Je2)ysexTim( lea) EXTIM(JeLISENXTIMC1¢3) EXTIM(JISEXTIM( Je) EXTIM(CJ+1SEKTIMC1¢5) EXTI*(J+2)SExTImue leo) EXTIM(J+3)sEXTIMC1%7) GO TO 1526 Cc STORAGE PLAN D $525 EXTIM(I+6)SExTIM(I42) ExTIM(I+7)SExTIm(le3) EXTIM( Jee)SsexTIn(le2) EXTIM( Jol) )SExTIM( lol) EXTIACJVSEXTIM(T) EXTIM( JL VSEXTIM(I91) ExTI4( Je2SseExTIve(Io2) EXTIM(C J+ SISEXTIM( 193) KIST+yg K221+40 1526 JisJe2 J2e5+3 ITEMSsITEMSeg ISK] PsI¢8 jstJeod Gc TO 1564 1520 CONTINUE ¢ END MAIN PROCESSING LOOP CHEROKEE KSEE ORES EERE ESEKERSESESES HESS SATS HETKAEESESEE SKE TH EEE CEST EE ETERS JeJel IFC Tee EU TTEMT)LES 1530 IF(LeGTelETEMS)GO TO 1340 JeJe! EXTIM(J)SEXTIMCL) Lele! GO 10 1530 1540 CONTINUE ME ertnene WERE ANY DELFTIUNS IN THIS P4&SS REPROCESS aLL REMAINING HIGnS AND LO#S TO Make FURTHER DELETIONS IF NEEDED. IFC(ITITEMS.GT,1TEMSIGO TO 1503 RETURN ND 5 Figure D-2. List of subroutine SMOOTH .--Continued 148 Point Exit 0 is within 3 points of record en processing loop 4 P number points so Point 0 is iS P that point accepted P 2 becomes point 0 ICASE = 1 ICASE = 4 Remove points Remove points Remove points 1 and 2 0 and 1 -1 and 0 Renumber points so Remove points. 2 and 3 that point 4 becomes point 0 Figure D-3. 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