Paes Sp A, STATISTICS ON WAVE HEIGHTS AND PERIODS FOR THE NORTH ATLANTIC OCEAN by R.L. Brooks, CDR, USN, and N.H. Jasper, Dr. Eng. v ) 2 STRUCTURAL | MECHANICS 4 0 a STRUCTURAL MECHANICS LABORATORY en RESEARCH AND DEVELOPMENT REPORT ~- WH WO ABSTRACT This report gives the frequency distributions of wave heights and wave periods obtained from weather ships stationed in the North Atlantic, together with an evaluation of the reliability of the visual wave-height estimates com- prising the basic data from which the distributions are derived. Visual esti- mates are compared with values determined from stereophotographs. An addi- tional check is provided by wave-meter measurements. It is shown that a log- normal distribution is applicable to the frequency distribution of wave heights experienced over a typical year and that this distribution is a useful guide to the determination of the incidence of a particular sea state at a given location. INTRODUCTION The David Taylor Model Basin is conducting a long-range research project! to evaluate present methods of ship structural design and to suggest improvements based on a realistic knowledge of the loads, stresses, and motions which ships experience in service. Instrumen- tation has been developed which measures the responses of ships to wave action in terms of stress, roll and pitch angle, and the corresponding accelerations. A large amount of data has been collected during voyages in the North Atlantic of aircraft carriers, destroyers, destroyer escorts, seaplane tenders, tankers, and dry-cargo ships. Typical of these studies is that con- ducted on the USCGC UNIMAK.?*3 Since the stresses and motions of ships are induced by wave action, these studies have included, as an important component, the problem of defining the surface variation of the sea, i.e., the waves. It is the purpose of this report to present the frequency distributions of wave heights and wave periods obtained from weather ships stationed in the North Atlantic together with an evaluation of the reliability of the visual wave-height estimates which comprise the basic data. Observations of wave heights and wave periods over a period of about six years have been made available by the U.S. Weather Bureau. In order to gain some idea of the reliability of the visual estimates made by observers, such as the Weather Bureau personnel, a special effort was made by the Taylor Model Basin to obtain stereophotographs of the sea surface at the same time that visual estimates were made. Such comparative data were gathered during extensive sea operations of the USS VALLEY FORGE (CVS 45) and the USCGC UNIMAK (WAVP 379) in 1955. A comparison of the visual estimates with values determined from the stereograms is given in this report. An additional check on the validity of the Weather Bureau data on wave heights is made by comparing them with wave measurements obtained by Darbyshire*»> at Ocean Stations I and J. IReferences are listed on page 51. The statistical presentation of wave data given here will show at a glance the probabil- ity of exceeding any given sea condition (as specified in terms of a characteristic* wave height) in an average year, for ten representative locations in the Atlantic Ocean. Such infor- mation can be utilized in the solution of design and operational problems connected with the strength, speed, and motion of ships at sea and in planning model tests of seaworthiness. SOURCES OF DATA WEATHER BUREAU DATA Wave heights and periods from the Weather Bureau records are presented in Tables 1 and 2. These data are visual estimates and were made every three hours by trained weather observers in accordance with instructions prescribed by the U.S. Weather Bureau.® Only one quantitative value for wave height and one for wave period were reported each time the sea was observed. These data cover a period of about six years and were made from weather ships at ten ocean stations, located as shown in Figure 1. 40 30 Figure 1 - Locations of Ocean Stations *The ‘‘characteristic’’? wave height is the average height of the larger well-defined waves. See the discussion under ‘‘Sources of Data’’ for a more specific definition. The term ‘‘characteristic’’ height should be differentiated from ‘‘significant’’ height. The latter term has a precise mathematical definition, the former does not. TABLE 1 Frequency Distribution of Characteristic Wave Heights Reported by U.S. Weather Bureau Wave Heights,* feet : Total Station| of Record | <1.0 | 1.0-2.5 | 2.54.1 | 4.1-5.7| 5.7-7.4 | 7.49.0 | 9.0-10.7 | 10.7-12.3|12.3-13.9] 13.9-15.6| Number of 15.6-17.2| 17.2-18.9| 18.9-20.5 | 20.5-22.1] 22.1-23.8| 23.8+25.4| 25.4-27.1 | 27.1-28.7|28.7-30.3] >30.3 | Observations oe 6/54] 103 891 2106 2403 a Mee me ase 528 186 175 151 134 164 12,891 ic 144 939 2601 2507 2484 1800 Fe ia 634 237 264 286 166 113 125 120 101 15,547 C | 1/49-12/54 105 860 2479 2964 ae 2125 re ne He 600 248 226 235 168 119 110 16,857 1/49-12/54 797 2861 3452 ay ot Me ae a oe ce 217 231 124 16,804 1/49-12/54| 280 4629 3371 2569 1335 ee oe 70 108 11 38 29 33 16,777 1/49- 6/54) 255 1863 2082 1310 1057 654 360 136 114 34 43 33 11 18 14,607 1/47- 6/53} 243 37 an ee 901 667 64 187 11,274 1/47- 6/53) 272 2271 eH ae 877 1028 763 270 684 196 196 27 21 105 12, re 1/49-12/53 144 743 1816 2146 ay re 401 506 116 93 89 29 S = 15 *Two wave height ranges are shown at the top of each column. Opposite each weather station, two entries appear in each column. The top entries apply to the top wave height ranges, and the bottom entries apply to the bottom wave height ranges. TABLE 2 Frequency Distribution of Characteristic Wave Periods Reported by U.S. Weather Bureau Ocean | Period Wave Periods, seconds Total [3 [|i is [ir ia a] bri | a _| vas. 6/54 | 2592 celmiat tt WL Bf a tka 1/49- 6/55 | 3655 | 6896 | 3918 | 1159 | 297 | 106 | 16,060 oii ve lae latina ae bare ae ar Hoi safer aorwbae Esfa ea 5/49- 6/54 Pane 12/51 1/49-12/51 Pee a aetetetate toot coe ie feyasrasa| aan] 2019 [ree [see | ae | 20 a Tw _[ vases [ies | ar [eo aoa ee Location of Stereocameras Figure 2 - Inboard Profile of USCGC UNIMAK Showing Location of Stereocameras Location of Stereocameras Flight~ ———— ey / — Fosc aa SS SSS SS SS SS= SS S= SSS SS SSS = _————e SSS SSS Ss SS SS as SS SS S55 ee 4th os SS ee ee en a a eee Ea ~ 4th 200 180 160 140 120 100 80 60 40 20 FP AP Figure 3 - Inboard Profile of USS VALLEY FORGE (CVS 45) Showing Location of Stereocameras Basic Characteristics Midship Section Moment of Inertia 27,502 ft* Midship Section Modulus 129,210 ft in.? Block Coefficient 0.585 Prismatic Coefficient 0.597 Midship Section Area Coefficient 0.980 Waterplane coefficient 0.743 STEREOPHOTOGRAPHS OF SEA SURFACE In order to evaluate the accuracy of the Weather Bureau wave estimates and their use- fulness in the statistical analyses of ship strength, stereocameras were installed on the USCGC UNIMAK (WAVP 379) and the USS VALLEY FORGE (CVS45). Figure 2 shows the camera locations on the weather ship, and Figure 3 shows those on the aircraft carrier. Table 3 gives identification and installation data for the cameras. Visual estimates of the sea surface were made by trained observers at the same times that stereophotographs were taken. The UNIMAK estimates were made by regularly assigned Weather Bureau observers. The VALLEY FORGE estimates were made by trained Navy aero- logical personnel; two or more of them made independent estimates twice each hour during most observation periods. By special arrangement, five or six of the Navy aerological per- sonnel made independent wave estimates three times each day. Subsequently, the VALLEY TABLE 3 Stereocamera Data Aerial K 24 Fairchild Eastman Kodak Manufacturer Serial Number 292 35,540 109,994 Instruction Manual AN-10-10AC-63 1 Aug 1947 10-10AB-1 over AP2315A 30 Jul 1943 Revised 1 Jun 1953 Revised 30 Oct 1943 Aft ee ee Equivalent Focal Length 153.57 mm 152.83 mm 179.66 mm 180.03 mm FORGE estimates were averaged and plotted against time of observation (see Figure 4), and Calibrated Focal Length a smooth curve was fitted to these points. In making their observations, the Weather Bureau personnel on the UNIMAK followed the instructions of Reference 6, from which the following sentences are quoted as they appear in the Ninth edition, pages 57 and 58: ‘‘Waves in the same system usually occur in a sequence of a few, large, well-formed waves followed by an interval in which only small and poorly formed waves appear, then another series of large, well-formed waves. To obtain uniform wave data from all ships, observers will record only the larger, well-formed waves, and omit entirely the low and poorly formed waves. ... The wave height as recorded. . . is the average of the estimated heights of the larger, well-formed waves.”’ The Navy observers were guided by Reference 7 which says: ‘‘In view of the consider- able variation in height between waves observed in a 7-minute period, reference is convenient- ly made to the significant wave height. This wave height is the average of the higher, well- defined waves present during the observation. Statistically, significant waves are defined as the average of the 1/3 highest waves observed in a given time. As the height is the most important wave characteristic from the operational point of view, care should be taken to ob- serve and report it accurately.’’* It is apparent that the estimating procedures specified by *Italics added. 6 360 x * g3 g 2= 350 S “=o Saf 340 efD A . . 1 Sig O@ Indicates time of strain Bee 330 and motion data on = = © Valley Forge and @ Sperry in seconds Average Period of Characteristic Waves @@ Indicates time of strain and Ae 2 | motion data on A 4 A ® Valley Forge and @ Sperry el 0700 0800 03900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 Ship's Time Average Height of Characteristic Waves in feet Figure 4 - Illustrative Example of Procedure Used to Average Visual Observations Note plotted points are averages of values estimated by seven aerologists on USS VALLEY FORGE for 12 October 1955. both the Weather Bureau and the Hydrographic Office for observers are essentially the same. Throughout this report the term ‘‘characteristic height’’ will be used to denote the aver- age height of the higher, well-defined waves. The statement in italics will serve as a defini- tion of characteristic wave height. The term ‘‘significant wave height’’ will be used only in its statistical sense. The ‘‘characteristic wave period’’ is the average period of the higher well-defined waves. The characteristic values may be estimated by shipboard observers, or they may be obtained by more precise measurements from stereograms or wave records. SEA SURFACE PROFILES Sea surface profiles were developed from the stereophotographs by means of planigraphs at the U.S. Naval Photographic Interpretation Center and at the U.S. Navy Hydrographic Office; see Appendix A. Then all profiles were analyzed at the Hydrographic Office by statistical methods as outlined in Appendix A. These analyses resulted in histograms of wave heights and wave lengths as well as in ‘‘significant’’ wave heights. The significant wave heights are listed in Table 4. Only waves of length greater than a certain ‘‘cutoff’’ length were in- cluded in the histograms. The cutoff length is a function of the wind conditions that gener- ated the sea. See Appendix A for an explanation of how the cutoff length was determined. The sea surface profiles were later used to determine the characteristic wave height by following the same general procedure as that used by the shipboard observers in making their estimates; thus these two independent determinations of characteristic wave height should be comparable. TABL iD a) 4 Correlation of Wave Lata Visual Observations Ir DTMB Stereoanalysis E in (ft)? Estimated from Wind Data* Stereophotographs Made Aboard Ship USN Hydrographic Office Stereoanalysis Time Position Stereophoto and Characteristic Characteristic iff 2 of injdeg Sea Surface |Number of | Wave Height, [ete Wave Heignt, | Waves. Wave Length WaeoHEatt Estimated Photo} Time: Lat, Long] Profile Number Estimators verses of Larger) waves (Average of Larger] in Sample ft ft from Wave Height® aves) Waves) Sea Swell |Sea |Swell Sea Swell CVS45 | 11 Sep 55) 1435P) 1200: 40N, 58W A-0119 3 5.0 12{ 12 54 6.5 12 Sep 55) 1701P} 2000: 40N, 51W A-0200 4 4.8 |) 7 44 6.9 20 Sep 55} 1113N] 1200: 45 N, 17W B-0163 4 OTS ame | LiL a5) 6.1 11.6 20 Sep 55] 1202Nj 1200: 45N, 17W B-0165 3 8.0 |12] 6 9.6 10.0 15 Sep 55109250] 0800: 43N, 37 B-0218 2 1.2 25 | 12 6.3 5.9 22 Sep 1200: 48N, 8W 3 10.0 11 7 5.9 11.2 28 Sep 2000: 42N, 13W 4 9.0 | 11] 14 5.5 9.0 30 Sep 1200: 40N, 15W 3 8.0 | 12] 21 6.6 8.6 30 Sep 1200: 40N, 15W 3 Ue |) 121) 32 8.1 8.2 30 Sep 1200: 40.N, 15W 3 7.8. [12] 15 9.1 7.5 1 Oct 55) 1245Z] 1200: 43N, 13W C-0495 6 9.0 | 12) 16 5.8 7.0 1 Oct 55} 14152] 1200: 43N, 13W C-0500 3 8.8 }10 | 11 6.3 8.0 2 Oct 55} 1300Z| 1200: 39N, 13 0-0017 3 4.4 |12] 8 2.8 4.5 11 Oct 55] 09482} 0800: 37N, 15W D-0085 5 6.2 | 15) 11 4.2 5.4 1] Oct 55] 16162] 2000: 37N, 18W 0-0122 7 8.0 | 16] 8 6.2 6.5 12 Oct 55]0812Z| 0800: 36N, 21W] 0-0148 6 uM we : 12 Oct 55} 09462} 0800: 36N, 21W 0-0159 5 5.0 | 13] 10 7.1 6.7 12 Oct 55} 16002} 2000: 35N, 22W D-0199 3 6.9 | 10} 14 6.5 6.3 13 Oct 55] 1046N} 0800: 35N, 260 E-0017 5 4.0 |10] 8 2.2 3.5 13 Oct 55] 1215N} 1200: 35N, 27W E-0021 7 3.7 | 12] 7 5.6 4.5 13 Oct 55} 1646N] 2000: 35N, 29W E-0034 7 2.3 13 5.0 14 Oct 55] 13150} 1200: 34N, 34W E-0062 3 6.1 10 § 7.2 7.1 14 Oct 55] 13450} 1200: 34N, 34W E-0064 3 6.1 | 15] 12 6.9 6.9 14 Oct 55] 14150] 1200: 34N, 34W E-0068 4 4.6 6.0 14 Oct 55] 14450} 1200: 34N, 34W E-0071 4 6.0 | 10} 14 6.9 5.9 14 Oct 55] 15150} 2000: 34N, 35W E-0074 3 SYM liza | a2) 7.3 6.5 16 Oct 55} 0645 P| 0800: 34N, 45 E-0145 1 10.3. ]13] 9 8.0 10.4 17 Oct 55} 0945 P| 0800: 35N, 57W F-0222 4 7.3 16} 10 4.1 5.7 18 Oct 55] 1245 P| 1200: 35N, 59W F-0289 4 6.1 | 15 5.6 19 Oct 55] 1015 Q| 1200: 35N, 67 F-0344 4 4.8 12] 9 5.8 5.1 1200: 35N, 67 = 4 Ga PT] 8 9.2 6.3 2000: 35N, 68W 1 71 13] 4 8.2 11.0 2000: 35N, 71W 5 3.7 | 15] 7 5.0 5.0 0800: 37N, 71W 4 5.9 | 14] 11 5.7 5.1 1200: 36N, 72W ZI 6 6.6 [15] 10 9.5 6.7 22 Oct 55] 1617R} 2000: 37N, 73W G-0150 2 65 | 14] 8 4.2 7.1 200 15 Nov 55] 0750R| 0800: 37N, 73 H-0047 2 By We ©) 3.6 3.9 15 Nov 55] 0950R} 0800: 37N, 73W H-0060 2 2.7 | 14] 10 3.6 3.7 15 Nov 55] 1055 R} 1200: 38N, 73W H-0078 4 2.7 | 14 3.8 15 Noy 55] 1115R] 1200: 38N, 73W H-0087 1 2.7 | 14| 10 2.9 3.7 15 Nov 55] 1155R| 1200: 38N, 73 H-0100 3 2.8 | 14) 12 3.5 4.5 é d 16 Nov 55] 0715R| 0860: 37N, 72W H-0155 6 3.6 | 14] 15 6.0 5.3 188 20 5.1 12.0 16 Nov 55) 1255R] 1200: 37N, 72 H-0201 2 7.0 | 16) 11 5.1 5.3 169 30 5.2 119 16 Nov 55] 1545 RJ 2000: 37N, 72W 1-0014 3 7.5 | 16} 13 6.4 7.9 148 32 6.9 22.0 17 Nov 55] 0815 R| 0800: 37N, 74W 1-0052 5 7.1} 20} 13 3.8 6.0 200 Sa 4.6 9.7 17 Nov 55] 1345R| 1200: 37N, 74W 1-0073 2 4.3 | 17) 12 4.3 6.0 200 = 4.6 10.3 9 Dec 55] 1530R] 1200: 36N, 70W J-0003 4 10.5 9.9 | 16) 11 7.3 9.3 139 34 6.8 215 10 Dec 55] 0945R| 0800: 36N, 720 J- 0035 3 12.5 12.7 | 18] 9 15.3 12.2 144 38 8.1 29.4 10 Dec 55} 1045R| 1200: 36N, 720 J-0040 3 12.0 12.1 | 13 16.8 150 40 10.9 58.8 WAVP 379] 16 Oct 54) 16292} 1500: 44N, 41W U- 359 1 13.0 14 11.6 146 9.4 22 Oct 54] 13182} 1200: 44N, 41W U- 415 1 5.0 11 7.1 31 5.5 16 Jan 55] 12402] 1200: 57N, 51W U- 505 1 14.0 10 14.7 91 8.1 16 Jan 55] 1628Z] 1500: 57N, 51W U- 517 2 19.5 10 21.6 87 9.0 28 Jan 55] 1536Z] 1500: 57N, 51W U- 526 2 5.5 8 8.4 66 (HY) 15.8 37 8.2 1 Feb 55} 1220Z] 1200: 57N, 51W 1 1 Feb 55} 12212) 1200: 57N, 51W U- 541 1 1 Feb 55} 12222] 1200: 57N, 51W B 1 * E is the mean square value of wave heights. 32.0 30.0 18.4 10.4 24.0 40.0 48.0 The data from the sea surface profiles worked up by the Hydrographic Office were used further to obtain the mean square of the wave height, FE. These quantities are also listed in Table 4. In order to check the utility of the theoretical method of Reference 8 for the prediction of wave heights, the Hydrographic Office also computed the mean square of the wave height F on the basis of the distribution of wind velocities for the sea area and sea sur- face profile in question; these values are listed in the last column of Table 4. Graphs of these various quantities are presented and discussed in Appendix B, Figures 34 through 36. STATISTICAL BACKGROUND Wave heights and wave periods estimated from the Weather Bureau data for ten ocean stations are presented in the form of their distribution functions. For example, all wave heights reported by the shipboard observers are considered to be members of a statistical ‘“population’’ of wave heights. The distribution function of wave heights indicates the rela- tive probability of encountering a wave of a given height as a function of that height. Figure 5 illustrates this distribution function. The area under the curve to a value z, is the integral of the function up to z, and is equal to the fraction of all members of the population of wave heights which have a height less than z,. Mathematically x oo P (a) = [eae and P(e > ») = il adh = I ty) 0 0.18 where p is the probability density and P is a function of x designated as the cumulative oe distribution function of z. P(z) is numeri- ol4 cally equal to the probability that a value ie chosen at random from the population is = less than z. 8 010 A detailed discussion of the statisti- ae | | cal methods utilized in this report is given & Experiment in Reference 9. Only a few of the major Otel concepts will be described here. The dis- Ge tribution applicable to a given sea condition is here called a ‘‘short-term’’ distribution, 0.02 12,365 observations whereas the distribution applicable when each of which represents a given sea state. O 4 a yl eo, SS x = Significant Wave Height in feet a wide range, such as over a year’s time, the sea conditions are allowed to vary over is called a ‘long-term’? distribution. Thus io 5 - Distributi ie i Sarees 3 Bigures: ie uten ener the long-term distribution is the result of a summation of a number of short-term distributions. Oceanographers have held that the short- term* distribution of wave heights 2 is approximately of the Rayleigh type (a narrow power spectrum is assumed) for which, 2 = E. P(z) = 22 orn as i where E, is the mean square of all the individual wave heights x corresponding to sea condi- tion 7. Note that numerically the value of FE computed for wave height will be four times the value of E computed for wave amplitude because wave height is taken equal to twice the wave amplitude. See References 8 and 10 for a discussion of the distribution of wave heights in terms of the power spectrum concept. In Reference 9, it was suggested that the long-term distribution of wave heights and wave periods is of the log-normal type, that is, that the logarithms of these heights and peri- ods are approximated by a normal distribution. Thus 1 2 2 p (log w) d(log zx) ae GMI LY) VAG Glee @)) log x P(x) = P (log «) p(log x) d (log z) log x =—oo where p(logz) is the probability density of the variate, log z, u is the mean value of log z, and o? is the variance of log z. Then the parameters u and g define this distribution completely. In this report log-normal distributions are fitted to the characteristic wave heights and periods reported by the Weather Bureau. The resultant graphs represent long-term distributions and give the probability with which a given value of the variate x will or will not be exceeded in an average year.** *The short-term distribution is approximately valid if measurements are taken over a relatively short period of time, of the order of one hour, during which interval the sea conditions do not change appreciably. It can be shown that this distribution is the same as that representing the wave heights, in the area under consideration, at one instant of time. ** Although the distributions given here are for a six-year period, study of the distributions for the individual years making up the six-year period indicates that a single year gives a typical sample of the distribution obtained for many years. Therefore, the six-year distribution may be considered valid for an average year. Characteristic Wave Height, determined from Stereo Wave Profiles, in feet 22 20 O CVS, Swell predominant O CVS, Sea predominant © UNIMAK, Swell and Sea Line visually fitted to the plotted points Me e) O TiN Sf- 7 e) 4 Line with 45 degree slope S |, | Oo @) — fa 1 2 4 6 8 10 12 14 16 18 20 Characteristic Wave Height in feet from reports of Shipboard Observers Figure 6 - Scatter Diagram for Comparison of Characteristic Wave Heights Determined Independently from the Wave Profile Analysis and by Shipboard Observers The plotted data are taken from Table 4, Columns 7 and 9, 10 EVALUATION OF RELIABILITY OF VISUAL ESTIMATES OF WAVE HEIGHT The U.S. Naval Photographic Interpretation Center and the U.S. Navy Hydrographic Office utilized the stereophotographs from the UNIMAK and the VALLEY FORGE to produce the sea surface profiles illustrated in Appendix A. Fromthese profiles, wave height data were determined as discussed in the section on ‘‘Sea Surface Profiles’’ and tabulated in Table 4. Note that this table includes waves associated with sea conditions in which seas predominated, others in which swells predominated, and still others in which both seas and swells were present. The characteristic wave heights obtained by the shipboard observers are compared with those derived from the stereograms for the same sea condition in Figure 6, where values from Columns 7% and 9 of Table 4 are plotted as abscissas and ordinates, respectively. If exact agreement existed between visual estimates and the results of photogrammetric analy- sis, then all points would lie on a straight line with a 45-deg slope. The points plotted in Figure 6 scatter fairly well about a straight line which has a slope somewhat greater than 45 deg. The average deviation of the points from the line is expected to decrease as the number of points is increased. It should be noted that each stereophotograph covers a limited field of view compared with the field of view of the shipboard observer; see Figure 7 for ocean areas included in the camera perspectives for both ships. It is considered that Figure 6 shows good correlation between the visual estimates and quantitative height determinations made from the stereophotographs. Individual estimates may not be accurate, but when the number of estimates is large the correlation is good. Scale Scale jeans 000 Itt, ._1000 ft, Ria he ee 6500 ft ——_—_____- 5000 ft from Cameras }-— s00 t— VALLEY FORGE CAMERAS See 0.58 Square Miles \ 1650 ft from Cameras UNIMAK CAMERAS See 0.026 Square Miles 250 ft 250 ft from Cameras from Cameras Figure 7 - Ocean Perspectives Seen by Stereocameras on UNIMAK and VALLEY FORGE Areas indicated are fixed by properties of cameras, film, stereoplanigraphs, and camera separation. 11 0.01 99.99 0.5 (From U.S. Weather Bureau Data 1947-53) (Data Truncated at 24% Feet) Station J Observations of Significant Wave Height (From U.S. Weather Bureau Data 1947-53) Stations I and J Measurements of Maximum Wave Height obtained by Darbyshire (Reference 9) during period Feb 1953 - Jan 1954 (Data Truncated at 5 feet) . yale a lai in ih aL AT A eV TT IL |) ie ae eee eee Probability of Exceeding Wave Height (Percent) Probability of Not Exceeding Wave Height (Percent) ake OcuL 7A Cro Note: The Straight lines are the oF ‘ sour cameo oe a BASS oc 99.5 computed directly from the ex- perimental data. #338 2 3 4 5 6 7 BY w 20 30 40 50 60 70 80 90 Wave Height, feet Figure 8 - Comparison of Wave Height Distributions Derived from Visual Observations and from Measurements of Wave Heights at Atlantic Ocean Stations. I and J The distribution fitted to the Darbyshire data corresponds to a standard deviation of 0.57 for log, (maxi- mum wave height) and a median value of the maximum wave height equal to 15 feet. 12 The UNIMAK data and the VALLEY FORGE data do not indicate divergent trends, that is, the methods®’” of wave estimation used by the observers on the UNIMAK and the VALLEY FORGE give approximately the same characteristic wave height. The U.S. Weather Bureau data, on which the long-term distributions given in this re- port are based, comprise between 11,000 and 18,000 separate observations for each ocean station. It is concluded that the errors associated with the visual observations are fairly well averaged out when such a large number of observations are utilized to define the distri- bution and that the reported characteristic wave heights are therefore proportional to the severity of the sea. Further evidence to support the validity of the Weather Bureau data can be drawn from an analysis of measurements of wave height recently made by J. Darbyshire* by means of a wave meter installed on a weather ship. These measurements were made over a period of about one year, February 1953 to January 1954, at North Atlantic Weather Stations I and J; see Figure 8. Darbyshire reported the maximum wave height for each 3-hr period for which visual wave observations were made while the ship was at sea. The visual observations made by weather observers are reported as the ‘“‘characteristic’’ wave height. According to Appendix B the characteristic height is proportional to the significant height. It is of interest to compare the visual observations with the measurements obtained with the wave meter. If the hypothesis is accepted that the short-term distribution of wave height follows the Rayleigh distribution, then the maximum significant and characteristic wave height for any given sea condition are related by a constant factor. Thus the long-term distributions of maximum and characteristic wave heights should be of the same type, log-normal in this case, and should differ only in their mean values. The U.S. Weather Bureau data indicates that the standard deviation* of log, (characteristic wave height) is 0.622 at Station J and 0.612 at Station I as compared with a value of 0.57 for log, (maximum wave height) for the measurements at Stations I and J reported by Darbyshire; see Figure 8. A log-normal distribution has been fitted to the wave-meter data on the assumption that the distribution of maximum wave heights is log-normal. The experimental data indicate excellent agreement with the fitted distribution, well within the accuracy of the measurements. The latter fact, together with the good agree- ment between the standard deviations of characteristic (visual estimates) and maximum (meas- urements) wave heights, supports the hypothesis that the distribution of wave heights may be approximated by Rayleigh and log-normal distributions for the short and long term, respective- ly. In a recent article> Darbyshire tests the applicability of the long-term log-normal distribu- tion to extensive data on maximum wave heights obtained by use of the British wave meter. He concludes that the logarithmic law appears to be a useful guide to determine the incidence of a particular wave state at a given location. *The numerical values given here apply for wave heights measured in feet. 13 It is concluded on the basis of the foregoing discussion that the visual estimates by Weather Bureau personnel of sea state, reported as a ‘‘characteristic’’ wave height, may be used with confidence in establishing distribution patterns such as are given in the following section. DISTRIBUTION PATTERNS OF WAVE HEIGHTS AND WAVE PERIODS FROM ANALYSIS OF U.S. WEATHER BUREAU DATA Cumulative long-term distribution patterns of the characteristic wave heights and periods are given in Figures 9 through 30 for the ten ocean stations shown in Figure 1. For each station the odd-numbered figure gives the wave height distribution and the even-numbered figure gives the corresponding wave period distribution. Methods for fitting a log-normal dis- tribution to the data are given in Reference 11. In Appendix C a sample calculation illustrates the method used for deriving Figures 9 through 30 from the data (Tables 1 and 2) furnished by the Weather Bureau. The rather good fit of the computed lines to the plotted data, in Figures 9 through 30, suggests that a log-normal distribution is a good approximation to the distribution pattern of characteristic wave heights and periods for values above the truncation point.* Distribution patterns for wave length can be derived from the data for wave periods by applying an approximate conversion** Wave Length = 5.1 (Wave Period)? This conversion has been made for all the weather stations. It is apparent that the distribu- tion of wave lengths will be log-normal if that for the periods is log-normal, since the conver- sion involves only a change in mean value and slope from the distribution of the periods. See Figure 16 for an illustration of the conversion to wave length. In Table 5 mean values and variances are given for the wave height and period data reported from each ocean station. Also the latitudes, longitudes, and observation periods over which the data were collected are shown. *The truncation point is that value of wave height or period below which no observations are available or are utilized. In this report only wave periods above 5 sec and wave heights above 2.5 ft were used. **This conversion is applicable to gravity waves in deep water. The numerical value of the factor, 5,1 in this case, does not affect the type of distribution; it only changes the value of the median. 14 TABLE 5 Statistics for Log-Normal Distributions Computed from Wave Observations Made in North Atlantic Ocean Characteristic Wave Heights Characteristic Wave Periods ——— Median Value of] Mean Value of | Variance (o) of| Period Number of Characteristic | Logarithm of Wave Height | Characteristic Wave Height Mean Value of | Mean Value of | Variance (a) of Logarithm of | of Records| Observations} Characteristic} Logarithm of Logarithm of Characteristic Wave Period | Characteristic] Characteristic Wave Height Wave Period Wave Period Ocean] Latitude | Longitude Station} deg, min Dates of Number of Observations | Observations 1/49-12/54| 16,777 1/49- 6/54| 14,607 1/47- 6/53| 11,274 A | 62°00°n | 33°00°W | 1/49- 6/54] 12,891 6.34 1.847 0.4524 | 1/49- 6/54] 12,342 1.840 0.0935 slow | 1749-12/54 | 15,547 1/49- 6/55| 16.060 | 1.857 0.0748 35°30" 1/43- 6/55| 17,471 | 14869 0.0873 44°00°N 1/49-12/54 1/49- 6/55| 17,310 | 1.831 0.0800 | 41°00°W. 35°00’ N | 48°00°W 36°00°N | 70°00°W 61°00°N | 15°20°W §2°30°N | 20°00°W 16°00°W 02°00°W - 6/55 16,896 5/49- 6/54 13,647 1/49-12/51 12,142 1/53-12/54 1/49-12/51 11,593 1/53-12/54 0.3033 6/49-12/53 11,906 0.2344 1/49-12/53 14,188 5.86 1.768 0.0939 ‘V/47- 6/53 45°00°N 66°00°N 1/49-12/53 1/49-12/53 14,324 oe Note: The statistical computations are based on truncated data. The truncation point is 2.5 ft for wave heights and 5 sec for wave periods. SUMMARY Frequency distribution patterns of wave heights and wave periods may be approximated by a one-parameter type of distribution function when the environmental conditions are steady, whereas they will tend to follow the two-parameter logarithmically normal distribution when the environmental conditions are allowed to vary over a wide range. It should be emphasized that the log-normal distributions in Figures 9 through 30 are influenced much more by the usual sea conditions than by the rare occurrences of very high or very low seas. Thus one should expect greater deviations from the fitted line for very small and very large wave heights and wave periods than for those heights and periods which occur more frequently. It is con- cluded that the long-term distributions of wave height and wave period may be approximated by the log-normal distribution. Reasonably accurate visual estimates of wave height can be obtained from trained ob- servers, provided a number of independent estimates are averaged. A single estimate may be considerably in error. ACKNOWLEDGMENTS This work would not have been possible without the cooperation of many persons on board the VALLEY FORGE and UNIMAK, on the Coast Guard and Navy staffs, in the Weather Bureau, at the Hydrographic Office, at the Naval Photographic Interpretation Center, and at the Taylor Model Basin. 15 Special thanks are due to Dr. R.W. James of the Hydrographic Office and Mr. W. Marks of the TMB staff who assisted in the oceanographic planning and analysis. The camera in- stallations were made by Mr. C.E. Lemich of the Taylor Model Basin. Mr. B.M. Wigle and Mr. R.J. Dominic, also the Taylor Model Basin staff, sorted and matched the many photographs and assisted in analyzing the stereograms and in developing the plotted figures. The stereo- grams were developed by Mr. John Davis and Mr. John Boyle of the Hydrographic Office and Photographic Interpretation Center, respectively. Much encouragement has been given to the overall program, of which this study is a part, by Mr. John Vasta of the Bureau of Ships. The paper was reviewed by Dr. E.H. Kennard, Mr. R.T. McGoldrick, and Miss M.C. Crook of the Taylor Model Basin staff. The authors wish to express their sincere appreciation to the persons mentioned here and to the many others whose assistance made this work possible. 16 Probability of Exceeding Wave Height - Percent ee Es te LE See eee nn mie te ee Le cee = Hit ANSE AE Pte el JU CEL 1 2 3 4 i) 6b 7 8 Op 0 30 50 60 70 8090 100 Wave Height —Feet Figure 9 - Distribution of Characteristic Wave Heights at Station A 17 ¥ a! { bate ye a | RL a ti Wy Sea se cecesscseteeerl ln ott Se sescssasss eh a = “tee 4 i Hh ie Fos 75 Say 25 00 AGT DH ae 0 OROUN cObd Osta 00 DOGN DORN ADEN a vt Bet an Ia Po oooscee sane jesoooa Gil Reese 3S SSS Ht a cae saeee ea os S. Weather Bureau Data aia 7 ee UiU1 Dias Wf i YE 100 (1-P) Probability of Exceeding Wave Period - Percent si veg Wave Period —Seconds Figure 10 - Distribution of Characteristic Wave Periods at Station A 18 f {1 t HAE [ ] i] 1H iho | Wiad ui oni Ce ac sto ee I tes Pm di : 50 ny i oa fi it Sl 4 | oh {ea mee sed adh /\ bos i 8 ywaoiad - JysIaH aAeM Bulpaaaxy Ayyiqeqolg Pee sereea 2 eae os ae ee "| + =>- H oe os 5 Wave Height— Feet Figure 11 - Distribution of Characteristic Wave Heights at Station B 19 BgS0E: s35S5 3 ae & ES ta F a fA FE i=] + yf o i =) } mop Pe Si a eG i eo} iif @ ‘ o 2 -E a[a [ iene | (2) | ett oe SS Gane © Sse HH = Saeipeeee mane zee =peran == BE SHIGE et Ma tee fe PSS Ee coos csscd dot crease 7 + le anaeeseta emaea0 Wua9!ad - poliad anem Suipaaaxy Jo Ayijigeqord (d-1) OT 20 eee Period -Seconds Figure 12 - Distribution of Characteristic Wave Periods at Station B uey | ssa] juadiag SemusiHEl SS ueyl Jajeai juaaiad Se oN Figure 13 - Distribution of Characteristic Wave Heights Station C at 21 fo} ic ae a waoiad - pollad adem Bulpaaaxy Jo Ayigeqod (d - 1) O01 99.99 60 70 80 90100 50 40 30 20 Seconds Wave Period - Figure 14 - Distribution of Characteristic Wave Periods Station C at 22 99.99 OOGUAROTTANLANN OG NY etal Ab OE Ere HH a BTFRELCU a Ge HE | {} TE GUND EAU MULT HO OLA LUTE macHU eat Percent Greater Than fercent Less Than ated Benes voses reas inemiutl BOB0 NDO00 OBTNE NNMINA HH Neamt I PH l vr 0.01 6 7 8 9 10 30. 40 50 60 70 80 90100 Feet Figure 15 - Distribution of Characteristic Wave Heights at Station D 23 yuaaied X U.S. Weather Bureau Data mea i tt ++} “iti tyHy ay aT i ae BATA my a uss 11 i Bayou aus igiel { north uy -)f-- + sre bd bake bps Powad ane Buipaadxy Jo Ayiyigeqold (d - 1) OOT 50 60 70 80 90 100 40 30 Period - Seconds Wave - Distribution of Characteristic Wave Periods Figure 16 Station D at 24 wey] SSaq] yada te Log Normal Distribution He (Fitted to the Experimental Da andl a eee ee ee ee eee se ele a ee ee eae, Gn ne a Oc ai eee eee camietem cea See ueys 12}e915 yuadiad Feet Figure 17 - Distribution of Characteristic Wave Heights at Station B 25 a Pe] w (a) < ° Fi -_ pe) () 5 5 | i {ea} nw A 3) ae o E = S é = P ~ 2) << laimial eI Ese 10 i i pagans i cooee eraes yh + +t tHethi sys. tet s4etheyd 50 60 70 8090100 40 30 20 Wave Period -Seconds 26 at Station E 6 7 8 910 5 [= a Jaa - Pollad anem Buipaaaxy yo Ayyigeqord (d- 1) OO] Figure 18 - Distribution of Characteristic Wave Periods 99.9 99.99 =H sEseatoes = — —— = SS —— ars Sh areas ag) Gaps aU ete ome uey | ssa7] uaa 50 60 70 80 90100 40 ALL BEL (Fitted to the Experimental Data) } { { Log Normal Distribution Pt AD as 0 SS ey (Beet Feros Wrens Wes |e Seay ft een 4 | Coe Bue ae ee uey J 19}e9I5 Juadlad Feet Distribution of Characteristic Wave Heights Figure 19 - Station H at 27 er Bureau Data Log Normal Distribution a 50 60 70 8090100 40 30 G6 7 8 BW 5 as Ge Be eal ey gt ae ean ea eee ee mae ra i 1 ic juaniad - pola anem Buipaax3 Jo Ayi|iqegold (d- 1) O01 99.99 - Seconds Wave Period Figure 20 - Distribution of Characteristic Wave Periods at Station H 28 Percent Greater Than ie eel an SIBUGrtua) (imam Aan J a ~ EBBTEGY AIER Percent Less Than atte HI | cha Gi thi eae +h t i + Figure 21 - Distribution of Characteristic Wave Heights at Station I 29 5 4 sb haba dd feliec hua PRCA tat elt tel obbts re i : ii 100 (1 -P) Probability of Exceeding Wave Period - Percent SS) eft ol sae as ae acid Fo a Oe - con se biwee sia Hi Sopa! ac bred soticl wana |ood Bas Gi oans NE SH uy a | Hl aii He nant rerrtrsty : Faees Seas eaogs bg90) Os Bars) 9 pouty Gs erry 1 7 - | i: 14 ame emaing Vas eA U . Sie ft ete f=t= ai balit: Hy: + hy Wave Period - Seconds Figure 22 - Distribution of Characteristic Wave Periods at Station I 30 uey | SSd] uaded teat = | Senneeeese Log Normal Distribution uey) 13}ea19 JuadIeg if i he tH PELE aggre pe i Feet Figure 23 - Distribution of Characteristic Wave Heights tation J at S 31 100 (1 -P) Probability of Exceeding Wave Period - Percent 1) ER BGE He SS) LA PE FG \ father Lhe t 4 tad i | | i | ae ieee : Ei a Wave Period -Seconds Figure 24 - Distribution of Characteristic Wave Periods at Station J 32 uey| ssa] juadiad BBE an Ce Are eter? RIT i Hy att Teter eis +t it {| li Hn Wa A hy | ! al etre Veneta See i4 i) cai TMi ibang Ged ie ty thy i eee peyibes i oon Neeettel bienicds i wey) 13}8a15 yuadiad 50 60 70 80 90100 Feet Figure 25 - Distribution of Characteristic Wave Heights Station K at 33 SSS eS SEuiEH iceman joa See ae + ‘ 4 tie fo ne Pa 1 Bureau Data tT 7 Weather 4x U.S. Log Normal Distribution egal f at flit t 50 60 70 80 90100 == 1U9919q - pola JAM BuIpaadx3 Jo Ayijigeqord (d- 1) OOT t + = t Ba c= = s Se Pea sd PA ee + — ae ips= Sal SS; “40 30 = oe Period - Seconds - Distribution of Characteristic Wave Periods 26 Figure at Station K 34 wey} ssa] uaaied 10.01 0 60°70 80 90 100 TI E - = © = + — i = iESe T= [a ar) = = = === 10 s — + ~ aE Se is] c SS: —6§ @ 3 = = = SS 2a OG w = Ss Sore a 3 ocd o — ——'_ -2 0 5 = Se ome uaz tol (2) 2 = 1S ies del 1] - = == oo ease & 2 + . fit fr Al @ | Seen eee) he F SS Ci o 5 i z (aie a) : | SIE Zeno) na P) H FA | ce | wo (=) i jl i i 2) S +t =] LS “= @ o - - = 2 ee: a == iSS== a = = =: = == : = = = = ~~ z a S==— — i pest = = 2 jetsi=e 3 sesterteete = = roto st S bs) — le ra I~ + ia maealilwlal nea RUNOONDN 1 Tht {+ a = 1 { + | = ~ : — r=} coo a -_ o 4 S = 3B & ae} S a so fo 2 eS rn a D> Gn ueys 19}28I5 yuadiadg Feet Distribution of Characteristic Wave Heights gure 27 - i F at Station M 35 que SS FTE nee tend Gesee Soe eee eeoee 100 (1 -P) Probability of Exceeding Wave Period - Percent aii rma f i X U.S. Weather Bureau Dataf He ayy itt eeeea aassie Le Bao ona Tt a ait oo.oa{_| | | uA i 7 W ave Period -Seconds a gael eal = Sone saooe codes sossoresmn HERE ia nL 07 Figure 28 - Distribution of Characteristic Wave Periods at Station M 36 P, Probability of Exceeding Wave Height - Percent 10a Wave Height - Feet Figure 29 - Distribution of Characteristic Wave Heights for Ten Ocean Stations | a aria SPS PSASH EAR AT Ie safle | {dq | { im hones canal can sl fears { | ; 100 (1 - P) Probability of Exceeding Wave Period - Percent a | ———. shH=E ma OC -zmoo® SSeS = Senin Pies = Period - Seconds Figure 30 - Distributions of Characteristic Wave Periods for Ten Ocean Stations 38 APPENDIX A PROCEDURE FOR ANALYSIS OF STEREOPHOTOGRAPHS SELECTION OF PHOTOGRAPHS Many hundreds of stereophotographs were taken during the sea trials of the VALLEY FORGE and the UNIMAK. Since a photogrammetric analysis of each photograph is time- consuming and expensive, a limited number were selected for analysis. The purpose of the stereo analysis was (1) to provide quantitative data against which the visual estimates of trained observers could be checked and (2) to provide a quantitative measure of the sea con- ditions for correlation with simultaneous measurements of the ship’s response to the Sea. Of the many stereophotographs available, sixty were selected for analysis; see Table 4. The selection was made to satisfy both the requirements just stated and to cover as wide a range of sea conditions as practicable. The accuracy of a wave profile varies with the dis- tance from the camera to the profile. The average accuracy is about +0.5 ft at a distance of 2000 ft and is better than this at shorter distances. ANALYSIS OF STEREOPHOTOGRAPHS Fach of the selected pairs of stereophotographs was converted into sea surface profiles by photogrammetric specialists at the Naval Photographic Interpretation Center and Navy Hy- drographic Office. The Wild A5 Audograph and Zeiss Stereo Planigraph Model C5 were used by the respective agencies, and vertical mapping techniques were adapted tothis horizontal application. Next the sea surface profiles were analyzed by the Oceanographic Division of the Hydrographic Office. The procedures devised for this analysis are given in the following sections. PROFILE DETERMINATION FRGM STEREOPHOTOGRAPHS Sea surface profiles were determined from the stereophotographs by the following pro- cedure: 1. Draw the first profile at a distance not less than 250 ft from the camera stations. 9. Draw successive profiles at increments of 125 ft. 3. Draw as many profiles as possible. The profiles should be approximately 1 in. apart on the manuscript. 4. Maintain the horizontal scale constant. Give horizontal scale factor. 5. Exaggerate the vertical scale (wave heights) as much as possible. Give the vertical scale factor. 6. Use number at the left of the profile to indicate the distance from camera station in feet. 39 7. Use number at the right of the profile to indicate the estimated accuracy of the vertical distance in inches. 8. Label doubtful profiles with ‘‘?’’ at the left. 9. Leave the masked portions of the profiles blank. 10. Indicate the dimensions of the sea surface over which the profiles have been drawn. PROCEDURE FOR SELECTION OF PROFILES TO BE ANALYZED Each set of profiles should be analyzed separately. The procedure is as follows: 1. Record the date and time of observation and any other pertinent information. 2. Label the most distant profile with the number 1 and the succeeding profiles 2, 3, 4 .n. Profile 1 is the first useful piece of information. 3. Compare Profile 2 with Profile 1. If there is a distinct similarity in shape between them for a distance greater than one-half the length of Profile 2 (for a noninterrupted distance), then discard Profile 2. Next compare Profile 3 with Profile 1 and test for acceptability in the same manner. Some Profile k is eventually examined which has the property that nowhere does half of its length coincide in shape with Profile 1. Profile k is the second usable piece of information. 4. Compare Profile k + 1 with Profile k in the manner described, and continue the process until all profiles have been exhausted. The net result is p usable profiles, where p | 2250' Fae ne WN Le Ne Ne ae x 22521 i oe ee 200'—+ ee Re a ee Na Oy ae 1750! se Om p-I75—4 EE 0 se a ee P= 150-5 1500' (—~ be— |25'—>} 1000 a ON EE ean [= 100] 750' 75 NI NIRA RI RL RII 500— ~s0-~ 500' 250' ins oe Prepared)by the) photogrammetry Model No J10 - 0040 Numbers at left of profiles repre- Branch, Division of Chart Construc- Compilation Date 1 Feb 1956 Bentidiletances fromtcomeratates tion, U.S. Navy Hydrographic Office tions in feet. Numbers at right of for the David W. Taylor Model Basin profiles representutheihorizontal using stereophotogrammetric methods Beater (stereoplanigraph). Vertical scale is exaggerated 5 times the horizontal scale. Profiles are arbitarily spaced 3/4 inch apart to facilitate calculations. Figure 32 - Wave Profile J-0040 43 TPS- STYO1g OACM - EE oINsIy 9S6I AVW Y31N39 NOMViSedy3iNI DIHdVESOLOHd NSN AG G31IdWOO SIWAYSLNI OO! LV S31I40ud alec |.2109S |DJUOZOH ’ PuDd |Ddy19/, no OF Gor moar Sor é440G2 uoZz140H 44 OSE 44 OSD 43001 uOSS 44 OS9 44 APPENDIX B COMPARISON OF WAVE STATISTICS It is of interest to utilize the wave data obtained during the sea tests of the VALLEY FORGE and the UNIMAK to gain some insight into the validity of a few of the assumptions often made in the forecasting and analysis of ocean waves. For this purpose the following items were computed: (a) The characteristic wave height was determined from the wave profiles. (b) The wave profiles were analyzed according to the method outlined in Appendix A to obtain the frequency distribution of individual wave heights and of the corresponding wave lengths above a certain cutoff length.* (c) The mean value of the squares of all individual wave heights corresponding to waves longer than the cutoff length was determined from the data obtained under Item (b). This value is denoted by the symbol £. (d) The average value of the upper third of the waves having the largest magnitudes (significant wave height) was determined from the data obtained under Item (b). (e) The average value of the characteristic wave heights determined by the shipboard ob- servers was tabulated. (f) The Hydrographic Office computed** a theoretical value of & on the basis of the dis- tribution of wind velocities that generated the sea. The method of Reference 8 was used, ac- cording to which £ is proportional tothe area under the power spectrum of the sea. These values are also given in Table 4. In Figure 6, Item (a) is plotted against Item (e). In Figure 34, Item (c) is plotted against Item (f). In Figure 35, Item (a) is plotted against the square root of Item (f). In Fig- ure 36, Item (d) is plotted against Item (e). Figure 6 indicates that trained shipboard observers can, on the average, estimate the heights of the predominant waves reasonably well. The value of F determined from the wind data should agree with the F obtained from the wave profiles provided the theory of Reference 7 is valid, a narrow sea spectrum exists, swell is a negligible factor, and the stereophotograph covered a representative area of the ocean. One may expect considerable deviations from these assumptions; for example, the sea surface profiles sometimes indicate considerable deviation from a narrow spectrum as well as the *The cutoff length is that wave length below which lies three percent of the area under the power spectrum. For a detailed description of the power spectrum concept and a method for computing E from wind data, see Ref- erence 7. The numerical values of F given in this report are four times those of Reference 7, since we are deal- ing with crest-to-trough wave heights rather than with wave amplitudes. **This computation does not take account of swell that may have been present in the wave system. 45 presence of swell. Nevertheless Figure 34 suggests a linear relationship between the f’s determined by two independent methods. It is concluded that the wind data may be used to determine the sea state, at least qualitatively. Figure 35 suggests a linear relation between the characteristic wave heights h, as de- termined by the method of References 5 and 6, and the square root of E, except for very se- vere sea conditions. Figure 3 would be expected to indicate a linear relationship between the significant wave height and the visual shipboard estimates of the characteristic wave height since the latter is presumably proportional to E in accordance with the indications of Figures 34’and 35. Figure 36 does not contradict such a linear relationship. The scatter of values is most likely due to errors in the determination of the significant wave height, inasmuch as Figure 6 shows that the visual shipboard estimates are reasonably correct. The computed value of the significant wave height, Item (d), is very much a function of the cutoff length. The UNIMAK stereophotographs did not furnish sufficient data, in the opinion of the Hydrographic Office oceanographers, to permit an evaluation of the significant wave height; and therefore these data were not available for the plots. From an overall point of view, consideration of Figures 6, 34, 35, and 36 suggests that 1. The methods of Reference 8 may be applied to make a rough estimate of wind waves. 2. Trained observers can, on the average, make reasonably accurate observations of the heights of the larger, well-formed waves that are present in a given sea. 3. The characteristic wave height reported by trained observers is proportional* to the square root of the statistic E, corresponding to the sea state considered, except for severe sea states. 4, The so-called ‘‘significant’’ wave height is not particularly significant since itis difficult to compute, although it is statistically well defined. The average height of the pre- dominant wave heights,** as reported by observers, is physically more meaningful and is more easily reproduced on repetitive estimates than is an estimate of the significant wave height. *The empirical relationship between the characteristic wave height h reported by the observers and the statistic VE, for the data plotted in Figures 34 and 35, is approximately as follows: h1.53./E when E is derived from wind data, and h~ 1.88 VE when E is derived from the wave data. **Here designated by the term ‘‘significant wave height.’’ 46 30 Figure 34 - Scatter Diagram for Comparison of Values E Derived from Wind and Wave Data nN (=) Each plotted point corresponds to the analysis of one stereophotograph. The computation of EF = = from the wind data neglects the presence of decay- E, ean Square ave Height in ft? (from the Hydrographic Office's Histogram of \/ave Heights) ing swell. The values plotted are taken from the last two columns of Table 4. 0 10 20 30 40 18 16 12 10 h, Characteristic Height of Sea Waves-ft, Determined from Stereo-Wave Profiles 0 1 2 3 4 5 6 7 E2 in ft, Computed from Wind Velocity Data Figure 35 - Scatter Diagram Showing a Plot of Characteristic Wave Height Against the Statistic E” The plotted data are taken from Table 4, Columns 9 and 14. 47 ‘TI pue £4 suuinjod ‘p aTqe.], wos uaye} ase ejyep pazjoyd ayy, JVYSIOF{ OAVM IIYSWOxOVIVYDO YSUIVIYV JYSTO]] OAM JUVOTJIUSIC JO JOT G B DuIMOYS WeIsVIG 10})89S - 9g oINdIY SlaMasqQ pleogdiys yo swoday wory ‘qj-YSIay aref\ WjsuajoeIeY 9 LT 91 SI bl el él I Ol 6 8 L 9 S b eas ‘SAD O Ia4S ‘SAD O 891JJO D1ydessopAH ayy Ag pasedasg SWeISOJSIH WOdJ ‘}jy - JYSIAH AAEM JURIIJIUBIS 48 APPENDIX C SAMPLE CALCULATION In fitting a log-normal distribution to the data on wave heights and wave periods given in Tables 1 and 2, a difficulty arises due to the fact that the lower limit of the lowest class is zero and inasmuch as the logarithm of zero is minus infinity, it is not possible to assign a mean value of the logarithm of the height or period for the lowest class. One way of circum- venting this difficulty would be to use the logarithm of the algebraic mean of the class limits. A less arbitrary solution, used here, is to omit the relatively few values falling into the low- est class and treat the remaining truncated data by the standard statistical method!! for fit- ting a truncated normal distribution. In the statistical sense used here, a truncation means that only values larger than a specified lower limit are used. To fit a log-normal distribution to the truncated data requires the calculation of the mean value and the standard deviation from the truncated data. The method and tables of Reference 11 are applied as indicated below. In the calcu- lations, the symbols used are o for standard deviation and y and z for parameters needed to enter Table IX of Reference 11, z being an estimate of the point of truncation. Following the procedure outlined on page 29 of Reference 10, we have from Table 6: _ (2Now?) (ZN) (459.864) (12,362) vu" Seles 7° p@usKaOs. y = 0.6623 and from Table IX, 2 = -1.293 at y = 0.6623 and g(z) = 0.6736. ENo (4) _ 2071.8 (0.6736) a = Sie 12,362 = 0.1129 oys= From Table II of Reference 11, at 2 = -1.2938, we obtain Theoretical percent of truncation = 9.80 Mean value of w=@=-2S 1.293 (0.1129) = 0.1460 Mean value of A = heo = he +o 0.6990 + 0.1460 = 0.8450 The value of h corresponding to P = 0.975 = hg, « = fg, + 1.96 (0.1129) = 1.0663 The value of 2 corresponding to P = 0.025 = 4, , = hg, — 1.96 (nD) = OZR 49 Taking the antilogarithms, we have The period corresponding to P = 0.975 = antilog (1.0663) = 11.7 sec The period corresponding to P = 0.500 = antilog (0.8450) = 6.99 sec The period corresponding to P = 0.025 = antilog (0.6237) = 4.21 sec Any two of these three sets of values (P, x) determine the straight line (log-normal distribution) plotted in Figure 28. The values of the mean and variance listed in Table 5 are in terms of natural logarithms. Using the conversion log, z = 2.3026 log, «, we have Standard Deviation (log, 7) = 2.3026 (0.1129) = 0.260 Variance (log, z) = (0.260)? = 0.0676 Mean Value (log, z) = 2.3026 (0.8450) = 1.946 TABLE 6 Long-Term Distribution of Estimated Wave Periods at Station M Mean value and standard deviation were calculated from data given in Table 2. The data are truncated at a wave period of 5 sec. Wave |Log,, a at |L8yo @ at |(L0B,, 2) — hy N Percent of Period| End of | Centerof | Measured Number of Class Class from Point Interval Interval | of Truncation o=h- hr Variations Cumulative Percent 9.80 400.369} 49.76 877.862] 81.68 612.839} 96.65 144.663} 99.44 30.774] 99.943 1.513} 99.965 0.556) 99.972 2.406] 100.001 0.623} 100.008 0.6990* 0.8451 0.9542 1.0414 1.1139 1.1761 1.2304 1.2788 1.3222 1.3222 459.864 |2071.605. *This value is the point of truncation A>. Note: This data is the basis for Figure 28. 50 REFERENCES 1. Bureau of Ships letter $29-7(442-440-330) of 21 June 1948 to David Taylor Model Basin. 2. Jasper, N.H. and Birmingham, J.T., ‘‘Sea Tests of the USCGC UNIMAK, Part 1 - Gen- eral Outline of Tests and Test Results,’’ David Taylor Model Basin Report 976 (Mar 1956). 3. Jasper, N.H. and Brooks, R.L., CDR, USN, ‘‘Sea Tests of the USCGC UNIMAK, Part 2 - Statistical Presentation of the Motions, Hull Bending Moments, and Slamming Pressures for Ships of the AVP Type,’’ David Taylor Model Basin Report 977 (Apr 1957). 4. Darbyshire, J., ‘‘Wave Statistics in the North Atlantic Ocean and on the Coast of Cornwall,’’ Marine Observer (Apr 1955), pp. 115-118. 5. Darbyshire, J., ‘‘The Distribution of Wave Heights, a Statistical Method Based on Observations,’’ The Dock and Harbour Authority (May 1956). 6. ‘*A Manual of Marine Meteorological Observations,’’ U.S. Weather Bureau Circular M, 8th Edition (1950), 9th Edition (1954). 7. ‘Instruction Manual for Oceanographic Observations,’’ U.S. Navy Hydrographic Office Publication 607, 2nd Edition (1955). 8. Pierson, W.J., Jr., et al, ‘‘Practical Methods for Observing and Forecasting Ocean Waves by Means of Wave Spectra and Statistics,’’ Hydrographic Office Publication 603 (1955). 9. Jasper, N.H., ‘‘Statistical Distribution Patterns of Ocean Waves and of Wave Induced Ship Stresses and Motions with Engineering Applications,’’ Transactions Society of Naval Architects and Marine Engineers (1956). 10. St. Denis, M. and Pierson, W.J., ‘‘On the Motion of Ships in Confused Seas,’’ Transac- tions Society of Naval Architects and Marine Engineers, Vol. 61 (1953). 11. Hald, A., ‘‘Statistical Tables and Formulas,’’ Wiley and Sons, Inc., New York (1952). 51 ‘ Reva sy ” eae a ua ant Copies 21 NAVY-DPPO PRNC. WASH. D.C INITIAL DISTRIBUTION Copies CHBUSHIPS, Library (Code 312) 1 5 Tech Library 1 Tech Asst to Chief (Code 106) 2 1 Nav Arch Planning Coordinator (Code 320E) 1 Appl Science (Code 370) 1 1 Noise (Code 375) 1 Shock (Code 376) 1 Vibration (Code 377) 2 1 Ship Design (Code 410) 1 Prelim Design (Code 420) 3 Hull Design (Code 440) 1 2 Sci-Struc & Hydro (Code 442) 1 Interior Commun, Fire Control & Naviga (Code 565) 1 2 Radar Br (Code 820) CHBUAER 1 CHONR 1 Deputy Chief & Chief Scientist (Code 102) 1 CO, USN Mat! 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Lunde, Skipsmodelltanken, Tyholt, Trondheim, Norway Mr. Lennart Swenson, Dir, Swedish Shipbldg Res Fdtn, Regnbagsgatan 2, Goteborg H, Sweden Natl Inst of Oceanography, Wormley, Near Godalming, Surrey, England, Attn: Mr. Tucker DIR, BSRA, 5, Chesterfield Gardens, Curzon St., London, W.1, England { ni py ar Wes avis afith LE0-LELSN “Til *H weulioy ‘iedsepe ‘]] “Ta ‘syoog “J (qj30N) wee2Q OUBT}Y — SOABM UBE8DG) “g BBP [Boy -s1j81g — qUeweinsveu Ao -uenboly — SOABM UB8d(—) “% Byep [Bo4sBIg — qYysIoj{ — SOABM UBODE) “T LE0-TSLSN “Ill "H uewsoyn ‘iedsepe [jy “Ta ‘syooig ‘J (q}I0N) UeeDG oUBYy — SOABM UBEeDGQ “g ByVp [Boy -sje}g — queweinseow Ao -uonbolg — SeABM UBEed(,) °Z eyep Teoysyneg — JYSIOY — SOABM UBEd,:) “T ‘U019890] DOAIT B 4B 07848 BOS JB[NONIed B Jo soUepIoUr ey] jo UONBUIWIEJEP OY} OF EpINd jnjosn eB Si UOTNGINSIp Sty} yByy pus reok eo1dsy B JeAO peoueliedxe szySIey eABM JO UOTyNGLIysSIp Ao -uenbeaj 04} 07 e[qeotjdde st uotynqi4jysip [ewsoU-so] B yey) UMOYS SI 4] “S]UeWloInsvoul 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