813 NAVY DEPARTMENT THE DAVID W. TAYLOR MODEL BASIN WASHINGTON 7, D.C. APPLICATION OF STATISHMGS TO THE PRESENTATION OF WAVE AND SHIP-MOTION DATA February 1955 Report 813 APPLICATION OF STATISTICS TO THE PRESENTATION GF WAVE AND SHIP-MOTION DATA by Alice W. Mathewson February 1955 Report 813 NS 731-037 ili TABLE OF CONTENTS INBXSSIDRVA (CHE cacescagecs0ec690s200008500950500550000000000500" JUNIE ROY DOL CFE KON cc10003080e6702580050000800000000000000000 PRESENTATION OF DATA .......-.-eeeeeeee OBSERVED WAVE HEIGHTS................ Weather Bureau Data .............ccscceceees Hydrographic Office Data ................ MEASURED WAVE DATA ........-.: eee Geet e eter e ret at acer en eres es aren esas nee ere cet esesever ese seeerereseseresene OBSERVED WAVE HEIGHTS AND MEASURED SHIP MOTIONSG..........cccceceeseessesess ANALYSIS OF DATA ..0...-esseeesesessseeeseteeeees DISTRIBUTION PATTERNS ..............- Breet teeta tee cnet eter e tater seen e tenes eter ets ee esas sees eres eeesenerererene FITTING MATHEMATICAL CURVES TO FREQUENCY DISTRIBUTION ................ MEANS AND STANDARD DEVIATIONS ..........cccsccccsssccecscececerecesarececstececssecesseasensreeeessers CONFIDENCE BANDB.......esesesesseeeseeess CORRELATION BETWEEN WAVE HEIGHTS AND PITCH ANGLES.......... cece DURATION OF SAMPLE .........--.:::seee (COINCIDE IUISIOINIS ‘ccaceccccospacecnoosodoonsdeca5sq0500000" ACKNOWLEDGMENTS .......--essceeeseeeseeeesceees 183) DLP) BIRD) BING Bf Slocgascccosoooescasoossecononoss0on959000000" ote n ener erences n een ener erenec eset er eset aren enes asses eesereeesesesesereee APPENDIX 1 - PEARSON-TYPE DISTRIBUTION CURVES ..........escceceeceeeeeeeeteteneeereee APPENDIX 2 - DURATION OF SAMPLE 15 Me NOTATION Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s Type I Curve Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s Type I Curve Number of distributions required to satisfy conditions assumed in Appendix 2 Deviation of the mean of a sample from the mean of a population Frequency of occurrence Wave height measured from crest to trough Root mean square value of wave heights Fraction between 0 and 1 Mean value of the first pn of n wave heights when arranged in descending order of magnitude Constant taken from a student’s ‘‘¢’’ table Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s Type I Curve Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s Type I Curve Total number of elements which make up the distribution Number of independent observations required to satisfy conditions assumed in Appendix 2. Probability Constant defined by an equation in Appendix 1 and used in evaluating Pearson’s Type I Curve Arithmetic mean of the population (7 = 1, 2, 3, 4) ‘‘zth’’ arithmetic mean of a sample Value of the abscissa in Pearson’s Type I Curve Value of the ordinate in Pearson’s Type I Curve Ordinate of the Pearson type curve at the mode Deviation of any value of f from the mean value of frequency distribution Criterion for the Pearson type curves and defined by an equation in Appendix 1 Criterion for the Pearson type curves and defined by an equation in Appendix 1 Gamma funetion Variable of integration in the evaluation of AP? h Hy Criterion for the Pearson type curves and defined by an equation in Appendix 1 (¢ = 1, 2, 3, 4) ‘‘cth’? moment about the mean, yp, = Instantaneous wave elevation Standard deviation of the population Standard deviation of a sample Sf 2 N Class interval Confidence bands Distribution Frequency Normal distribution Population Probability Probability density Probability level Random Sample Standard deviation Standard error Statistic “*¢” distribution Variability Mode Significant wave height vi DEFINITIONS A grouping of possible values of a variable The interval within which the ‘‘true’’ distribution will fall with a certain probability An arrangement of numerical data according to size or magnitude The number of times a value occurs or is observed A bell-shaped curve, symmetrical about the mean and defined by the mean and standard deviation The entire data from which a sample was drawn if all of it were available The likelihood of occurrence A quantity which, if integrated over the independent variation, is equal to 1; see probability A number which indicates the degree of confidence that can be placed on a given result, i.e., probability level 0.90 means that 90 times out of 100, a given hypothesis will hold The method of drawing a sample when each item in the population has an equal chance of selection A finite portion of the population A special form of the average deviation from the mean, a measure of dispersion, o = Y(2 f 22)/N The standard deviation of a distribution of means The estimate of a number describing the numerical property of a popu- lation The distribution of student’s ¢, defined by ¢ = (X, = X) VN/o where X; is the mean of a random sample of size N from a normal population with a mean X and a is the estimate of the standard deviation of the normal population as estimated from the sample. The variation of the data; the lack of tendency to concentrate The most frequent or most common value; its value will correspond to the value of the maximum point of a frequency distribution. Generally defined as the mean value of the one-third highest waves. Reference 12 and correspondence with the Hydrographic Office indicate that th- wave heights estimated by observers approximated the ‘““‘significant’’ wave heights. ABSTRACT Available observations of wave heights have been assembled and evaluated in terms of statistical methods in connection with the study of the service strains and motions experienced by ships at sea. Curves have been fitted to the distri- bution patterns, and confidence bands, averages, and standard deviations have been computed. Distribution patterns for wave heights observed in different parts of the world are all of the same type with a peak displaced toward the lower wave heights. Pitching motions measured on a ship at sea also follow this same pattern. INTRODUCTICN The David Taylor Model Basin is making a study of the motions and strains in ships at sea for the purpose of evaluating and improving methods for the design of the ship girder and its structural components. It is probable that the frequency-distribution patterns of strains and motions of ships at sea will be similar to those of wave heights. It is also expected that the year-to-year variability in the distribution patterns of wave heights will be of the same order of magnitude as the year-to-year variability in the distribution patterns of ship motions and dynamic hull-girder stresses inasmuch as the latter are, to a large degree, functions of the wave heights and wave lengths. To verify these expectations, observed wave heights have been obtained from the Weather Bureau and the U.S. Hydrographic Office. These data and data measured by the Model Basin on the USCGC CASCO have been studied to determine (1) the type of distribution pattern, (2) the variation in this pattern over a period of time, and (3) the mathematical function which will best fit these data. The results of the third phase of the study are presented in this report. PRESENTATION OF DATA Figures 1, 2, and 3 are frequency distributions of wave heights, that is, depth from crest to trough, obtained from various sources. These distributions are presented in the form of bar-type graphs or histograms. The ordinates of these histograms give the percent of total observations or measured values that fell between given limits of wave height as indicated by the abscissa. OBSERVED WAVE HEIGHTS Weather Bureau Data Figure 1 shows yearly and combined wave-height data which were furnished by the U.S. Weather Bureau at the request of the Taylor Model Basin.! These data represent wave-height DRererences are listed on page 13. og = 20.52 ft? of = 26.01 ft? Qo) — er If Seat Tey asses a Hep 1949 1950 ] 20 ic _ Total Observations = 2832 Total Observations = 2741 | | 21 Observations > 30,3 ft 17 Observations > 30.3 ft 5 i — kane X= 7.50 ft | | $ 10 i | r £ | oO 5 [| | 3 ) ~ © ao Qa c Tos Sia 1951 1952 3 90 Total Observations = 2797 Total Observations = 2820 a 29 Observations > 30.3 ft 6 Observations > 30.3 ft 3 15 X =8.45 ft X =7.40 ft = 2 of = 34.81 ft? z| of = 20.52 ft? = 10 8 fa) a5 fo) 5 10 15 20 29 30 35 (0) 5 10 IB oO) 25 SOMaSo Significant Wave Height, ft Figure la - Frequency Distributions of Yearly Samples distributions determined from observations made by weather ships at ocean station ‘*Charlie’’ (52° N, 37° W) in the North Atlantic from 1 January 1949 to 31 May 1958. The observations were made every three hours by trained weather observers in accordance with instructions prescribed by the United States Weather Bureau.” The observations are reported as the average of the significant* wave heights. Only one quantitative measurement was recorded each time the sea was observed. Hydrographic Office Data Figure 2 shows combined frequency distributions of wave heights for periods of 2, 7, and 40 years tabulated by the U.S. Hydrographic Office? at the request of the Model Basin. These observations, also at station ‘‘Charlie,’’? were made by German merchant ships from 1901 to 1939. The data are not as reliable as the data presented in Figure 1; because routes *Generally defined as the mean value of the one-third highest waves; see definitions page vi. Percent of Total Observations per Class Interval 1949 Sampling Period = 1 year Total Observations = 2832 21 Observations > 30.3 ft X = 17.36 ft o2 = 20.52 ft? + 1949 - 1950 Sampling Period = 2 years T r Total Observations = 55731 38 Observations > 30.3 ft Ie, Ks TS it | SSB ie 1949 - 1951 Sampling Period = 3 years Total Observations = 8370 67 Observations > 30.3 ft Maye i og = 27.26 ft2 — 1949 - 1953 Sampling Period = 4% years Total Observations = 12,365] 73 Observations > 30.3 ft Mz Tod ih val ge = Bd IE 15 QQ YQ Es) O 5 10 15 20 25 30 86035 Significant Wave Height, ft Figure 1b - Combined Frequency Distributions Figure 1 - Frequency Distributions of Samples of Significant Wave Heights Observed at Ocean Station “‘Charlie’’ by U.S. Weather Observers X is the mean and, Ox is the variance. The observations greater than 30,3 ft were included in the totals given but are not shown on the histograms. MEASURED WAVE DATA were often avoided at times of high seas. Fewer extreme values were recorded. Figure 3a is the frequency-distribution pattern of measured wave heights produced by the wavemaker at the Taylor Model Basin. These were simulated to represent a characteristic confused sea. Only 43 measurements were made. Figure 3b shows a frequency distribution of wave heights measured at sea by means of a pressure recorder. These data were tabulated on a form that shows the relation between wave heights, lengths, and periods.* Since measurements were made for a period of only 30 minutes, it may be assumed that they represent the characteristics of the sea at that time and 1939 - 1943 1934 - 1943 Total Observations = 5358 i——j Sampling Period = 7 years Total Observations = 528 Sampling Period = 2 years eo ih X = 3.99 ft o,= 11.77 ft? o2 = 20.06 ft2 i lo) 5 10 15 20 «25 30 35 40 Significant Wave Height, ft 1901 - 1943 Total Observations = 18,627 Sampling Period = 40 years Figure 2 - Combined Frequency Distributions eVGA Ge of Significant Wave Heights Observed at Ocean Station ‘‘Charlie’’ by Merchant Ships Percent of Total Observations per Class Interval a= 28.45 ft? These Data were obtained from the U.S. Hydrographic | Office. X is the mean and a is the variance. 0 5 10 15 2025 30 35 40 Significant Wave Height, ft at that particular geographic location. It should be noted that the waves were of very small height (less than 140cm = 4.6 ft). OBSERVED WAVE HEIGHTS AND MEASURED SHIP MOTIONS As pointed out in the introduction of this report, wave-height distributions could reasonably be expected to have the same type of pattern as ship motions and stresses. This similarity was evidenced by the weather ship USCGC CASCO.° Figure 4 shows the frequency distributions of wave-height observations and pitch-angle measurements made during this test. Both were made at 3-hr intervals over a period of one month at station ‘‘Charlie.’’ No waves less than 1 ft nor pitch angles less than 1 deg were recorded. 357 : 9 Total Observations = 43 '] Total Observations — 688 230) ReUOS | Rees PA of= 1.99 o2 = 883 i) S) 28) | & Beoh 1 @ 157 ian ye) fo) S 10+ + e oe ; S 1S) ane) | 2 3 4 5 6 tr 8 O 20 40 60 £80 100 120 140 Wave Height, in. Wave Height, cm Figure 3a - Frequency Distribution of Measured Wave Figure 3b - Frequency Distribution of Wave Heights at Heights Produced by TMB Wavemaker to Simulate a Sea as Measured by a Pressure Recorder Couiusee! Bea All wave heights for a 30-min period are included. Figure 3 - Frequency Distributions of Measured Wave Heights ELA Total Observations = 237 Total Time Duration = 1 month of Sampling _ X = 6.29 ft a” = 7.28 ft? Total Observations = 3090 Total Time Duration = 1 month of Sampling X = 2.45 ft a= 188 te b fe) oO oO ey fo) ~ oa 20 Percent of Total Observations per Class Interval (al (0) 2 4 6 8 10 12 14 16 18 (e) 2 4 6 a” © 14 16 18 20 Significant Wave Height, ft Pitch Angle, deg (Peak to Peak) Figure 4 - Frequency of Wave Heights Observed and Pitch Angles Kecorded on the USCGC CASCO at 3-hr Intervals over a One-Month Period at Ocean Station ‘‘Charlie’’ 2 X is the mean and 0; is the variance ANALYSIS OF DATA DISTRIBUTION PATTERNS Examination of Figures 1 through 4 shows that all have a similar type of frequency distribution, that is, distributions peak towards the lower wave heights. Similar distributions presented in Figure 11 of Reference 6 also showed such patterns. The data presented there® were compiled from charts of observations made by Japanese merchant ships in the North Pacific during the 15-yr period from 1924 through 1938. These charts’ are available at the U.S. Hydrographic Office at Suitland, Maryland. Areas of observations were broken down into 2-deg squares, that is, 2 deg latitude by 2 deg longitude. A study of these charts, which present the data in the form of histograms, leads to the conclusion that, in general, these histograms are also peaked in the direction of the lower wave heights. On the basis of a study of the experimental data thus far available to the author, there is a strong indication that the frequency distributions of wave heights may be approximated by a straight line when plotted on logarithmic probability paper. Figure 5 shows some of the patterns obtained from the data of Figures 1, 2, and 3 and Reference 6. This approximation of the wave-height distributions by a straight line means that they approach a logarithmically normal distribution, that is, if the frequency is plotted as a function of the logarithm of wave height, the distribution will be normal. FITTING MATHEMATICAL CURVES TO FREQUENCY DISTRIBUTION In addition to the log-normal curve two other types of curves have been fitted to the Weather Bureau data (Figure 1) inorder to find a suitable mathematical function which might be used to represent the observations. The Weather Bureau data were chosen because they appeared to have been obtained by the most reliable and consistent sampling procedure. The fitted curves are shown in Figures 6 and 7. The first is a Pearson Type I Curve whose shape is based on the values of the moments yp, of the given frequency distribution and whose origin is taken at the mode computed from the measured distribution. The curve is defined by the equation given in Figure 6 and discussed in Appendix 1. The second fitted curve, Figure 7, is of the form known as the ‘‘random walk’? distribution. It has been shown® that if the sea elevation é may be represented by a narrow spectrum, the probability that at any fixed location the wave height / lies between / and A + dA is approximately —p2 2 P (h)dh =28 e * dh (W h where h2 is the mean of h2. If the sea has a narrow spectrum, the elevations & of the wave surface have a normal distribution, see Figure 8. PajDdIpPu] AN|DA AADM UD) SSEeq S}YyBIAaH aADM JO jUadIaqG JayeusaeM GWL Aq peonposg syyZtoy{ aAeM pemseaw Jo uorynqrystq Ay]TqGeqosg - 9g em3Iy ‘ulsyBlayy BADAA Painsoey\ SPY IO OAM JO SuOTNGINSIG AIITIGuqorg - g eINdIG potieg IA-¢ & Joy Sdtyg JUeYoIay Aq OTsTOeg YWON UT SeT}TVOOT Mo je paAsasqgO s}yStIoH sAeM JUBSTHIUSTS Jo suoTINgI3ysTq Aj]IGQeqosq - qg emsty 13°44YB19H @ADM jUOdIJIUBIS SIOAIBSGO JoyJeaM pue sdrysg JuRYyoIeW Aq ¢ePTTIVYUD,, UOT}EIS U2 }e paAresgO sj}4sTEeH sAeM yuRorsustg Jo suotnqIystq AyIqeqosg - eG ems o¢ (ord o6s29c6 Pv € Zé o¢ 02 o6ésl9aoGc vv ¢ c Of (oy ol6é8s829 GC » € é g pes T 1p G6 Ol r in Z. 06 (S|DAsajUI 4Y-E 40 [ @pOW SUO!}OAsaSqo) 02 si0ak Z/| p=polsed ajdwos | V 08 SdiyS 194jD9M o¢ 27 Ou Ov Ir T A. o9 Os + Ve (oho) 09 + | 4 ir Ob MoOSI Ol; = (oh 08 + 02 5300S) 06 35 Ol MoS‘ZZ1 No8Z 7 G6 = — vig (Suoljonsasqo wopuos)—4 : SJDAA Ob = poluad ajdwoS 3500S! NoGZEe sdiyuS judyqEW 86 4L =I Ir + + t t + @ (SuO!}OA4aSqO WOpuUDA) 66 4 si0ek Z=poiag ajdwos— | sdiyS junyoua | iN GO 866 ia} + | | | 2:0 666 | aes | "0 Po}DIIPU) AN|DA |aADM UD! aJOW sjyBIayH aADM JO juaD70q MSEC Pearson's Main Type | Curve Pa 0.12 il 7 Ws) my 1m mi y-y(1+2) (1-2) ; on % 0.10} coer a, ie CI a, = 20.2 008 a m= 0.441 defined and evaluated in Appendix 1 2 | M,= 3.49 Bios \L_| Yo = 1357, ordinate of mode in number of ie | Las observations per ft = \ 3 ax = distance from mode, ft © TIN I a 0.04|-+ +t + x x + IL | 0.02 | i | Mode=3.99 ft | cl I 0 Bie 25 30 10 15 20 Significant Wave Height, ft Figure 6 - Pearson’s Main Type I Curve® Fitted to Probability Density Distribution of Significant Wave Heights Observed at Ocean Station ‘‘Charlie’’ by U.S. Weather Observers from January 1949 to June 1953 ol [aa | oe FEEEEH | |_|**Random Walk’’ Distribution} } | | | | aD 2 r ee 72 = 60.0 BEB Ie) Probability Density, P(h) fo) ro) a 2) {e) B 0.02 a 1 Ls (0) 5 10 15 20 25 30 h Significant Wave Height,ft Figure 7 - ‘‘Random Walk’? Distribution Fitted to Probability Density Distribution of Significant Wave Height Observed at Ocean Station ‘‘Charlie’”’ by U.S. Weather Observers from January 1949 to June 1953 | Figure 8 - Wave Record Showing Elevation ¢ and Wave Height A It is not necessarily true that a sea for which the wave heights follow the probability density function [1] will have a normal distribution of &(¢), where €(¢) is the instantaneous wave elevation. Reference 8 gives the ratio co BO i, aay eri 1 [2] hp hp hp h®) denotes the mean value of the first pN of the N wave heights when arranged in descend- ing order of magnitude, where p is a fraction between 0 and 1. Thus the average of the ‘significant waves’? is A“1/3), It should be noted that, experimentally, it is difficult to find A from measurements of inasmuch as the average depends to a considerable degree upon the lower limit to which the wave heights are measured; see Reference 4 for a discussion of this effect. The distribution plotted in Figure 7 is fitted with the mathematical curve given in Equation [1]. The value of 42 which gives the best fit is h? = 60.0." As stated on page 10 the random walk theory holds only if the sea has a narrow spectrum. It may well be that the spectrum of the sea for the wave height distribution shown in Figure 7 will not remain narrow due to the fact that the sampling extended over a period of years. MEANS AND STANDARD DEVIATIONS Means and standard deviations were computed for the distributions shown in Figures 1 and 2 and are given in Table 1. It will be observed that the average of the data obtained from merchant-ship observations is much lower than the average of the Weather Bureau data. From the standard error of the mean (the standard deviation of the distribution of the means of samples) of the four yearly Weather Bureau samples, it may be stated that there are 99.7 *The best fit was determined by a Chi square test. 10 TABLE 1 Means and Standard Deviations for Frequency Distributions of Wave Heights Standard Standard | Average of cea Number of Sample Observations — Computed from Weather Bureau Data (Figure 1a) 1949 2811 7.36 4.53 1950 2724 7.50 5.10 7.69 0.45 1951 2768 8.45 5.90 1952 2814 7.40 | 4.53 Computed from Weather Bureau Data (Figure 1b) J 1 yr 2811 7.36 4.53 | 2 yrs 59390 7.51 4.81 3 yrs 8303 7.82 5.22 mae 12,272 7.76 5.07 | 4830 18,627 chances out of 100 that the average mean computed, 7.69 ft, will be no further away from the true mean than 1.35 ft.? (3c = 1.35 ft) for the period 1949 to 1952. CONFIDENCE BANDS Figure 9 shows confidence bands fitted to the probability density distribution of the Weather Bureau data. These confidence bands, computed according to Kolmogorov’s statis- tic,!° show the interval within which the “‘true’’ distribution will fall at a probability level of 99 percent, that is, in 99 cases out of 100 random sampled distributions, the distribution will fall within these bounds. The requirement for the use of Kolmogorov’s statistic is that the sampled wave heights be random and that the distribution of wave heights be continuous. A plot of the data on probability paper is shown in Figure 9a. The encircled points were computed from the observed wave heights and the solid line represents a logarithmically normal distribution. In this figure the confidence bands were fitted to the observed points. The curve fitted to the probability density distribution shown in Figure 9b was obtained by taking the average probability density of the class intervals at their centers and fairing a curve through these points to make the area under the curve equal to the area under the (qq oins1q ul UMOYS OSTe) ESET ouNnL oO} GFGET ArenueeP WI ssoArosqgQ JeyIVEM *S'Q Aq ,,EeTTIVYO,, UOTI}LIG UBdDO IB PoeAJOSAQO SYSOP] OAV JUROTJIUSIG Jo UONNGI4SIG JO] spueg soUeplUuoD UCdJeg BUIN-AJOUIN - 6 CANS TY ita uot Nqrs}stq uorngIystq Ajtsueq A} ][Iqeqoig Jo} spueq_ aouaptyuo| - q6 amaty AZITIGe qoJq JWSTOH VAM JOJ spueg aousptjuOD - e6 oindIy 43446198H aADM 6 gz 2 02 ey Zl 1 4UOIBH BAOM se. ee : Z Wie @ og 02 o6sz9S b € 2 aes ee str = =e = 8 Ol + 068 | : | 2 ibd one : =f + ft —+—+ a —| Z0'0 = 02 + 08 z | = of | 02 2 | = Ob 09 z 3 = r0'0 3 os oss jiWi7 a@duapljuoD juadsEd GEG 4aMO7 v 5 09 Ov = | fe) . g & OL oe }ilwWi7 aduapljuoD juadsed GEG seddp Si > > co (9005 | ee oee o 5 < — 8 3 06 Ol < a) < < + a 800< = c6 S = a 86 23 8 66 18 a | & (o) Ke) S"66 \— sO | 4 JL = 4 G9E‘Z! “SUO|OAIASQO |D}O) , gle 808K 2/| b ‘polled Buljdwos | ke 12 histogram. The confidence bands were computed for this curve, utilizing Kolmogorov’s statis- tic CORRELATICN BETWEEN WAVE HEIGHTS AND PITCH ANGLES The scatter diagram* of Figure 10 shows the pitch angles measured on the CASCO plotted as a function of wave heights. Except for scattered observations, this diagram indi- cates a correlation between pitch angle and wave height which may be approximated by a straight line. The figure also indicates that the most probable combination is that correspon- ding to about 3-deg pitch angle and 5-ft significant wave height. Oo (Peak to Peak) (For a given observation) Average of One-Third Highest Pitch Angle, deg Significant Wave Height, ft Figure 10 - Scatter Diagram of Pitch and Wave Height Data of Figure 4 A straight line was faired by using pitch angle measurements when ship was headed into the waves; these are indicated by ©; the numbers give the number of observations. DURATION OF SAMPLE An estimation of the period throughout which samples must be taken in order to permit a statistically valid prediction is often necessary. The details of the computation involved for two such methods are given in Appendix 2. *A scatter diagram? is a method of showing the relationship between two associated variables. In this form the independent variable is placed along the abscissa while the dependent variable is placed along the ordinate. It is obvious that if the relationship between the two variables were perfect, every given value on the abscissa would indicate a value of the ordinate. If there is a direct simple relationship between the variables plotted, the points will tend to fall on some curve, possibly a straight line. 13 CONCLUSIONS 1. Frequency distributions of wave heights are not normal but tend to peak toward the lower wave heights. They do, however, have a pattern which is approximated by a logarith- mically normal distribution. 2. The frequency distribution of pitch.angle for the USCGC CASCO has the same general form as that shown to be applicable for the wave heights. It is reasonable to expect that this pattern will also hold for other ships. 3. The frequency distribution patterns of the pitch angles measured on the USCGC CASCO show a correlation with those of the wave height observations. 4. The Pearson Type I Curve may be fitted to frequency distributions of wave heights. ACKNOWLEDGMENTS The studies made in conjunction with this report were done under the supervision and guidance of Mr. N.H. Jasper. Members of Statistical Engineering Laboratory of the National Bureau of Standards, namely Mr. I. Richard Savage, Mr. Marvin Zelen, and Dr. Edgar King made suggestions as to treatment and presentation of data. The wave height data presented in various figures was obtained with the cooperation of members of the Division of Oceano- graphy, U.S. Hydrographic Office and Aerology Branch, CNO. Finally, the author is indebted to Dr. George Suzuki for review of the statistical mathematics in this report and the suggestion of Appendix 2. REFERENCES 1. TMB ltr $29/12 (732:AWM:cae) to CNO (Aerology Branch) of 30 July 1958. 2. ‘*A Manual of Marine Meterological Observations,’’ U.S. Weather Bureau Circular M, 8th Ed (1950). 3. U.S. Hydrographic Office ltr Code 541-LBB/rvg of 10 July 1953. 4. Ehring, H., ‘‘Kennzeichnung des Gemessenen Seegangs auf Grund der Haufigkeitsver- teilung von Wellenhoehe, Wellenlaenge und Steilheit,’’ T.B. 4 (1940), pp 152-155 (Translation: U.S. Hydrographic Office, S10 Wave Project, Report 54, Contract Nobs 2490). 5. Jasper, N.H., ‘‘Study of the Strain and Motions of the USCGC CASCO at Sea,’’? TMB Report 781 (May 1953). 6. Harney, L.A. et al, ‘‘A Statistical Study of Wave Conditions at Four Open-Sea Locali- ties in the North Pacific Ocean,’? NACA Technical Note 1493 (Jan 1949). 7. ‘Waves in the North Pacific Ocean,’’ U.S. Hydrographic Office Jan-Dec, H.O. Misc 11, 117-1. 14 8, Longuet-Higgins, M.S., ‘‘On the Statistical Distribution of Heights of Sea Waves,”’ Journal of Marine Research, Vol XI, No. 3 (1952). 9. Arkin, H., and Colton, R.R., ‘‘An Outline of Statistical Methods,” 4th Ed., Barnes and Noble, Inc., New York, p. 116 and p. 75. 10. Birnbaum, Z.W., ‘‘Numerical Tabulation of the Distribution of Kolmogorov’s Statistic for Finite Sample Size,’’ Journal of the American Statistical Association, Vol. 47, No. 259, p. 425 (Sep 1952). 11. Elderton, W.P., ‘‘Frequency Curves and Correlation,’’ 4th Ed., Harren Press, Washing- ton, D.C. (1953). 12. Sverdrup, H.U., and Munk, W.H., “Wind, Sea and Swell: Theory of Relations for Fore- casting’’, H.O. Publication No 601. 15 APPENDIX 1 PEARSON-TYPE DISTRIBUTION CURVES A set of curves that may be fitted to different frequency distributions was compiled by the statistician Karl Pearson. The theoretical derivation and calculations necessary for fitting these curves are described in Reference 11. The curve type which best fits a frequency distribution may be identified from criteria calculated on the basis of the values of the moments p;. The steps for identifying and computing the constants for, the fitting of the curve shown in Figure 6 are given here. The numerical values are those for the frequency distribution of the Weather Bureau data for the 414-yr period, Figure 1. The moments measured about the mean value of the distribution are: py, -= 2 -0 [3] x f 2 Hy =~ = 18.68 [4] wie By =—— = 40.27 [5] See hq =< = 593.6 [6] where z represents the deviation of the actual value f from the mean, f is the frequency, and N is the total frequency of the sample. The criteria 8,, 8,, and K computed from the preceding moments are: ue B, =— = 0.6336 [7] BS By 2 = Bale [8] 2 Ho B,(B, +3) eS 103090 [9] 4(46, be 38) (2B, 7 38, Te 6) Since the criterion K is negative, it identifies Pearson’s Main Type I Curve as the most suitable one. This curve is defined by the equation y= (1+=] 2 [10] 1 16 where y, is the ordinate at the mode,” z is the distance from the mode, and Taieyue2 are computed from the equations which follow and a 2 pp &a m, ™ pad yet [11] lap First a parameter 7 must be evaluated 6(2,- -l pm peel aie Ha [12] (6 + 38, - 28,) a8 - ollie (B, (r + 2)? + 16(r + 1)} = 22.71 [13] [14] When y, is positive, m, is the root corresponding to the plus sign; if , is negative, m, is the root corresponding to the minus Sign. For our numerical example m, = 0.441 mM, = 3.493 Finally m m igo Mo S My 2 : P(m, +m, + 2) [15] i arita, m+m, U(m, + 1)0(m, +1) (m, + my) 1 Tables of the gamma function are given in the reference 11. With the use of logarithms, y, was computed to be 1404, and the mode of the distribution was found to be at 3.99 ft. Equation [10] becomes: 0. b pole (t-—2 el ae ae [16] 2.55 20.2 This gives the frequency distribution in terms of a class interval of unit length. Therefore for a class interval of 1.6 ft, the frequency would be 2171. Since the probability density distribution was desired, y was divided by N. *The mode is the most frequent or common value; it will correspond to the maximum ordinate of the frequency distribution. 17 APPENDIX 2 DURATION OF SAMPLE Assume in the first approach that for the specific locale indicated, one of the wave- height distributions has a mean X = 7.76 ft and o, = 5.07 ft Then, by standard statistical procedure, the sample size necessary to obtain a sample mean which differs from the true mean by no more than 5 percent with a confidence coefficient of 90 percent can be obtained by solving for n in the equation ee = [17] 0.05 X vn where + & is the particular abscissa on the ‘‘¢’’ distribution with n defined such that the area under the ‘‘¢’’ distribution between * & is 90 percent. By substitution pe [SOO age ORC || that is approximately 467 indcpendent and random observations are necessary. Weather observations are characterized by the lack of independence in successive ob- servations when the time interval between observations is relatively short. The duration of interval necessary to insure independence cannot be determined. If one independent obser- vation can be obtained every 10 days, then by the above calculations, more than 13 years are necessary to obtain a sample fulfilling the stipulated conditions. If 7 days are sufficient, then about 9 years are necessary. As another approach, suppose that the means of the wave heights obtained for each year (Table 1, page 10) represent independent observations. In this treatment the statistical “‘population’’ is the totality of these independent observations. Basing the following compu- tations on the observed mean values, it is found that n = 7. This implies that 7 years of rather extensive observations are necessary to fulfill the conditions imposed. Problem: Find the number of samples which are required to make |2-1) <0.0% [18] x; with a probability of 0.90, where ¥ is the mean of the population and X, is the mean of the ith sample. 18 Procedure: 1. Find the n means of n distributions, XeP Mop nee 2. Assume the mean of the population to equal the average of these means and compute xX ap Ae) #P 000 ah [19] 3. Assume that the standard deviation of the means of n yearly distributions is equal to the standard deviation of the population of means of yearly distributions and compute _ Vc 7 - Vad [20] where d is the distance of the mean of the sample from the mean of the population. 4, The mean of any one distribution of c yearly means may take on a range of values X is the mean value of the population, o is the standard deviation of the population, aa = Ska [21] xe is a sample of c means of the population, é k is taken from a student’s ‘‘z’’ table at probability level 0.90 and (c — 1) degrees of freedom, and c is the number of distributions required; in other words the number of independent observations. 5. From Equations [18] and [21] obtain ko Ve < 0.05 |X,| Thus, solving for /c, gives k “ [22] > = vez 0.05. t 6. Values of c are assumed until Equation [22] is satisfied. Example: Assume each Weather Bureau yearly distribution to represent one sample. Then, from Figure 1 (with n = 4) se X, = 7.36 (year 1949) P< | 7 = 1.50 (year 1950) X, = 8.45 (year 1951) x, = 7.40 (year 1952) 19 Therefore Xx - 30.71 _ 769 4 a -\/2 d? [23] = It o =] /0-10 + OBE 0.591 + 0.08 _ 9 xo and 0.52 i ’ ec > 14" _— 1.4% from Equati 22, Ve> aap (7.36) guaonie when x, is used. As a first approximation, assume c = 5. The value &(¢) from the ‘‘2’’ table at a 0.90 probability level and four degrees of freedom is 2.13 V5 > 1.4 - 2.18 As a second approximation, assume c = 6 V7 = 1.4 - 2.02 Therefore c = 7, indicating that measurements would have to be made over a period of 7 yr to establish a distribution pattern which would be valid at a probability level of 0.90 such that Equation [19] is satisfied, providing that no year-to-year bias exists in the data. " ~ ht" TTA Bers <0 Sy pereeeray acer , (ESTA ORE} ET ute 7 i eet Copies 3 21 INITIAL DISTRIBUTION Copies Chief, Bureau of Ships, Library (Code 312) 10 5 Tech Library 1 Tech Asst to Chief (Cade 106) 1 2 Applied Science (Code 370) 2 Prelim Des & Ship Protec (Code 420) 2 2 Hull Des (Code 440) 2 Scientific-Structural & Hydro (Code 442) 1 CNO 1 Aerology Br 1 CO & DIR, USNEES, Annapolis, Md. 1 DIR, NACA DIR, Natl BuStand 1 2 Statistical Engin Lab Hydrographer, USN Suitland, Md 1 1 Div of Oceanography (Code 541) Military Sea Transport Service, Navy Dept, 1 Washington, D.C. U.S. Weather Bureau, Washington, D.C. COMDT, USCG Headquarters, Washington, D.C. 1 2 Sec, Ship Structure Comm Maritime Admin, Dept of Commerce, Wash., D.C. l 1 Mr. Vito L. Russo, Chief, Div of Prelim Des E.V. Lewis, ETT, SIT, Hoboken, N.J. 8 J.H. McDonald, Nav Arch, Bethlehem Steel Co, 1 Shipbldg Div, Quincy, Mass. 9 E.L. Steward, Mgr, Construction & Repair Div, 3 Marine Dept, Standard Oil Co, New York, N.Y. F.L. Pavlik, Hull Tech Asst, Sun Shipbldg & Drydock Co, Chester, Pa. Scripps Inst of Oceanography, Univ of Calif., La Jolla, Calif. Woods Hole Oceanographic Inst, Woods Hole, Mass. Newport News Shipbidg & Drydock Co, Newport News, Va. 1 Chief Nav Arch Admin, Webb Inst of Nav Arch, Long Island, N.Y. 1 C. Ridgely Nevitt NAVY-DPPO PRNC WASH DC Sec, American SNAME, New York, N.Y. Prof. K.E. Schoenherr, School of Engin, Notre Dame Univ, Notre Dame, Indiana A. Gatewood, Chief Engineer, American Bureau of Shipping, New York, N.Y. J.B. Letherbury, Asst to Nav Arch, N.Y. Shipbldg Co, Camden, N.J. Prof. John W. Tukey, Dept of Math, Princeton, N.J. DIR, Expmt Nav Tank, Dept of NAME, Univ of Mich., Ann Arbor, Michigan Prof. Maxwell Woodbury, Univ of Penn., Phila- delphia, Pennsylvania Capt. R. Brard, Directeur, Bassin d’Essais des Carénes, Paris, France Sr. M. Acevedo y Campoamor, Director, Canal de Experiencias Hidrodinamicas, El Pardo, Madrid, Spain Prof. J.K. Lunde, Skipsmodelltanken, Tyholt Trondheim, Norway Dr. J.F. Allan, Superintendent, Ship Div, Natl Physical Lab, Middlesex, England ALUSNA, London, England BSRA BJSM (NS) CJS Rite eae elie Ht elenay ety PANEL GRE ANY Sin epee a ve a an ie el batt jared 4 paaites 4 ey B Hochevuli? id i a rh Hh L i ibis ant a Sepp 4 yin PD YAY eR Ts EAN LS) MET PR eit 39 pba |! cate Aa bers 0 Nene i io! At glean ror #4 bait nee 7 iar’ y PEPE Fray HN aan ; Ve a Maids i eu SWE AS eTehe VRC: Sila Beate cali Deiat nee yo fea RARER ORIG, NAY? Ris ae ; 0) { ae a Pare: ‘ Bites VCs a te Wits iy (Gi aM i Poe eee iehoge! 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