and a large aircraft carrier. Figs. 6 through 11 

 show typical distributions obtained from the sea 

 trials of the destroyer and carrier. It is again 

 evident that there is no significant difference be- 

 tween the measured data and the hypothesis that 

 the data are samples taken from a Rayleigh dis- 

 tribution. 



It will be of interest to compare the largest 

 measured value out of a given number of meas- 

 urements with the most probable largest value 

 predicted by the statistical theory. Longuet- 

 Higgins shows that the largest probable value 

 Xm,x out of iV-measurements is -y/E times a con- 

 stant if the population is of the Rayleigh type 

 where the constant is a function of N only. For 

 N large this constant becomes approximately 

 equal to V loge N. Table 4 gives the comparison 

 of predicted and measured values for a number of 

 tests. There appears to be satisfactory agree- 

 ment between measurement and predicted values. 



On the basis of these data, applicable to a wide 

 variety of sea conditions and ship types, it is 

 concluded that the wave-induced motions and 

 stresses of ships are distributed according to the 

 Rayleigh distribution, for any given set of steady 

 conditions of the sea, ship speed and heading. 



Analysis of Long-Term Distribution 

 Functions 

 Background 



It is the main purpose of this section to show 

 that wave-induced hull stresses and motions of 

 ships may be approximated by a logarithmically 

 normal distribution if the environmental condi- 

 tions under which the data are secured include a 

 wide variety of sea conditions, ship's headings 

 and speeds, such as a ship would be likely to 

 encounter over a period of weeks, months or years. 

 It is a secondary objective to show that the 

 heights of ocean waves may be represented by a 

 log-normal distribution. 



When this study was initiated several years 

 ago, there were few data available on the motions 

 and hull-girder bending moments which ships 

 experience in service. It was realized that large 

 quantities of data would have to be collected in 

 order to provide a reasonably firm basis for the 

 objective of this program which is the specifica- 

 tion and prediction of the service stresses and 

 motions of ships. Statistical methods were to be 

 utilized in the planning of the tests and the col- 

 lection, presentation, and analysis of test results. 

 It was felt desirable, early in the planning stage, 

 to have a preliminary idea of the general type of 

 frequency distributions which might be applicable 

 as well as of the length of time over which meas- 



urements would have to be made in order to estab- 

 lish the desired distribution patterns. 



Because the stresses and motions of ships are 

 caused by the passage of the waves past the ship, 

 it was thought desirable to study the extensive 

 wave-height observations that had been collected 

 over a period of many years both in the Atlantic 

 and Pacific Oceans with the expectation that the 

 frequency-distribution patterns of ship response 

 would, to some degree, be similar to those appli- 

 cable to the wave data. Inasmuch as the wave- 

 height data comprise visual observations, the dis- 

 tributions will be evaluated in a qualitative 

 sense only; that is, it will not be possible to assign 

 statistical significance levels. The wave observa- 

 tions will be analyzed first, followed by the hull- 

 girder stress and ship-motion data. Most of the 

 analysis will be concentrated on the hull stresses 

 because much more stress than ship-motion data 

 have been measured to date. 



Heights and Lengths of Ocean Waves 



Fig. 12 shows a chart on which the several 

 weather stations, manned by weather ships, are 

 marked. For each of these stations the U.S. 

 Weather Bureau has collected observations of the 

 significant wave height^" and wave period, re- 

 ported every 3 hr, over a period of about 6 years 

 in accordance with instructions of the U.S. 

 Weather Bureau (8). Detailed study of these 

 observations indicate that if the data are plotted 

 on log-normal probability paper and if a log- 

 normal distribution is fitted to the plotted data,'' 

 very good agreement is obtained. Fig. 13 shows 

 the distribution for Station C. This curve is 

 typical of the distributions obtained from the 

 U.S. Weather Bureau data. Station C is the one 

 at which the sea trials of the USCGC Casco (1) 

 were held. The essential statistical data for sig- 

 nificant wave height are given in Table 5. Com- 

 parison of the distributions obtained for each 

 year with the sum of the observation over a period 

 of about 6 years indicates that one year's observa- 

 tion gives a good approximation to the distribu- 

 tion obtained over the longer period of time. 

 The details of the comparison will not be given 

 here. 



Recently J. Darbyshire (9) has obtained meas- 

 urements of wave heights by means of a wave 

 meter installed on a weather ship. The measfire- 

 ments were made over a period of about a year — 

 February 1953 to January 1954— at the North 



" The significant wave height is the mean height of the upper 

 third highest waves. 



•1 The fitted log-normal distribution corresponds to the value of 

 the mean and variance computed directly from the wave data. 



15 



