ing a truncated distribution was applied to ob- 

 tain the results shown in Figs. 20 and 21. The 

 classification used for the motion data (acceler- 

 ations) was coarser than that used for the stress 

 data on this ship. Kolmogorov's statistic was 

 the only measure of significance applied in these 

 cases. An inspection of Figs. 4 and 5 indicates 

 that the log-normal distribution fits the data 

 fairly well. 



Applications of Statistical Distribution 

 Patterns 



A knowledge of the distribution of wave- 

 induced ship motions and stresses together with 

 the height and length of ocean waves is of con- 

 siderable interest in many applications such as 

 engineering specification and design and oper- 

 ational studies of merchant and combatant ships. 

 Following is a discussion of a few representative 

 examples illustrating some problems the solution 

 of which may benefit from this study. 



Specification of Engineering Requirements 



There are numerous occasions where it is de- 

 sired to specify the environmental conditions to 

 which shipborne equipment, personnel, and struc- 

 tures are subjected. In many cases the ship 

 motion will be among the more important items 

 that need to be specified. Several methods of 

 presenting such information have been given in 

 this work; for example histograms, basic and 

 cumulative probability distributions. 



In the design of a ship-stabiUzation system it is 

 necessary to estimate the required capacity of 

 pumps, motors (22), and so on. The cost of the 

 installation is very much a function of the required 

 capacities. The designer must therefore be able 

 to predict the percentage of time that the roll 

 angle will be less than any specified value in order 

 to evaluate the desirability of the installation of 

 the proposed stabilizer. The distribution pat- 

 terns given here are directly applicable to this 

 problem. 



In the design of fire-control equipment, air- 

 craft launching and landing apparatus, it may be 

 necessary to predict the maximum ship motion 

 that may be expected under given sea conditions. 

 Although it may not be practical to design for 

 operation under rare, extremely severe conditions 

 it is desirable to have an estimate of the condi- 

 tions under which satisfactory operations can be 

 expected. The distributions of significant wave 

 heights and wave lengths together with the dis- 

 tribution of ship motions for various combinations 

 of sea condition, heading and ship speed should 

 make an evaluation of the operational possibili- 

 ties practicable. 



Often it is desired to estimate the extreme value 

 of stress or motion that a ship structure may be 

 expected to experience over a specified period of 

 time. For example, the extreme value of the 

 wave-induced, hull-girder stress that can be ex- 

 pected over the service life of a ship is of interest 

 inasmuch as the safety of the ship depends upon 

 keeping the total hull-girder stress below the ulti- 

 mate strength of the structure. 



Therefore, it will be of interest to make an esti- 

 mate of the largest stress to be expected in a 

 given period of time. As an example such an 

 estimate will be made on the basis of the under- 

 lying log-normal distribution for the Esso Ashe- 

 ville shown in Fig. 18. For this purpose a formula 

 will be developed which will give the fraction / of 

 all samples of size N (belonging to a given under- 

 lying distribution) which will have at least one 

 (extreme) value oi x > Xmi- The formula is as 

 follows 



y = 1 — e-«^[i - pcim)] . 



[2] 



This formula may be developed as foUows (as 

 was pointed out to the author by Dr. Lieblein of 

 the National Bureau of Standards) : 



Let P{xm\) be the fraction of the underlying 

 (basic) distribution of x with x < Xmi. 



Take a sample of N values of x in such a way 

 that each x is chosen at random from the basic 

 distribution, and repeat such sampling many 

 times. Then the probability of choosing a value 

 less than Xmi when one single value of x is chosen 

 from one sample of size N, is Pixmi) and con- 

 sequently the probability that some members of 

 the sample of size N exceed Xm\ is 



1 - \P(.x^.)Y 



Therefore the fraction / of all (a very large num- 

 ber) samples of size N which will have one or more 

 values of x greater than Xm, is 



/= 1 - [P{Xr.)V 



This formula is exact, but for large N it is in- 

 convenient. Now for large N it is practically 

 certain that in any given sample, x^i has such a 

 value that P{xmd is close to 1. For values of 

 Xmi such that 1 — P{xm^ is small, a more con- 

 venient approximate formula can be found. We 

 have 



log.(l -/) = 7Vlog.P(x„.) = 



A^log, |1 - [1 - P(xv)]' 

 log. (1 -/) « -N[\ - P{x„d\ 



by use of the series 



log, (1 — x) = —x — X-/2 .... 



33 



