limited by the assumptions required by their 

 mathematical analysis. 



The purpose of this paper is to present evidence 

 that the wave-induced pitching, rolling, and 

 heaving motion of ships, as well as the associated 

 hull-girder stresses follow simple, mathematically 

 defined statistical frequency or probability dis- 

 tributions. It will be shown that the distribution 

 functions of motions and stresses are of the same 

 type as those applicable to the height of ocean 

 waves.' In particular it will be shown that the 

 distribution function approximates the single 

 parameter "Rayleigh" type for a given state of 

 the sea, ship speed, and course. On the other 

 hand, when distributions of wave height, ship 

 motion or stress experienced over a wide range 

 of conditions are considered, the two-parameter 

 "logarithmically normal" distribution is shown 

 to be applicable over the range of conditions 

 observed thus far. The latter distribution is the 

 result of a summation of many distributions of 

 the first type. Finally, the extreme values of 

 ship motions and stresses will be examined to 

 evaluate the feasibility of the prediction of ex- 

 treme values. 



For the sake of brevity, the distribution ap- 

 plicable to uniform conditions of the sea, ship 

 speed, and course wiU be called "short-term" 

 distribution, whereas the function which repre- 

 sents the distribution obtained when the seas, 

 ship speeds, and courses are allowed to vary over 

 a range of conditions, will be designated as a 

 "long-term" distribution. 



The conclusions arrived at in this paper are 

 based on numerous observations of wave height, 

 hull-girder stress"" (for seven ships) and ship 

 motions (for five ships). The experimental data 

 were obtained over a period of years and cover 

 a wide range of service conditions. 



There are many applications in which a knowl- 

 edge of the frequency distribution of hull mo- 

 tions, stresses, and the heights of ocean waves can 

 be used to advantage. For example : 



1 Prediction of the most probable amplitudes 

 of roll and pitch motion of ships under given 

 environmental conditions. 



2 Estimation of the extreme values of ship 

 response or of wave heights encountered over 

 given periods of time. 



3 Statistical estimation of the capacity for 

 which shipboard stabilization equipment must be 

 designed. 



« The study of ocean waves originated as a by-product of the basic 

 investigation of the ship's response to the sea. 



* Wherever the term stress is used in this paper, a stress computed 

 from the measured strain is to be inferred. 



4 Estimation of the endurance strength of the 

 ship structure. 



The experimental work carried out in connec- 

 tion with this problem has been a co-operative 

 effort of many people and several organizations. 

 In particular the U. S. Navy, The Society of 

 Naval Architects and Marine Engineers, the 

 U. S. Coast Guard, the U. S. Weather Bureau, 

 and the Esso Company have contributed much 

 to make this research possible. The special 

 instrumentation and methods used to collect the 

 data, as well as some of the sea tests are described 

 in references (1-4).^ This paper will not be con- 

 cerned with the methods with which the data were 

 obtained except in so far as they are pertinent 

 to the statistical analysis. 



The general plan of presentation will be as 

 follows: Some pertinent statistical background 

 information will be given first, followed by a short 

 outline of the sources of the data employed in this 

 study. The analyses of short-term and long- 

 term distributions will be given separately. Each 

 analysis applies statistical tests of significance to 

 the hypothesis that the experimental data are 

 samples from a specified distribution. The re- 

 sults of the analyses are discussed and applica- 

 tions of the results to practical problems are 

 illustrated, followed by a summary of the major 

 conclusions. 



Statistical Background 

 The wave heights, the variations in ship mo- 

 tions, and hull-girder stresses experienced under 

 a given set of conditions will be described in terms 

 of their distribution functions. The statistical 

 methods applied in the present study comprise 

 techniques which were used to obtain, classify, 

 analyze, and present large masses of experimental 

 data. In particular they were used to : 



1 Collect the data by sampling procedures. 



2 Classify and condense the data. The data 

 were classified according to the magnitude and 

 frequency of the measured variations. Each vari- 

 ation is understood to mean a fluctuation in the 

 magnitude of the variable from the largest posi- 

 tive value to the succeeding largest negative 

 value; for example in the case of a rolling motion 

 the variation would denote a motion of the ship 

 from the extreme starboard position to the ex- 

 treme port position. 



3 Present the data. The data are given in 

 tabular form as well as in the form of statistical 

 distribution patterns, such as histograms and dis- 

 tribution functions. 



4 Analyze the data. With the data presented 



"Numbers in parentheses refer to the Bibliography at the end 

 of the paper. 



