Understanding and Prediction of Ship Motions 



shallower. The density function for the waves observed as a function of time at 

 a fixed point will then have a certain amount of skewness that could be investi- 

 gated in terms of bi- spectra. 



Wind waves have been carefully measured by Kinsman (1960) and found to 

 be non- Gaussian. His data have been used by Longuet-Higgins (1963) to verify 

 a theory for the probability structure of the waves and this probability structure 

 has been adequately represented by a modified Gram-Charlier series. The 

 mathematical techniques of Longuet-Higgins would be applicable to the study of 

 the nonlinear aspects of certain ship motions. 



Another example of great interest to this symposium is the example pro- 

 vided in the comments of Dr. Yamanouchi. He has solved the very complicated 

 problem of the nonlinear damping of the rolling motion of a ship in irregular 

 waves in terms of a random process and second order nonlinear correction to 

 the motions. Just as the work of Dalzell was cited by Dr. Ogilvie as establishing 

 the principle of linear superposition assumed by St. Denis and Pierson, some 

 investigator now needs to study the rolling motion of a ship in long crested beam 

 seas in order to see if it is possible to verify the nonlinear probabilistic theory 

 of roll damping propounded by Dr. Yamanouchi. It is quite likely that this non- 

 linear theory will verify quite well and that the spectra of the rolling motion as 

 predicted by this theory will agree with the observations. Nonlinear roll in short 

 crested waves requires very careful control of resolution, sampling variability, 

 and coherency calculations in the analysis of the time series that would be re- 

 corded. Still missing is the probability structure of the rolling motion. Per- 

 haps the techniques of Longuet-Higgins (1963) could be applied here to obtain it. 



CONCLLTDING REMARKS 



The essential feature of the work of St. Denis and Pierson now appears to 

 be that of expressing the short crested waves and the resulting ship motions in 

 terms of a probabilistic description instead of in terms of a deterministic one. 

 The assumption of linearity so convenient in order to obtain results on the 

 probability structure of the resulting ship motions is no longer absolutely es- 

 sential toward the further understanding of the motions of ships at sea. Insofar 

 as the actual waves that force the ship satisfy nonlinear differential equations, 

 it should be possible to model these waves with these essential non-linear fea- 

 tures as accurately as desired by means of continued efforts along the path out- 

 lined by Tick in the work cited above. At the same time whenever it should turn 

 out that the differential equations that describe a particular phenomenon are 

 nonlinear, a perturbation technique such as the one described by Dr. Yamanouchi 

 should make it possible to obtain the spectra of these motions and certain of the 

 statistical properties of these motions. Even at times the probability density 

 functions that provide considerable information about the phenomenon can be ob- 

 tained by analogy to the results on the forces of waves on a pile. The strength 

 of the techniques that have been developed quite obviously then lies in the as- 

 sumption of randomness and the use of the very powerful tool of probability for 

 the derivation of results and the very powerful tool of statistics in the analysis 

 and interpretation of observations. 



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