scribe the actual data with a degree of significance 

 comparable to that shown by the author. 



By reanalyzing the data according to the power- 

 spectrum concept, the total area under the spec- 

 trum was obtained and this was compared with 

 the author's parameter E which defines the 

 Rayleigh distribution. The maximum discrepancy 

 found was of the order of 7 per cent. This, of 

 coiu-se, means only a 3.5 per cent error in the de- 

 termination of the significant or average height 

 of the motions. 



One sample might be cited in particular for 

 which the test conditions corresponded to those in 

 Fig. 6 of the paper with the exception of the signifi- 

 cant wave height which was 10 ft instead of 14 ft 

 (the significant wave height appropriate for Fig. 

 6). The Rayleigh distribution for this case 

 showed that the pitch angle having the highest fre- 

 quency of occurrence was about 1.9 deg. If 

 linearity of response can be assumed, the most 

 probable pitch angle when the ship is running in 

 waves whose significant height is 14 ft would be 

 2.7 deg. This appears to be in close agreement 

 with the results of the author's Fig. 6. 



One further remark may be of some interest. 

 Data obtained on the pitching motions of a trans- 

 port ship also were fitted by the Rayleigh dis- 

 tribution. The data were collected at 3^-hr in- 

 tervals over a 36-hr period. The significant wave 

 height varied from approximately 5 to 15 ft and 

 the wave period varied from 5 to 12 sec. The 

 wave direction during this period changed by 3.5 

 deg. The average speed of the ship was 17 knots. 

 While, according to the author a log-normal dis- 

 tribution should more accurately describe the 

 data, the Rayleigh distribution was found to pro- 

 vide a very reasonable fit in this case also. 



Finally, one might remember that while the 

 author's method does seem to offer a convenient 

 method for organizing a large amount of data, 

 the power-spectrum technique, while more labori- 

 ous, appears to offer a solution of even greater 

 practical significance. 



Prof. W. J. Pierson, Jr.," Visitor: This very 

 fine paper will certainly provide the naval archi- 

 tect with important design criteria in his work. 

 The words of warning in the author's discussion 

 are particularly apt, and they should be heeded! 

 The author of these comments agrees with a 

 major portion of the results of the paper and be- 

 lieves that quite a few years will elapse before 

 they are superseded by more refined analysis. 



A few points need to be made, however. 



27 Department of Meteorology and Oceanography, Research Divi- 

 sion. College of Engineering, New York University, New York, 

 N. Y. 



The author states, "the experimental approach 

 taken in this paper should complement the theo- 

 retical work of St. Denis and Pierson, although the 

 present study was developed independently of 

 theirs and is not limited by the assumptions re- 

 quired by their mathematical analysis." In my 

 opinion, all of his results are strictly limited by the 

 assumptions of the mathematical analysis em- 

 ployed by Mr. St. Denis and the writer. In par- 

 ticular our assumption that the sea surface could 

 be represented by a stationary Gaussian process 

 and that the response of the ship to the waves is 

 linear is at the very root foundation of the Ray- 

 leigh distributions studied in this paper. It also 

 follows that these assumptions are at the founda- 

 tion of the log-normal distribution. 



It is interesting to note that the Rayleigh distri- 

 bution and the log-normal distribution fit so weU. 

 However since a stationary Gaussian process is a 

 linear process, since waves are definitely nonUnear, 

 and since extreme ship motions are nonlinear, it 

 follows that neither the log-normal nor the Ray- 

 leigh distribution will truly describe conditions at 

 extremely high values. That no such discrep- 

 ancy was detected in this work is probably due 

 to observational error and sample size, but never- 

 theless it must exist. 



As an example, consider a sea generated by a 30- 

 knot wind blowing for an infinite length of time 

 over an infinite fetch. A wave record from such a 

 sea would have a definite E value. If one waited 

 long enough, according to the extreme-value 

 theory of Longuet-Higgins, a wave higher than 

 any preassigned arbitrarily high value would pass. 

 However, the sea surface is nonlinear, and the 

 spectrum appears to have no energy below a cer- 

 tain frequency. Therefore waves in excess of a 

 certain height cannot exist because they would 

 break. Our knowledge of the process is not suffi- 

 ciently precise to tell us whether the limiting non- 

 linear wave is given by the highest wave out of 

 1000, 10,000, 100,000 or 1,000,000 according to 

 the statistics of Longuet-Higgins. Similar re- 

 marks are true about any other quantity with a 

 Rayleigh distribution in this paper. 



Similarly there is an upper bound to the wind 

 speeds possible over the ocean, to the duration of 

 the wind and to the length of the fetch over which 

 it blows. And there is a highest possible wave 

 due to nonlinear considerations for these condi- 

 tions. The log-normal distribution may weU be 

 too high for higher values for the foregoing 

 reasons. 



The Rayleigh distribution fits a great deal of 

 these data quite well. Since the "Chi Square" 

 test gives the probability that a sample truly 



49 



