no influence on the merit of the paper or on the 

 results, because only the probability function is 

 used. 



If we define the significant wave heights as being 

 the mean third highest wave, then the distribu- 

 tion of the third highest value is a modification of 

 the distribution of the largest value. The proba- 

 bility points for the distribution of the third 

 highest values are given in the Bureau of Stand- 

 ards Tables on extreme values. 



The distribution of the largest value and of the 

 third largest value have some geometric similarity 

 with the logarithmic normal distribution. But 

 in our case, it is more natural to use the distribution 

 of the third largest value for the significant wave 

 heights. The means of the third highest value 

 should be normally distributed, just as the means 

 of the largest value are normally distributed. 



I agree with the final statement : If the stresses 

 are logarithmically normally distributed, then the 

 largest stress has the first asymptotic distribution 

 of the largest value. These remarks are given 

 only from the statistical standpoint and should 

 not be considered as a critique of the intrinsic 

 technical value of this paper. 



Dr. Jasper: The wide interest shown in this 

 paper is greatly appreciated and the author wishes 

 to thank the discussers for their contributions 

 which will add considerably to the value of the 

 paper. Several discussers (Professors Korvin- 

 KjTOukovsky and Pierson, Dr. Szebehely, Messrs. 

 Press and Marks) indicate the desirability of pro- 

 viding a theoretical foundation for the proposed 

 distribution patterns. The author fully agrees 

 that this would be desirable but is neither in a 

 position to do this adequately nor to evaluate 

 available theories of ship response to waves, such 

 as have been proposed by St. Denis and Pierson, 

 on the basis of the meastu-ements reported here. 

 Therefore, there would appear to be little reason 

 to elaborate on these well-known theories at this 

 time. 



The distributions given here are based on 

 physical measurements, i.e., facts; the statistical 

 theory is utilized to describe these facts concisely 

 and to test if there are significant differences be- 

 tween the assumed statistical distribution and the 

 factual data. Therefore the proposed distribu- 

 tions are derived from experiment alone, that is 

 they are empirical and therefore are not limited 

 by prior assumptions such as always are the basis 

 of theories. This is contrary to the opinions ex- 

 pressed by some discussers. Dr. Chadwick's 

 comments on this matter are very much to the 

 point and are fully concurred in. 



It is not precluded, as pointed out by several 

 discussers, that mathematical distributions may 

 be found which fit the data better than those sug- 

 gested here, though no doubt at the expense 

 of additional complexity. Undoubtedly the true 

 physical distribution is much more complex than 

 any discussed here but this does not prejudice 

 the practical utility of a much simpler distribu- 

 tion. 



It has been stated by Mr. Lewis and others 

 that the Rayleigh distribution applies only to 

 time-stationary processes with narrow spectra 

 and that therefore the author's work indicates 

 that most sea and ship-response spectra are 

 "narrow." Objections can be raised to the term 

 "narrow" spectrum even if the assumption of a 

 stationary time series process is accepted, al- 

 though the applicability of this or other dis- 

 tributions- is obviously not restricted to time- 

 stationary processes. A paper by Cartwright and 

 Longuet-Higgins^^ has demonstrated that, for 

 the representative cases studied by them, no sig- 

 nificant deviation between the Rayleigh distribu- 

 tion and test data was apparent for spectra with a 

 "root mean square width" e of less than 0.48, 

 where e = corresponds to the exact Rayleigh 

 distribution and e = 1 to the normal distributions. 

 Although they did find significant deviations from 

 the Rayleigh distribution for some ocean-wave 

 records no such deviations were indicated for 

 the ship's response to the waves. We might then 

 define "narrow" spectra as those for which e < 

 0.50. 



Several discussers (Marks and Pierson) have 

 mentioned the recent paper by Cartwright and 

 RydilP' which makes some comparisons be- 

 tween distribution patterns obtained by theory 

 and by sea tests. The sWtistical function used by 

 them to represent the short-term distribution of 

 ship motions is more general than the Rayleigh 

 pattern. They applied it to the distribution of 

 the maximum deviations from a mean value 

 rather than to the peak-to-peak variations dealt 

 with in the present paper. It should be noted 

 that these two variables in general would not be 

 accommodated by the same function. The authors 

 claimed that there were no significant differences 

 between the theoretical distribution and the ex- 

 perimental histograms but did not give the basis 

 for this claim. Personal correspondence with 

 the authors indicates that the question of good- 

 ness of fit had been somewhat neglected and that 

 there were indeed significant departures from the 



'= "The Statistical Distribution of the Maxima of a Random 

 Function," by D E. Cartwright and M. S. Longuet-Higgins, Proc. 

 RoyalSociety, vol. 237, 1956. 



54 



