scale could be made directly in terms of the 

 parameter E rather than measuring the variable 

 as a function of time. 



Dr. Szebehely's question regarding the long- 

 term distribution applicable to a destroyer shown 

 in the paper can be answered as follows: (a) The 

 line would be different for different operating con- 

 ditions or for a substantially different destroyer. 

 It would not be necessary to measure stresses for 

 long periods of time to evaluate the relative merits 

 of various ship designs. One would need to find 

 the parameter E for representative operating con- 

 ditions utilizing either model or full-scale tests or 

 theory. The long-term distribution could then 

 be synthesized as noted before. Running several 

 destroyers side by side in the same sea for 30 

 min would give a comparison for the test con- 

 ditions but not for other operating conditions un- 

 less recourse can be had to a valid theory. Further- 

 more, it would appear to be a rather impractical 

 solution. The best approach, at this time, would 

 appear to be in the use of model tests to determine 

 and compare the desired distributions. 



Dr. Szebehely's suggestion of determining the 

 transfer functions by model tests in regular waves 

 and then utilizing the energy spectrum of the sea 

 to find the behavior of the ship is, the author be- 

 lieves, probably the best immediate solution to 

 the entire problem. However it requires that the 

 method be shown to be applicable over a suffi- 

 ciently wide range of conditions. 



It is heartening to see the good checks Mrs. 

 Bledsoe was able to provide between the direct 

 application of the Rayleigh distribution and the 

 autocorrelation technique. If a rapid computer 

 such as the UNIVAC is available then the use of 

 the autocorrelation techniques may be just as 

 quick and certainly more informative than the 

 direct method of computing the mean square 

 value of the measured variation. However the 

 availability of such computing facilities is a 

 severe restriction on the general utility of the 

 autocorrelation technique. 



Mrs. Bledsoe indicates that significant and 

 average heights, and so on, can be obtained from a 

 knowledge of the total energy of a power spectrum. 

 This is not believed to be generally true; it is of 

 course true for a "narrow" spectrum. 



The physical example of a distribution problem 

 of pitch motion was cited in which the significant 

 wave heights varied over a range of from 5-15 ft 

 and the Rayleigh distributions still gave an ade- 

 quate representation. This is of course interest- 

 ing and significant but it does not in any way con- 



flict with the applicability of the log-normal dis- 

 tribution for representation of long-term dis- 

 tribution. The variation of the environmental 

 conditions simply did not cover a wide enough 

 range. This illustrates the fact that although a 

 narrow spectrum gives rise to a Rayleigh dis- 

 tribution the reverse is not necessarily true. 

 This point seems not to have been given emphasis 

 thus far and is pertinent to the comments of 

 Professor Pierson. 



The comments of Commander Brooks as well 

 as the considerable personal contributions made 

 by him to this program diu-ing his tour of duty at 

 TMB are deeply appreciated by the author. 

 Commander Brooks has rightly called attention to 

 the possibility that with the advent of nuclear 

 power it may occur that full utilization of the 

 available speed potential will be limited by the 

 lack of adequate seaworthiness characteristics. 

 This may weU become a major problem facing the 

 naval architect in the not too distant future. 



Dr. Gumbel is a well-known authority on ex- 

 treme value theory and the author is pleased to 

 receive the approval of so well-qualified a statis- 

 tician on his approach. The author regrets he 

 did not appreciate the fine distinction between the 

 logarithmic normal distribution and the normal 

 distribution of the logarithm. For the record, 

 the definition from Cramer (reference 6) will be 

 given : If log x is normal (m, o-) then the variable x 

 itself has the frequency function 



p{x) 



aX yJ%T 



This distribution is called the logarithmic normal 

 distribution. Thus the same physical situation is 

 equally well described by the normal distribution 

 of the logarithm of x and by the logarithmic nor- 

 mal distribution of x. The only difference is in 

 the mathematical expression utilized; i.e., seman- 

 tics. In order to keep to the usage of statis- 

 ticians the author has in the paper now given the 

 expression for the probability density of x rather 

 than the density of log x. 



Dr. Lieblein's comments are greatly appreci- 

 ated because, coming from a statistician, they 

 should help engineers evaluate the adequacy of 

 the statistical treatment. 



In closing, the author wishes to express his sin- 

 cere thanks to all for their interesting discussions. 

 It is hoped that some of the questions raised have 

 been answered. Any further questions that may 

 arise will receive the serious attention of the 

 author. 



58 



