cal situation." A paper by Williams was re- 

 ferred to as an illustration of a physical basis for 

 the Rayleigh distribution. The mathematical 

 expression given by Williams for a "fluctuating 

 oscillation" together with the stipulated assump- 

 tions made by him does give rise to the Rayleigh 

 distribution. But what is the advantage of 

 carrying out such a derivation? If the mathe- 

 matical model did describe the physical situation 

 correctly and generally, this would indeed prove 

 the Rayleigh distribution to be the applicable dis- 

 tribution function. However, there appears to 

 be no reason to believe that this mathematical 

 model is the correct one, and if it is not, then the 

 mathematical development is superfluous. 



It was the intention in this study to work di- 

 rectly with service motions and stresses of ships 

 thus arriving at results which do not depend upon 

 the validity of a priori assumptions. Thus there 

 has been no attempt made here to relate cause 

 and effect. If a theory relating the sea to the 

 ship's response should be proven practical to 

 apply, then distribution functions would still have 

 to be established. Thus it seemed prudent to 

 start by making an attempt to establish these 

 probability distributions directly since they would 

 be of immediate utility, irrespective of the avail- 

 ability of any suitable theory on waves and the 

 ship response to waves. 



The comments of Professor Pierson have, for the 

 most part, already been discussed. The author 

 agrees that the proposed distributions cannot be 

 valid for very extreme values; this limitation was 

 stated in the paper. The limitation is however 

 not serious in many practical applications. For 

 example, steady-state conditions of ship speed, 

 heading, and so on, wUl not generally be main- 

 tained long enough to result in predictions which 

 are too extreme. 



The remarks of Dr. Chadwick are much ap- 

 preciated especially so because he, in a way, 

 represents the consumer. He is in a position to 

 utilize the information presented here and it is 

 pleasing to hear his fresh and forthright com- 

 ments on the paper and on some of the points 

 raised by others. In fact, Dr. Chadwick has 

 effectively answered some of the points raised by 

 other discussers. 



Mr. Erickson's problems are typical of those 

 met more and more often in recent years. The 

 question of specification of environmental condi- 

 tions as to rigid-body motion, vibration, and 

 shock may well benefit by consideration of statis- 

 tical methods for the collection, presentation, and 

 analysis of the data. In some of these areas we 

 are not as yet much concerned with a high degree 

 of accuracy in specifying the environmental con- 



ditions but rather with assigning a reasonable 

 order of magnitude and the probabilities asso- 

 ciated with them. 



Dr. Szebehely is amazed "by the fact that such 

 a large amount of data could be organized accord- 

 ing to relatively simple basic laws." He probably 

 means to imply that the basic laws are really 

 much more complex. The author does not doubt 

 that this is true and has not claimed that the 

 Rayleigh and log-normal laws are applicable basic 

 laws. He merely takes the position that these 

 simple laws give results which are good enough to 

 be useful and possibly just as good as would be 

 obtained if the correct law were known, consider- 

 ing the statistical variations together with the 

 various unknowns and inaccuracies that enter the 

 problem. 



The author is reasonably satisfied to accept the 

 good fortune of having simple laws available which 

 describe the situation fairly well without knowing 

 why this is so. Life and Nature are full of mys- 

 teries — if we should refuse to accept their mys- 

 teries unless given a scientific explanation, prog- 

 ress would be at a snail's pace. 



I shall attempt to answer several of Dr. Szebe- 

 hely's questions. A number of statistical laws 

 were tried before testing the log-normal law. The 

 latter was tried because it has been found appli- 

 cable in a number of other physical phenomena. 

 As far as I can determine there is no definite re- 

 lation between the short-term Rayleigh distribu- 

 tion and the long-term log-normal distribution on 

 a purely mathematical basis unless one assumes 

 an a priori distribution of Rayleigh distributions. 

 Nature of course provides a distribution of short- 

 term distributions and it may eventually be possi- 

 ble to establish the connection then. 



The question raised as to the manner of apply- 

 ing the results of this paper to the evaluation of 

 ship performance could be discussed at length. 

 However, briefly the following may be suggested: 

 (a) Assuming the model test procediu-es are 

 shown to be valid then model tests may be utilized 

 to determine E for short-term distribution. The 

 parameter E is an index of performance. Thus a 

 single number may be a good criterion of per- 

 formance rather than a graph or a table of values. 



(&) Long-term distributions of ship response 

 may be prepared by synthesis, utilizing typical 

 short-term distributions of ship response each 

 specified by the parameter E, together with the 

 long-term distribution of sea conditions given in 

 the paper. This has been done in TMB reports. 



(c) Expected extreme values can be given on 

 the basis of the parameter E and the expected 

 number of variations. 



{d) Measurements on both model and full 



57 



