58 



THEORY OF SEAKEEPING 



1.0 



0.5 











0.5 



1.0 



1,5 



2.5 



H, 



50 per cent 

 Fig. 66 Values of parameters a and which define shape of autocovariance function (from Voznessensky and Firsoff, 1957) 



were presented by Jasper (3-1956). Reference was 

 made to H. Cramer, "Mathematical Methods of Statis- 

 tics," Princeton University Press, Princeton, X.J., 1946. 

 The histograms describe an ol)served sea without dis- 

 closing functional or causal relationships. By plotting a 

 series of histograms (as for instance in Fig. 62 for a series 

 of wind speeds), an idea of the functional relationship 

 may be achieved, but not a mathematically defined 

 quantitative relationship. Since the plan of this mono- 

 graph is to trace the C[uantitative effect of wa^•es on ship 

 motions and on ship stresses, the histogrammatic ap- 

 proach will not be di.scussed further. Attention will be 

 concentrated on spectral analysis, which permits the 

 formulation of functional relati(}nships. 



8 Mathematical Representation of the Sea Surface 



8.1 Probabilistic Versus Deterministic Representa- 

 tion. In the first five sections of this chapter, the reader is 

 acquainted with the great amount of time and effort that 

 has been expended in the development of the \'arious 

 theories relating the generation and growth of ocean 

 waves to the winds blowing o\-er the sea .surface. It is 

 ciuite natural that the reader should come to expect at 

 least some attempt at a partial unification of all this 

 work with perhaps a view toward formulation of a deter- 

 mmistic representation of the sea surface. Unfortu- 

 nately, and regrettably, this is j'et to be done. Notwith- 

 standing the wealth of ideas and material revealed in 

 these five sections, a champion is yet to appear on the 

 oceanographic scene who can specify the precise behavior 

 of the sea surface for all time and/or space simply on the 



basis of a set of initial conditions. The fact is, we just 

 don't know enough about the interaction of the wind 

 with the sea surface to do this. 



This is not difficvilt to imderstand if one but looks at a 

 time history of the waves passing a fixed pohit. Fig. 61. 

 E\'en this simplified description of the seaway illustrates 

 finite clearly the absence of any regularity; high waves 

 follow low wa\-es in no ,set pattern and even "period"' 

 measurements fail to demonstrate any regularity. To 

 make matters worse, the appearance of extra small wig- 

 gles in the record defies classification and many high- 

 freiiuency wa\'elets are not even sensed by most trans- 

 ducers. Examination of very long records (1 hr dura- 

 tion) reveals that no portion of the record repeats itself 

 throughout. It is natural therefore to assign a property 

 of "randomness" to ocean wa\-es. On the basis of this 

 assumption and fortified with ignorance of any deter- 

 ministic method for repre.'^entation of the sea .surface, the 

 appeal of a probabilistic definition could iiardly be re- 

 sisted. 



The suggestion of prot)al)ilistic methods was not im- 

 mediately embraced becau.se there was a ntitural suspi- 

 cion (b3' those not well grounded in this branch of mathe- 

 matics) of the implication of qualified .statistical aver- 

 ages. Howe\'ei-, when the \-irtues of probabilitj' theoiy 

 are examined, in terms of defining the state of the sea, 

 the matter takes on a positive hue. Consider again, the 

 random (stochastic) process in Y\g. 61. One fact be- 

 comes strikingly clear. This precise e\'ent, recorded in 

 the open sea, may be expected never to occur again. 

 Consequently, any description of this event as a particu- 

 lar realization of the seaway is only useful in a very 



