The correct way tothinkof a wave record is to think of it as composed 

 of a very large number of very low sine waves with phases all mixed 

 up and with different periods in such a way that figure 7 represents 

 the wave record. 



If any one of the above pieces is generated for a long enough time 

 in a wave tank, the waves propagate without change of shape and have 

 classical wave lengths and classical phase speeds after initial transients 

 have died out. Since, as a first approximation, the system combines 

 linearly when all waves are produced simultaneously, the behavior 

 of the sum equals the sum of the behaviors of the individual sinusoidal 

 components. 



Figure 7 explains why it is so difficult to observe the visual properties 

 of waves or to analyze a wave record statistically. The variation 

 in wave amplitudes described at the start of this paper is caused by 

 the complicated effects of phase reinforcement and cancellation of this 

 large (infinite) sum of small (infinitesimal) amplitude true sine waves 

 combined in random phase. 



It also explains the difficulties involved in determining the "periods", 

 since a "period" is the time interval between two successive zero 

 up- crosses. When a sum of, say, fifty or sixty true sine waves is 

 written out and when they are assigned amplitudes according to some 

 spectral law and phases at random, it then becomes difficult, if not 

 impossible, to solve for those times in the record produced where the 

 record adds up to zero and to compute the time intervals between the 

 zeros. These "periods" thus are produced by an interference effect. 

 This is why the probability distribution function of the "periods" 

 is not known theoretically. Mathematicians simply have not yet been 

 able to solve this problem. 



Intuitively, at least, the reason why the average "wave length" 

 is given by equation (37) in a fully developed sea can nov^ be explained. 

 If the wave crests were infinitely long, then corresponding to each sine 

 w^ave in the sum as observed as a function of time at a fixed point, 

 there would be a sine wave on the sea surface as a function of distance 

 along a line. 



Each wave length in feet would be given by 5.12 times the square of 

 the true period of the sine waves in the sum which goes to mcike up the 

 sea surface along the line. The wave lengths are related to the square 

 of the periods. The more rapid oscillations in the record as a function 

 of distance for periods less than the average "period" outweigh the 

 effect of the much less rapid oscillations for periods greater than the 



40 



