It appears from Figure 3-1 that the wave field might be better de- 

 scribed by a sum of sinusoidal terms. That is, the curves in Figure 3-1 

 might be better represented by expressions of the type 



N 

 77(t)= S a. cos(w.-0.) , (3-11) 



j= 1 J J ' 



where n(t) is the departure of the water surface from its average posi- 

 tion as a function of time, aj is the amplitude, toj is the frequency, 

 and (jjj is the phase of the j*^ wave at the time t = 0. The values 

 of oj are arbitrary, and cj may be assigned any value within suitable 

 limits. In analyzing records, however, it is convenient to set 

 (iijj - ZiTJ/D, where j is an integer and D is the duration of the obser- 

 vation. The dij will be large only for those w^- that are prominent in 

 the record. When analyzed in this manner, the significant period may be 

 defined as D/j , where j is the value of j corresponding to the 

 largest a^-. 



It was shown by Kinsman (1965) , that the average energy of the wave 

 train is proportional to the average value of [n(t)]^. This is identical 

 to a^ where o is the standard deviation of the wave record. It can 

 also be shown that 



1 ^ 

 o' = - S a?. (3-12) 



Experimental results and calculations based on the Rayleigh distri- 

 bution function show that the significant wave height is approximately 

 equal to 4a. Thus, recalling that 



and 



then 



". '^ v/2 H^^^ 



H^ « 4a 



or 



a == 0.25 \/2^H^^^, (3-13) 



^rms * 2 V2' o. (3-14) 



The a? may be regarded as approximations to the energy spectrum 



function E(oj) where 



a? 



E(cj) Aw = -^ . (3-15) 



3-12 



