Davis and Oates 



$^^f) . -^e-'^^/- f (ft)Vcps (28) 



where f is the wave frequency in cycles per second, v is the generating wind 

 speed in knots, and Cj and c^ are constants. A typical wave elevation spectrum 

 is shown in Fig. 9. 



Implicit in the description of the seaway as a stationary Gaussian random 

 process is the assumption that the instantaneous surface elevation at any point 

 results from the superposition of an infinite number of small sinusoidal compo- 

 nents of different frequency, phase and direction of propagation. Analytically, 

 the wave elevation can be expressed as a stochastic integral of the form 



Z(t) = COS [ojt - 0(aj)] V2<I>2(f) df 



•'0 



(29) 



where co = l-ni and $(cj) is a randomly chosen phase angle uniformly distributed 

 in the range (o , 277). While this is not integrable in the ordinary sense, it can be 

 expressed in the form of a Fourier sum (see St. Denis and Pierson). 



This representation can be extended to include the effects of distance by 

 using the wave equation for transverse wave motion for each sinusoidal compo- 

 nent. 



Thus, if X is the distance measured in the direction of the wind from a 

 fixed point on the earth, the wave elevation can be expressed by 



Z(t,x) = cos (ojt - fix - <p) J2<^(i) df (30) 



Jo 



where 



n = wavenumber = a/c = 277/^, 



\ = wavelength in feet, and 



c = crest speed (wave celerity) in knots. 



If each of the small sinusoidal components is assumed to propagate as a 

 gravity wave, then, in addition, 



g c2 



The validity of this assumption has been confirmed by the general success of 

 the wave forecasting methods based on Pierson' s theory. The wave elevation 

 can then be expressed by 



634 



