If the spectrum function, A(a)), is appreciable only within a single, 

 narrow frequency band of wavelength 2tt/wq around frequency w then 



,(t} = e^}'ot f - A(^) e^(^-'^o) * do 



e"-"^* B(t) 



(B-6} 



Here, the factor e ^ represents a carrier wave of wavelength 2tt/(j0q and 

 the integral B(t) represents an envelope function around the carrier wave 

 [see Fig. B-2) . The values of the amplitudes at the maximums and minimums of 

 n(t) are approximately equal to the values of the envelope function at those 

 points. Thus, the probability distribution of the amplitudes, and hence of 

 the wave heights, is the same as the Rayleigh probability distribution of the 

 values of B(t). Therefore, the probability, P(H), that the wave height H 

 is between H and H + dH is given by 



P(H) dH = -d [e ^^'/^Ws^ J 



^rmsJ dH 



CB-7) 



Envelope Function, B (t) 



•^Time,t 



Water Surface Level Function rjif) 



P'igure B-2. Definition of envelope wave function for nCt) with 

 single narrow frequency band. 



For N waves the fraction, p, of the wave heights which are larger than 

 a given wave height, H, is equal to the probability that a wave height will 

 exceed H and is given by 



P = /^ P(n) dH 



CB-E 



27 



