Ill- 109 



For the rest of the analysis, it is assumed that all terms involving a 

 factor a^, where n s 3, may be ignored. It is difficult to determine whether 

 this assumption is warranted; however, if the m = 2 specular correction is small 

 compared to the m = component, we might have confidence in the adequacy of 

 3 terms. The wave number k is given by k = 2nf/c where c = 150,000 cm/sec 

 is the speed of sound in water. For f = 1000 cps and small waves (h = 100 cm), 

 we have a fc:4. We will have a> 1 whenever fh > 50, 000, which happens for 

 most sea states and sound frequencies of interest. Thus we see that this trunca- 

 tion is valid only when the magnitudes of the §n(^> u) decay rather rapidly- -e.g., 

 exponentially. 



Using this truncated form for I , the amplitude of the scattered wave 

 in the plane z = 0, is 



00 00 



P3^(§,ri)= \ I e'^^^^+^^^) $a,u)dXdn 



^-i(a?+3ri)^2.^^^^^^^^^-i(a?+0n) ^^^^,^3^^ 



-2va^C(?,ri)[0(5,,)*[c(5,,)e-^<^?^^^| 



where v is the Fourier transform of v . The autocorrelation function Y (?, t]) of 

 p' ( §. Tl) is given by 



,(5,^).e-^^°^^+^^>[l+4Y=a-Z(?,n)] 



(III- 140) 



00 00 



4yc:^ I \ va,u) Z(X,u) dXdu 



-00 -00 



where Z(?, r\) is the autocorrelation function of C ( ?. tl) and Z is the Fourier trans- 

 form of Z . The Fourier transform of Y (?, v) then gives A(^ , u). the intensity, as 



S-7001-0307 



