IV -9 



The last expression in (IV-31a) may be interpreted as follows: the gradient of 

 the logarithm of the relative amplitude is a measure of the rate of change of the 

 relative logarithmic amplitude with distance. If this is multiplied by the wave- 

 length of the undisturbed wave, we obtain a measure of the change of the relative 

 logarithmic amplitude over one wavelength. The change in the relative log- 

 arithmic amplitude encountered in going one wavelength must, therefore, be small 

 compared to 4tt. This condition is always satisfied for any of the typical wave- 

 lengths and scattering strengths encountered in the ocean. 



The condition (IV-31b) is considerably more restrictive. To understand 

 its implications, we observe that the surfaces Sq(x) = constant andS(x) = constant 

 are the wave fronts, respectively, of the undisturbed and the perturbed waves. 

 The gradient vectors v Sq and '^ S will point in the direction of wave propagation in 

 both cases. The magnitudes of VS and vSq must be of the order of magnitude of the 

 wave number k, since S and Sq are the local phases of the waves, and since the 



phase changes by an amount 2tt in going one wavelength, i.e. , -r r — - k. The 



Ax A. 



meaning of (IV-31b) is, therefore, that the difference between two unit vectors 

 pointing in the direction of propagation of the undisturbed and perturbed wave must 

 be very small compared to 2, even though the phase and the amplitude of the two 

 waves may differ greatly. 



If we make these two approximations and remove the corresponding terms 

 from (IV-26) we find that the correction to the complex phase 'i/i is governed by: 



2 «J k^ + i v2 ,1(1 - 2 (V ijf . V ,|ii ) = (IV-32) 



o 



We may reduce this to the ordinary inhomogeneous wave equation by introducing 

 the function W according to: 



"^* W 



<|ii = e ° W = — (IV-33) 



Pq 



If we substitute (IV-33) into (IV-32) and use (IV-25) to simplify the expression, we 

 obtain: 



v^ W + k^ W = 2iuk^ p (IV-34) 



In the homogeneous case (i.e. , u = 0), we know that the appropriate solution of 

 (IV-34) is W = 0, since in that case we desire that \|Ji = 0. We are, therefore, only 

 interested in the inhomogeneous solution to (IV-34) which, subject to the condition 

 of giving an outgoing wave, must be: 



S'^i 



W(x) = 2k^i I df e^'^^u(?)p^(?) (IV-35) 



V 



Arthur B.littleJnr. 



S-7001-0307 



