REFLECTION COEFFICIENTS AND OTHER AUXILIARY OUTPUTS 



Once the depth functions of a mode have been determined, it is relatively easy to 

 compute reflection coefficients at any interface. Therefore, a subroutine called RCOEF has 

 been added to the program which will compute and print out reflection coefficients if 

 requested by the use of control key 3. If key 3 is set to 1 , the reflection coefficients at all 

 interfaces are computed. If set to a number, n, greater than 1 , the coefficient is computed 

 at the nth interface only, where the surface is the first interface. 



The printout includes the phase as well as the ampHtude of the reflection coefficient 

 and the grazing angle. The grazing angle, 6 , of the equivalent rays is computed from the 

 mode phase velocity and the sound speed, c, at the bottom of the layer, by Snell's law: 



d = cos" (c/v). 



The grazing angle is computed only if the phase velocity is greater than the sound speed at 

 the interface, since otherwise the equivalent ray does not reach the interface. 



The reflection coefficient is derived, following Bucker (ref 9), by assuming that an 

 isospeed layer exists for a small depth just above the interface. In this layer the depth func- 

 tion can be written as 



f(z) = Aei'^ + Be~il^, (43) 



where 1, the vertical component of the mode wave number, is given for mode n by 



1 =k -X (44) 



n 1 n 



and 



ki = oj/ci,i, 



where c^j is the sound speed at the bottom of layer i. The derivation now consists of identi- 

 fying A and B as the pressures of the upgoing and downgoing waves at the bottom of the 

 layer; thus the reflection coefficient 



R = A/B. 



A and B are evaluated by making f and its derivative at the interface between this small iso- 

 speed layer and the regular profile continuous with the normal mode depth functions. The 

 thickness of the isospeed layer is then allowed to approach zero, giving the desired value of 

 R. If F and F' are the normal mode function and its depth derivative at the interface depth 

 defined by eq (9) and (19), the reflection coefficient resulting from the above derivation is 

 as follows: 



R = (ilF + F')/(ilF-F'). (45) 



This coefficient is a complex number. Loss per reflection is given by 20 times the 

 log of the absolute value. The phase gives the phase shift that an equivalent ray would 



9. Sound Propagation in a Channel with Lossy Boundaries, by HP Bucker; J Acoust Soc Am, vol 48, 

 p 11 87-1194, November 1970. 



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