where 



and 



some instances the physical conditions of the problem lead to a simpler solution in terms of 

 rays. In others, a solution in terms of normal modes is more satisfactory. In any case, it is 

 clear that the sediments that stratify the ocean bottom interact with the sound field through 

 the bottom reflection coefficient, R, in the derived theoretical expressions which follow. 



RAY THEORY 



In working with active sonar problems at frequencies of approximately 2.5 to 

 15 kHz usually we use ray theory. In ray theory, the sound field is made up of contributions 

 of rays that travel from the source to the receiver as shown in Figure 15. On the right hand 

 side of Figure 15 we show two neighboring rays that bracket the receiver at range r. Thus 

 there is an eigenray somewhere between these that travels precisely from the source to the 

 receiver. Call this the n^^ eigenray. The magnitude of the ray is Aj^ and its phase is d^ as 

 expressed in the equation for the velocity potential, \p 



n 



An-|R|[cos7s5Ts/(r5h)]l/2 



d^^j (aj/c)dC + arg(R)-m7r/2 . 

 •^ o 



R is the reflection coefficient, gj, the angular frequency, c, the sound speed, dC, a path 

 length along the ray and m is the number of times the ray has touched a caustic. 



The reflection coefficient R is defined for reflection of plane waves and is a complex 

 number. The pressure carried by the ray is reduced by a factor equal to the modulus of R 

 and the phase of the ray is advanced by the argument of R. In experimental work, Aj^ can 

 be measured and the |R| determined from the equation for Aj^. At high frequencies, the 

 plane and spherical coefficients essentially are equal. At low frequencies, the ray theory 

 breaks down and the sound cannot be separated into packets that have a well defined 

 trajectory. As shown in the following equations, the direct and bottom reflected sound 

 field can be written at any frequency in terms of the plane wave reflection coefficient. 



WAVE THEORY 



The general form of the sound field can be written as a sum of cylindrical waves in 

 the form (Bucker, 1970) 



iP = I -2 UCZq) V(z) W-1 Jo(kr) kdk (Zq < z < Zfa) 



•^o 



46 



