STICKLER: NORMAL MODES IN OCEAN ACOUSTICS 



, 1 f 



■,z,z ) = — / 

 o 2 J 



p(r,z,z ) =^ / P(z,z ,k) H ^"^ (kr) kdk. (3) 



" ' ' o o 



C 

 o 



y2 2 / 2 2 

 ^..^ ^..^«.^^>^^ ^..„^„^..„ . k - k and Vk - k that 



Li S 



introduce the branch point singularities in P(z,z ,k) . Their 

 presence is due to the isovelocity termination of the longitudinal 

 and shear sound speeds. 



Integral Evaluation 



The evaluation of the integral in Equation 3 can be performed in 



several ways. To obtain a normal-mode expansion, however, it is 



necessary to identify the singularities of P(z,z ,k) as a function 



of k and to close the C integration contour around these singularities. 



o 



For the class of sound-speed profiles described in Figure 1, the 



singularities of P(z,z ,k) are of two types, an infinity of poles 



o 



plus two pairs of branch points. 



In these ocean models the depth coordinate extends to infinity; 

 therefore, the representation of the pressure field is always a finite 

 sum of proper modes (which are defined below) plus a contribution of 

 the continuous modes. For the profile described in Figure 1, the 

 continuous modes are represented by an integration around the branch 

 cuts mentioned above. For other terminations, this contribution can 

 take a different foinn. This point will be discussed more completely 

 below, but, roughly speaking, the continuous modes represent energy 

 that does not remain ducted, and they will form part of the repre- 

 sentation of the field when the sound-speed profile allows energy to 

 be lost to infinity in the z direction. It is not a loss mechanism 

 in the sense that acoustic energy is transformed to thermal energy 

 but, rather, it represents acoustic energy radiated to infinity. 



133 



