STICKLER: NORMAL MODES IN OCEAN ACOUSTICS 



The pressure field representation for the Pekeris branch is thus 

 described by the symbolic equation at the bottom of Figure 5. For 

 this choice the integration at the contour at infinity can be shown 

 to be zero only if r is sufficiently large or z sufficiently shallow. 

 This can present serious practical as well as theoretical problems. 

 In addition, when the contribution of the integral at infinity is not 

 zero, the sum of the trapped modes is divergent. When this representa- 

 tion is convergent, it can be used and, furthermore, when it does 

 converge, we see that 



E (leaky) + / Pekeris = / 



EJP. 

 BR BR 



The foregoing analysis is, of course, not restricted to the 

 Pekeris model. The simple plane-wave interpretations are model 

 dependent, but the differences in representations due to the two 

 choices of branch cut are not. There are two small differences: 



• In the Pekeris model each mode has only one turning 

 point and it occurs at the ocean-bottom interface. 

 In a refracting ocean this is not the case. There 

 may be more than one; however, as in the Pekeris 

 model none can occur in the isovelocity half-space. 



• The critical-angle concept depends on source loca- 

 tion and the sound-speed profile, and it is defined 

 by the grazing ray. 



General Comments 



In the next several paragraphs I will make several comments 

 for general profiles, neglecting for the moment the effects of shear. 

 It is convenient to return to a point discussed in the introduction, 



namely, the dependence of the representation on the teirmination of 



2 



the sound-speed profile. If c (z) -> sufficiently rapidly, then 



139 



