STICKLER: NORMAL MODES IN OCEAN ACOUSTICS 



there are no proper or trapped modes ; this has been pointed out by 

 Labianca (1973) in his study of a surface-duct model. The result 

 follows directly from a theorem by Titchmarsch (1946) . In this case 

 there is only a continuous superposition of modes. This integral can 

 be evaluated, when the range is sufficiently large or the z sufficiently 

 small, by summing the leaky modes — as has been done by Pedersen and 

 Gordon (1965) . 



2 



Ifc ->--'»asz->°°, then it is clear that no energy can propa- 

 gate to infinity in the z direction. For this termination, Titch- 

 marsch (1946) has shown that there are only trapped modes. Such a 

 termination has been used by Fitzgerald (private communication) . 



For the isovelocity termination, numerical examples show that 

 the branch cut integrals can, in general, be expected to be important 

 to a range of one water depth and sometimes more. Physically, they 

 can be expected to be important when there is a constructive inter- 

 ference of the lateral wave and proper modes. This occurs for a set 

 of modes near cut-off. An example will be presented later to illus- 

 trate this point. 



When convergence is not a problem, one can ask, "When does a 

 finite set of the leaky modes offer a good approximation to the EJP 

 branch?" Niamerical experience shows that this sum is not always a 

 good approximation. This point will also be illustrated in a later 

 example . 



It can be established that the EJP branch decays roughly alge- 

 braically with range and faster than 1/yJr:; thus, it is not surprising 

 that the sum of leaky modes alone, which decay exponentially with 

 range, is sometimes a poor approximation to the EJP branch. Returning 

 to an earlier point, this also suggests why it takes an infinity of 



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