STICKLER: NORMAL MODES IN OCEAN ACOUSTICS 



An analysis of the integrand for this model indicates only a 



finite number of such poles and that they lie between k and k . This 



L J. 



result can be anticipated physically and is not restricted to this 



elementary model. Since no loss mechanisms are present, we expect 



these proper wavenumbers to be real, to give rise to a standing wave 



field in the depth coordinate in the ocean layer, and eventually to 



decay with depth. This leads directly to the condition k < k < k . 



L n X 



Note that these poles lie in the region of real incidence angles. 



These proper roots have, in addition, the following properties 



and physical interpretations (see Figure 5) : the phase velocity c 



satisfies c^ < c < c ; that is, the phase velocity in the radial 

 1 n L 



direction is faster than that in the ocean layer and slower than that 

 in the bottom. 



These modes can also be thought of as being formed by a pair of 



plane waves traveling in the plus and minus z directions at an angle 



^ with respect to the z axis. This angle I" satisfies, through the 

 n n 



simple polar transformation described earlier, 



^J* > = sin cVc^ . 

 n c 1 L 



Recall, further, that for such an incidence angle, the plane-wave 

 reflection coefficient has a modulus equal to one. That is, at 

 these angles, no energy is transmitted into the bottom — all of the 

 energy is trapped in the ocean layer and the fields must, therefore, 

 decay with depth into the bottom. This is the origin of the term 

 trapped mode. 



Finally, the modes k near k, (i.e., c near c. or 'i' near it/2) 



n 1 n 1 n 



correspond to the low-order modes, while those k " k correspond to 



n L 



the high-order modes. The turning points for each of these trapped 

 modes occur at the interface between the ocean and the bottom. 



136 



