STICKLER: NORMAL MODES IN OCEAN ACOUSTICS 



these improper modes to approximate the algebraic decay associated 

 with acoustic energy radiated to infinity. 



Finally, it is interesting to bring the virtual-mode concept of 

 Labianca (1973) into this framework. For the profile in Figure 1, 

 the virtual-mode sum is obtained by an approximate integration of the 

 EJP branch integral. This approximate integration accounts for the 

 proximity of the leaky poles to the integrand of the branch line 

 integral. 



Effects of Shear 



This section is concluded with the description of some of the 

 effects that the presence of shear introduces into the representation 

 for the pressure field. These effects are summarized in Figure 7 and, 

 for completeness, the two corresponding cases neglecting shear are 

 also included. These are at the top of Figure 7 with the case just 

 considered being on the right. The case on the top left represents 

 a case in which the "bottom" speed is less than water speed. It is 

 not particularly useful since, in any model of the bottom, the sound 

 speed eventually becomes greater than that in the water. However, for 

 this case, there are no proper modes and, hence, the representation 

 consists of either a single EJP-type branch integral or a Pekeris- 

 type branch plus an infinite sum of improper modes. The convergence 

 of the improper-mode sum can, again, be guaranteed only when z is 

 sufficiently small or r is sufficiently large. 



Continuing to the cases where shear is present, it is quite 

 straightforward to show that only when c is larger than c is it 

 possible to have trapped modes. In the other two cases, one is 

 either faced with the evaluation of the EJP branches or of the Pekeris 

 branch and determination of the leaky-wave modes. The convergence of 



142 



