STICKLER: NORMAL MODES IN OCEAN ACOUSTICS 



Finally, the representation for the EJP branch is given by the 

 symbolic equation at the bottom of Figure 5: a finite sum of trapped 

 or proper modes plus an integral around the EJP branch plus an integral 

 around a semicircular contour at infinity. This latter integral can be 

 shown for this branch to be arbitrarily small for all observation and 

 source coordinates. 



The representation arising from the Pekeris branch (see Figure 4) 

 can be thought of as being formed by pushing the EJP branch to a 

 vertical position. When this is done, some of what had been on the 

 second sheet of the EJP branch is exposed. That is, in the unshaded 



vr^T-2 



region the condition Im Vk - k < 0. Any residue that arises from 

 a pole in this shaded region will eventually grow exponentially with 

 depth and, thus, will represent a mode with infinite energy. Such a 

 mode will be called an improper mode. 



An infinity of these improper poles has been found and the 



reason will be clear later. Some of the properties of these improper 



modes are shown in Figure 6. For these modes, the real part of 



k , k ', satisfies k ' < k^ and their phase velocity in the radial 

 n n n L 



direction is greater than the phase velocity in the high-speed bottom. 

 They are sometimes called fast waves. In addition, for the plane- 

 wave incidence-angle analogy, 4* ' < . For such angles the reflec- 



n c 



tion coefficient |r| < 1, and a plane-wave incident at such an angle 

 will have some of its energy transmitted or leaked into the bottom. 

 This is the origin of the term leaky mode. 



These leaky modes not only eventually grow exponentially with 

 depth and, hence, do not represent fields with finite energy, they 

 also have another rather unphysical property: they decay exponen- 

 tially with range. This decay suggests physically that some absorp- 

 tion mechanism rather than a radiation- type mechcinism is present. 



137 



