and yet may never be captured in the appropriate deformations of the 

 integration path which will have to be made, and such poles then 

 represent no distinct and identifiable structure of the field. Pole 

 contributions also serve, in acoustics problems, to represent the 

 abiupt changes that would occur according to geometrical optics as one 

 crosses boundaries between illuminated, rrflected, and shadow wave 

 zones. These pole contributions have to be supplemented in more 

 complicated probleas by integrals arouud the branch cuts which join 

 br&uch points. There are many techniques for estimating the contributions 

 from branch cut integrals [9,10,11,12] including cases where various 

 kinds of singularities come close together, and even coincide. ' enerally 

 branch cut integrals represent forced near-field behavior, which decays 

 algebraically away from a junction or discontinuity of the two-part 

 system usually studied by the W-H technique, leaving only the natural 

 propagating modes at large distances. 



22 



—— ^^.^..^ „.^__ Ji . J1 __^. 



