residue calculus, since only simple poles are involved, leading again 

 to results which can be confirmed using elementary methods. 



The purpose of this section has not been to solve a particular 

 string problem, but to illustrate how the W-H method, applied to three- 

 part problems, leads in general rot to a solution , but to a pair of 

 coupled Integral equations. For an even kernel K(s) we can decouple the 

 equations, and deal with a pair of similar ^.ndepencent integral equations 

 for the functions D (s) , S (s) . In our case the integral terms ctn 

 actually be evaluated explicitly in terms of unknown constants D (k ) , 

 S (k ), which can then be determined by setting s - kj in the Integral 

 equations. A precisely similar situation exists in "near-resonance" 

 problems, such as the scattering of acoustic waves by a long tube, open 

 at both ends, at frequencies near resonance. There it is argued that, 

 although the function K (t) in the integral equation (7.33) has branch 

 point singularities, the dominant contribution near resonance comes 

 from a pole terra. WpII away from resonance it is anticipated that the 

 dominant contribution to the integral comes from a branch point 

 singularity which represents the rather weak acoustic interaction between 

 the ends of the tube. The functions are expanded about the branch point, 

 and the integral term can then again be evaluated (approximately) as the 

 product of D (-k) say (where -k is the branch point) and an integral 

 which is expressible in terms of Whittaker functions (which can be 

 further approximated in most cases) . Again the constant D (-k) can be 

 found by setting s ■ -k in the integral equation. 



49 



