~ '%&&fmmmi 



For y > wa can apply Fourier transforms to (8.5), and can 

 integrate the 9 2 <{>/3x 2 exp isx tenn by parts twice with no contribution 

 from x » ± °° provided s lies in D. This gives 



(w-il) «•» 



,y) - o (8.14) 



\OJ Of 



where 



l 



Y - (s 2 - k 2 ) T , (8.15) 



introducing the square root function whose behavior in the complex 



s-plane holds the key to many aspects of acoustic diffraction and 



scattering processes. 



_L i 



The function Y ■ (s - k ) l (s +k ) has branch points at s ■ ± k , 

 s o o o 



and branch cuts must emanate from these points to form a barrier which 



must not be crossed. We can either make a cut from + k to -k , or 



o o 



we can make a cut from + k to °° (in any direction) and a cut from 



1 i 



-k to °= (in any direction). If the values of (s + k ) 2 , (s - k ) T 

 o oo 



are specified at any point in the s-plane (not necessarily the same 



point for the two functions) and the branch cuts are fixed, then a 



unique value of (s + k ) , (s - k ) 2 is obtained by starting at the given 



point and moving to any desired point without crossing any branch cut , 



and insisting that the function change continuously from its initial 



value. Figure 4 gives various possible choices of branch cuts. In 



addition to the choice of branch cuts we shall take each of (s i k ) 2 



i 1 ° 



to be the branch which behaves like + s 7 " (rather than - s 2 ) when s is 



large and positive. 



Now in our problem we know that $ must be analytic in D, and since 



the general solution of (8.14) is 



54 



^mm^^^^m^^ss^&n^^M^^^ 



