K(s) Y + (s) + Y_(s) - -±£ (2.26) 



where the kernel is 



s z - k z 

 K(s) - -J |- . (2.27) 



The W-H equation relates a linear combination of the unknown half- 

 range transforms Y (s) f. a function related to the incident wave field 



exp ik x. The equatlo-i holds in the strip of overlap D, and the coaffi- 

 o 



dents, K(s) and (s + k , -1 , are analytic and non-zero in the interior 

 o 



of the strip in the simple ?t cases, though cases in which K(s) has 



zeros or poles in the interior of D can also be handled straighforvardly. 



T n the next section we show how the general situation of the W-H 

 equation «.an be obtained by inspection and how the Fourier transform 

 integrals can be inverted to yield explicit solutions for the trans- 

 mitted and reflected waves. Following that we look at differences 

 which *rise when the boundary conditions at x ■ are changed, and when 

 the strings are replaced by elastic beams. 



3. SOLUTION OF THE W-H EQUATION 



Assuming, that we have obtained an equation (2.26) with the coeffi- 

 cient of one or other of Y (s) reduced to unity, the first crucial point 

 lies in the W-H FACTORIZATION of K(s) . In this we express 



K(s) - K + (s) K_(s) (3.1) 



as the product of two functions of which K (s) is analytic and non-zero 



in R and of at most algebraic growth at infinity in R , while K_(s) is 



analytic and non-zero in R and of -it most algebraic growth at infinity 



13 



