been scattered into other subsonic modes and also into supersonic 

 radiating modes by the discontinuity in the surface y ■ 0. 



To analyze the process of scattering , or wavenumber conversion , by 

 the plate edge, we define a scattered field $ by 



y total 



— exp(iqx - y y) + 



q 



for y >. 0. Because the derivative 3* ^/dy has the same value on 



y - o + . 



K total 



must be an odd function of v, and so it is enough to 



consider only y >. 0. The scattered field <J> is a solution of 



\^X 2 + 3y 2 o) 



with 



14.0 



8y 



and with 



on y » 0, x < 



exp iqx =0 on y « 0, x > 0. 



(8.4) 



(8.5) 



(8.6) 



(8.7) 



This is a typicil two-part mixed boundary value problem which we may 

 expect to solve by the W-H technique. Two further conditions are 

 needed, however, to get a unique solution for <)>. One comes from con- 

 ditions expected to hold as \x\ •*■ °°, and defines the domains R + of 

 analyticity of half-range transforms and the strip D of overlap. The 

 other comes from conditions at the plate edge, x ■ y » 0, and determines 

 the behavior at infinity in the transform s-plane, and hence determines 

 the entire function arising in the W-H method. We shall leave the 

 matter of edge conditions until we need to look at it in detail. For 

 the moment we just assume that all functions with which we deal have 



at most integrable singularities at x ■ y ■ 0. 



52 



