V 8 >°^i 1 -f^ dt • ' 7 - 28a > 



F - (s) ■ m I -f^i dt • (7 - 28b > 



where in (7.28a) the path runs from - °° to + °° in the strip D passing 



below the point t « s while in (7.28b) the path passes above t ■ s. 



Applying (7.28) here we have, for (7.23b) and (7.27b) in particular, 



Y (s) F e 1 ^ Z, (t) 

 G - (8) " iT(sT + 2il J K.(t)(t-8) dt " ° < 7 - 29 > 



Z (s) f e" itA Y (t) 



-V 8)+ K78T + mJ V t)(tls) dt -° > 



(7.30) 



the forcing fields G_(s) and H (s) being known, in principle. Clearly 

 (7.29), (7.30) are a pair of coupled integral equations for the unknown 

 functions Y_(s) and Z (s) which determine the reflected and transmitted 

 fields in x < 0, x > £., respectively. Once these are known, the field 

 Y (s) in the middle portion of the string can be found directly from 

 the generalized W-H equation (7.18), for example. 



To see the structure of these equations, suppose that the integral 

 term in (7.29) were zero. Then we would have 



Y_(s) - K_(s) G_(s) 

 and noting the definition (7.21) of G (s) end comparing with (3.11) 

 we see that this Y (s) is precisely the field in x < if the wave 

 exp(ik x) were incident upon a semi- infinite, rather than finite, 

 string to the right. Similarly, the situation Z + (s) ■ K (a) H (a) 

 Is the solution for a semi- infinite problem of reflection at the 



43 



a! Wffi^ffWMgil»liaftHITJ«g^^ -jH.'rf-«i : -ri^i^»^.'m.-.^j- 1 f:. tf -, l| . | .^ji gjg^a^jjjgjg^j , r ,- aii t ,t — riif M af^wirifrTtini 



