Figure 176 — A semi-infinite obstacle 

 AB B' A', or an offset in an 

 infinite wall ABOC. 





' 



A 



B 







^ X C 





h 



A' 



B' 



t= 1 



real t axis. Then the exterior angle at B is -tt/2 and at is +77/2. The values of i at these 

 two points can be chosen arbitrarily as -1 and +1. Then, in Equation [31a] of Section 31, 

 Qj =-l,aj =- 1/2; a^ = 1, a 2 = 1/2; hence Equation [31a] becomes in the present case 



dt 



K{t+ 1)1/2 (^_ i)-i/2^ 



[107a] 



Integrating, 



z^K\{t' -lY^' +\r,{t + {t' -lY^'^\ + L. 



To fix the amplitudes, assume that, for the values of ;; required, < amp t < n. Then amp 

 {t + 1) and amp {t - 1) can be understood to lie in the same range, so that amp {t^ - 1) will 

 range from to 2 77, and amp {P' - 1)^^'^ from to 77. Then, to preserve continuity, for negative 

 real t<-l,{t^ - l)^/^ ^ _ y^T^ < 0. Also, < amp [t + {t^ - 1)^^^] ^ n, for use in de- 

 fining the logarithm. 



The constants K and L are chosen so as to make t = - 1 at B or z - ih, and ^ = 1 at 

 3 = 0: 



ih = K log (- 1) + L = Z77/C + L; = + L. 



Hence L - 0, K - h/n and 



= _ \(t^-l)'^^ + lnU + {t^ -l)'/^]\. 



[107b] 



263 



