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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



classical work deals with the general case of finite conductivity, the 

 series we use are special cases of the ones discussed by him. 



The parabolic coordinates (^, 77) which we shall use are related to the 

 rectangular coordinates (.r, y) and polar coordinates (r, (p) as follows: 



x + iy = (^ + i-nfm = r e'\ 



X = ^r] = r cos <p, y ^ (r?" - f )/2 = r sin <p, 

 r = ix' + y'r- = (^' + v')/2, 

 dx' + (hf = (r + V-) (df + dr,') = 2r(c?r + dr,'), 

 ^ = (2ry" cos(^/2 + 7r/4), 

 77 = (2r)"^ sm(<p/2 + t/4). 



The lines 77 = constant are a series of downward-curving confocal 

 parabolas having their focus at the origin. The parabolic cylinder .r" = 

 4:h(h — y) is given by 77^ = 2h. This special value of 77 will be called 770: 



(4.1: 



Vo 



Clhf" ^0, h = 1)1/2. 



(4.2) 



When 770 = 0, the cylinder reduces to the half-plane .r = 0, ?/ ^ 0. The 

 lines ^ = constant are halves of upward-curving confocal parabolas 

 having their common focus at the origin. Outside the cylinder ■»? > 770 ^ 0, 

 so 77 is always positive in our work. ^ is positive in the half-plane x > 

 and negative in .r < 0. 



For much of our work we shall assume the incident w^ave to come in 

 at the angle ^, ^ ^vr, as shown in Fig. 4.1. As mentioned in Section 



4 = CONSTANT 



T] = CONSTANT 



n^Vo 



Fig. 4.1 — This diagram shows the angle 6 of the incident wave and the surface 

 of the perfectly conducting cylinder x^ = Ah{h — y) (or r] = 770). 



