310 BELL SYSTEM TECHNICAL JOURNAL 



Fig. 1 ) walls are to be specified by x = and a: = a we set x = p — pi + a/2. 

 Thus, the two sets of coordinates are related by 



p = X -\- pi — a/2 = X -\- Pi 



<P = - 2/pi (1.2-1) 



y = y 



where pi, pi and a are constants. 



We again choose A and B in (1.1-1) to be parallel to the y axis. In the 

 cylindrical coordinates, 



^J_al4_ia6 _ 1 a^l 1 d'B 



iuie dpdy p dip p d(p iccp, dpdy 



i(jO€p Oipoy op op to)fjLp Oipoy 



1 f^ A 1 f)^ R 



Ey = -iojiiA + -^- VT ^y =" -i(^^B + -r— — ■ 



to)e oy^ twfjL oy^ 



p dpL dp J p^ 5^2 



+ ^ = <r^^ (1.2-3) 



where now, from (1.1-2), A satisfies the wave equation 



ji 



p dpL dp J p^ 5^2 5y2 



and likewise for ^. 



One method of dealing with (1.2-3) which is sometimes used is to assume 



A = e^'^ X (sine or cosine function of y) X /(p) (1-2-4) 



where f(p) turns out to be a Bessel function of order p with its argument 

 proportional to p. However, we shall proceed in a different direction. 

 The change of coordinates (1.2-1) transforms (1.2-2) into 



fi' _ ;, _ 1 ^'^ , Pl^^ r, rr Pi ^^ > 1 ^'^ 



i<0€ oxdy p oz p oz icop. oxoy 



Ey = -toifiA + ^- T-^ Hy = -loieB 4- — -— (1.2-^) 



tQ)€ oy^ iu3yL dy^ 



E. = -£, = .^ |!^i - f //. = -//, = ^/ + /' i;f- 



icocp ozdy dx dx iwfxp dzdy 



and (1.2-3) into 



& 



dx^ ' dy^ ' p2 522 ■ p dx 



^"^ + ^;4 + P; ^^ + 1 M _ /,! = 0, (1.2-6) 



