REFLECTIONS FROM CIRCULAR BENDS 311 



where pi is a constant and p = -v + pi — a/2 is to be considered a function of 

 .V. To solve (1 .2 6) and the corresponding equation for B we assume 



A = zl «mn(3) sin (irmx/a) cos (irny/b) 



(1.2-7) 

 w = 1, 2, v3, • • • ; // = 0, 1, 2, • • • 



B = z2 /3mr.(s) COS (irmx/a) sin (irny/b) 



(1.2-8) 

 /« = 0, 1, 2, • • • ; w = 1, 2, 3, • • • 



these expressions being suggested by (1.1-3) and (1.1-4). The expressions 

 (1.2-5) for the electric intensity show that this choice of A and B make its 

 tangential component vanish at the walls of the guide. Thus the boundary 

 conditions are satisfied. 



In order to determine am„(z) so that the differential equation for A is 

 satisfied, we substitute (1.2-7) in (1.2-6). The resulting left hand side of 

 (1.2-6) may be regarded as a function, say/(.r, y), of x and y with the a's 

 and their derivatives entering as parameters. We must choose the a's so 

 as to make this function zero. Relations which must be satisfied by the a's 

 may be obtained by expanding/(.v, y) in a double Fourier series for w^hich the 

 typical term is a coefficient times sin (irmx/a) cos (irny/b), and then setting 

 the coefficient of each term to zero. This form of expansion is suggested by 

 (1.2-7). However, it should be mentioned that such an expansion is best 

 suited to a function which vanishes at x = and .v = a, a condition not 

 fulfilled by/(.v, y) because of the term p~^dA/dx in (1.2-6). This causes no 

 real trouble because our region of representation runs only from x = to 

 A- = a and hence our series is no worse than the Fourier sine series for the 

 periodic function (of period 2a) which is — 1 for — d^ < .r < and + 1 for 

 < X < a. 



To carry out the procedure outlined above, we multiply (1.2-6) (after 

 putting in (1.2-7)) by sm(irpx/a) cos(irty/b) and integrate x from to a 

 and y from to b. Using the expression (1.1-5) for F^n and reducing gives 



00 



-Tl(a,((z) + Z iPprnC^lf.(z) - S,ma„Az)] = (1.2-9) 



m=l 



where p may have any one of the values 1, 2, 3, • • • and the double prime on 

 a denotes the second derivative with respect to s. The P's tind 5's are 

 constants given by 



P^„. = (2/a) [ (pl/p') sin (irpx/a) sin (irmx/a) dx, (1.2-10) 



•'0 



S^,n = —2irma'~'' I sin (irpx/a) cos (irmx/a) dx/ p (1.2-11) 



