. The Bell System technical journal . lar 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 + yr- = (^ + v)/2,dx + (hf = (r + V-) (df + dr,) = 2r(c?r + dr,),^ = (2ry cos(^/2 + 7r/4),77 = (2r)^ sm( 770 ^ 0,so 77 is always positive in our work. ^ is positive in the half-plane x > 0and negative in .r < 0. For much of our work we shall assume the incident w^ave to come inat the angle ^, 0 ^ 0 ^vr, as shown in Fig. As mentioned in Section 4 =CONSTANT. T] = CONSTANT n^Vo Fig. — This diag
. The Bell System technical journal . lar 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 + yr- = (^ + v)/2,dx + (hf = (r + V-) (df + dr,) = 2r(c?r + dr,),^ = (2ry cos(^/2 + 7r/4),77 = (2r)^ sm( 770 ^ 0,so 77 is always positive in our work. ^ is positive in the half-plane x > 0and negative in .r < 0. For much of our work we shall assume the incident w^ave to come inat the angle ^, 0 ^ 0 ^vr, as shown in Fig. As mentioned in Section 4 =CONSTANT. T] = CONSTANT n^Vo Fig. — This diagram shows the angle 6 of the incident wave and the surfaceof the perfectly conducting cylinder x^ = Ah{h — y) (or r] = 770). DIFFHACTIOX OF RADIO WAVKS HV A IA l{ A |{< )I,I( CVMXDKK III 2, the field quantities are assumed to depend upon the time through thefactor e\p(icoO where co is the radian frequency. The wave equations for horizontal and vertical polarization are,respectively, S + § + <« + ^^ = 0 () f + 0 + «^ +) = ° () where, as explained in Section 2, the unit of length has been chosen sothat the wa\elength X = 2ir. On the surface of the perfectly conductingcylinder, for r] = rio, we must have E = 0 and dH/drj = 0. WhenE and H are kno^\Ti the remaining components of the field may becomputed from Maxwells solutions of () (and ()) are exp {i(r, - f)/2] U,X^i) U„(r,i-), () exp [iin - r)/2] Uni^i) W„(nr), () where i^~ stands for exp (zV/4) and Un(z), Wn{z) satisfy the equation ^^^^ - 2z
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