DIFFHACTIOX OF RADIO WAVKS HV A I'A l{ A |{< )I,I(' CVMXDKK III 



2, the field quantities are assumed to depend upon the time through the 

 factor e\p(icoO where co is the radian frequency. 



The wave equations for horizontal and vertical polarization are, 

 respectively, 



S + § + <«' + "'^'^ = (4.3) 



f + + «^ + "') " = ° (") 



where, as explained in Section 2, the unit of length has been chosen so 

 that the wa\'elength X = 2ir. On the surface of the perfectly conducting 

 cylinder, i.e. for r] = rio, we must have E = and dH/drj = 0. When 

 E and H are kno^\Ti the remaining components of the field may be 

 computed from Maxwell's equations. 

 Special solutions of (4.3) (and (4.4)) are 



exp {i(r,' - f)/2] U,X^i'") U„(r,i-'"), (4.5) 



exp [iin' - r)/2] Uni^i''') W„(nr"'), (4.6) 



where i^'~ stands for exp (zV/4) and Un(z), Wn{z) satisfy the equation 



^^^^ - 2z '!^ + 2nTM = 0. (4.7) 



Another solution of (4.7) with which we shall be concerned is Vn{z). 

 These three solutions are defined by contour integrals of the form 



{2Triy^ I exp [/(/)] (If where /(/) = -T + 2^/ - (n + 1) log f. 



The path of integration for Un(z) comes in from — x where arg t = —t, 

 encircles the origin counterclockwise and runs out to — oo witli arg 

 t = IT. The path for Vn{z) runs from — x where arg < = tt to + x where 

 arg t = 0, and the path for Wn{z) runs from + oo to — x whei'e arg 

 t = —IT. The integrals are Avritten at greater. length in equations (9.1) 

 and the paths of integration are shown in Fig. 9.1. Since the paths may 

 be joined to form a closed path containing no singularities of the inte- 

 grand it follows that 



lL(z) + VrXz) + Wn(z) = 0. (4.8) 



When n is a non-negative integer 



r7„(.) = Hdz)/n! = ^-^ (''" f- e-'" (4.9) 



n! dz" 



