A separation of the logarithmic singularity in the exact kernel of the cylinderical antenna integral equation

Abstract

A. I n t r o d u c t i o nThe well-known c y l i n d r i c a l a n t e n n a i n t e g r a l equation h / I ( z , z ' ) dz' = C cos kz + B s i n k z -h + % 7 EZ(g) s i n k ( zg) dg 5=0 ( 1) possesses a kernel of the form [l]: where The function I ( z ) = IT J ( z ) i s t h e n e t l o n g i t udinal current flowing onsthe tube. The i n t r i n s i c impedance of t h e medium i n which t h e c l i n d e r i s immersed i s denoted by T I and k = u(uc)'f2 is t h e wavenumber i n t h e medium. The e x c i t a t i o n E, e i t h e r i s t h e f i e l d a l o n g t h e c y l i n d e r s u p p o r t e d by circumferential feed gaps for the antenna case o r i s t h e axial component of an impinging field f o r t h e s c a t t e r i n g c a s e . -( j u t ) time dependence i s assumed and suppressed. Definitions of geom e t r i c q u a n t i t i e s are given i n Figure 1. When t h e above equations are u s e d f o r t h e s c a t t e r i n g problem, t h e c y l i n d e r must b e s u f f i c i e n t l y t h i n t h a t t h e e x c i t a t i o n from the impinging wave be approximately rotationally symmetric. The c u r r e n t I ( z ) i n (1) sum of c u r r e n t s flowing on t h e e x t e r i o r and i n t e r i o r of a hollow tubular antenna. For thin and moderately thick s t r u c t u r e s , t h e i n t e r i o r c u r r e n t i s small compared w i t h t h e e x t e r i o r c u r r e n t and s o l u t i o n s t o ( 1 ) may be i n t e r p r e t e d as t h e e x t e r i o r c u r r e n t . King and Wu [Z] have derived a m o d i f i c a t i o n t o (1) which The reader i s r e f e r r e d t o r e f e r e n c e [ 2 ] and t o s e p a r a t e s t h e e f f e c t s of t h e i n t e r i o r c u r r e n t . Chang [ 3 ] f o r d e t a i l s r e g a r d i n g i n t e r i o r c u r r e n t s .F o r t h i n s t r u c t u r e s ( a << h.and a << A ) t h e reduced kernel approximation G ( z , z ' ) = K ( Z , Z ' ) = e x p [ -j k r ( z , z ' ) l / r ( z , z ' ) with i s widely used. For larger radius cylinders, this approximation breaks down and t h e e x a c t kernel (2) must be used. Chang 131, Duncan and Hinchey [ 4 ] and Harrison e t a l . [5] have given ways of t r e a t i n g t h e e x a c t k e r n e l ( 2 ) so as to o b t a i n s o l u t i o n s t o ( 1 ) .The work reported here e x t r a c t s t h e s i n g u l a r c h a r a c t e r of t h e e x a c t k e r n e l (2) as a s i n g l e t e r m of t h e form -1nlz -2 '