Translational addition theorems for spheroidal scalar and vector wave functions

Abstract

The translational addition theorems for spheroidal scalar wave functions R m n ( i ) ( h , ξ ) S m n ( h , η ) exp ⁡ ( j m ϕ ) ; i = 1 , 3 , 4 R_{mn}^{\left ( i \right )}\left ( {h, \xi } \right ){S_{mn}}\left ( {h, \eta } \right )\exp \left ( {jm\phi } \right ); i = 1, 3, 4 and spheroidal vector wave functions M m n x , y , z ( i ) ( h ; ξ , η , ϕ ) , N m n x , y , z ( i ) ( h ; ξ , η , ϕ ) ; i = 1 , 3 , 4 M_{mn}^{x, y, z\left ( i \right )}\left ( {h; \xi , \\ \eta , \phi } \right ), N_{mn}^{x, y, z\left ( i \right )}\left ( {h; \xi , \eta , \phi } \right ); i = 1, 3, 4 , with reference to the spheroidal coordinate system at the origin O O , have been obtained in terms of spheroidal scalar and vector wave functions with reference to the translated spheroidal coordinate system at the origin O ′ O’ , where O ′ O’ has the spherical coordinates ( r 0 , θ 0 , ϕ 0 {r_0},{\theta _0},{\phi _0} ) with respect to O O . These addition theorems are useful in acoustics and electromagnetics in those cases involving spheroidal radiators and scatterers.