A transformation of a formula of Sommerfeld

Extract

Sommerfeld has shown that the electromagnetic field set up by a Hertzian oscillator placed on the infinite plane surface separating two media normal to that surface may be obtained from the Hertzian function of cylindrical coordinates r, z,

where the suffixes 1, 2 refer to the two media occupying the regions z>0, z<0,

and

where ɛ₁, μ₁, σ₁ are the dielectric constant, permeability, and conductivity (in Heaviside units) of the first medium, k₂, k₂ like expressions for the second, c is the velocity of light, and n the frequency in radians per second; +(λ²−k²)^½ is taken to have its real part positive.