# Goubau F(γ'ρ) Function

April 15, 2014

Goubau (1950) analyzes the TM propogation wave for a coated wire configuration. To calculate the power extension radially of the wave in the external medium ρ > a2 , the calculation, which involves numerical solution of a transcendental equation, is performed initially assuming an ideal conductor (perfect conductivity) to determine the real propogation constant βr. This leads to a purely imaginary radial propogation constant jγ' for the magnetic field in the external medium with dependence ~ H11(jγ'ρ). Goubau introduces a function F(γ'ρ) which contains the radial dependence of the integrated axial power flow. This function can be rewritten equivalently in terms of K0 and K1 Bessel functions:

there the axial power flow between radius ρ and infinity, normalized to total current squared in the wire cross section is:

where βr and k3 are very nearly the same for the TM mode. The normalized total power flow between a2 and infinity is, since γ'a2 <<1 always:

The fraction of the total external power flowing axially between the outside of the coating layer and an external radius ρ is:

(Note: Pz(ρ) above is the time-averaged power which contains an extra factor of 1/2 compared to Goubau's result).
A calculator can be used to solve for the TM mode and calculate the power extension.

The graph below shows the dependence of F(x) along with approximations that apply for small and large values of x:

Reference:
• Electromagnetic Theory, J. Stratton, 1941, McGraw Hill, pp. 524-536, 546-554
• Surface Waves and Their Application to Transmission Lines, G. Goubau, J. Appl. Phys, V21, 1950, p.1119
• Electrodynamics, A. Sommerfeld, Lectures on Theoretical Physics, 1952, VIII Academic Press, pp 177-185
• Field Theory of Guided Waves, R. E. Collin, 1991, IEEE Press, pp.697-700, 718-720