Derivation of the magnetic field on a metal cylinder excited by a longitudinal magnetic dipole transmitter: II. Cylinder in a two-layer dissipative dielectric medium

We derive an exact convergent analytical solution for the complex frequency-dependent magnetic field on the surface of an infinitely long and perfectly conducting metal cylinder situated in a cylindrically layered dissipative medium. The inhomogeneous medium consists of two exterior cylindrical layers that are concentric with the cylinder. The magnetic field on the cylinder is excited by a longitudinally oriented oscillating magnetic dipole transmitter on the cylinder surface. An exact analytical solution to this problem has not been previously published and is of theoretical as well as practical importance, e.g., in modeling the responses of electromagnetic wave propagation well logging tools. It is shown that the magnetic field on the cylinder surface can be expressed as a real-axis integral; however, the integrand oscillates rapidly and diverges for large values of the integration variable. The real-axis integral is replaced by the sum of two convergent branch line integrals and a sum over the residues of the complex poles in the integrand of the real-axis integral. The poles correspond physically to waveguide modes that propagate with discrete wave numbers. A pole search algorithm is developed to locate the positions of the poles and compute their residues. Phase shifts and attenuations of the magnetic field between receivers for a 1.1 GHz well logging tool are computed to elucidate the pole spectra and the relative contributions of the waveguide modes and the branch cut integrals for different thicknesses of the innermost dielectric layer and for different layer properties.