Wave Impedance Matrices for Cylindrically Anisotropic Radially Inhomogeneous Elastic Solids

Abstract

Impedance matrices are obtained for radially inhomogeneous structures using the Stroh-like system of six first-order differential equations for the time-harmonic displacement-traction 6-vector. Particular attention is paid to the newly identified solid-cylinder impedance matrix Z(r) appropriate to cylinders with material at r = 0, and its limiting value at that point, the solid-cylinder impedance matrix Z₀. We show that Z₀ is a fundamental material property depending only on the elastic moduli and the azimuthal order n, that Z(r) is Hermitian and Z₀ is negative semi-definite. Explicit solutions for Z₀ are presented for monoclinic and higher material symmetry, and the special cases of n = 0 and 1 are treated in detail. Two methods are proposed for finding Z(r), one based on the Frobenius series solution and the other using a differential Riccati equation with Z₀ as initial value. The radiation impedance matrix is defined and shown to be non-Hermitian. These impedance matrices enable concise and efficient formulations of dispersion equations for wave guides, and solutions of scattering and related wave problems in cylinders.