On the wave motion and stability of dynamic systems with imperfect boundaries

Abstract

General normal-mode methods and variational expressions are developed which are useful for studying linear wave motion, diffusion and stability of a wide variety of dynamic systems having non-uniform internal energy dissipation and imperfect boundaries. The motion of these systems is here restricted to that representable by one-dimensional 2 N-th-order equations of motion and a set of boundary impedance equations. Orthogonality relations are developed among eigenmodes to facilitate eigenfunction solutions of transient motion, and of motion excited by either deterministic or random forces. Variational expressions are established which facilitate approximate solutions of the eigenvalues and eigenfunctions when exact calculations are not feasible. The normal-mode solutions obtained are general in the sense that for the same system under different boundary conditions (including those perfect ones) one need only change values of boundary impedances in the general solutions. The variational expressions obtained are stationary with no restrictions on the boundary conditions of the trial fields. The methods are directly applicable to a wide variety of electromagnetic, acoustic, elastic, thermal and chemical systems. Three specific examples are considered, namely the influence of imperfect boundaries on a vibrating string, on plane magnetohydro-dynamic waves and on the ‘flute’ instability at a confined plasma boundary. General remarks are made about the significance of imperfect boundaries on plasma waves and instability, the range of validity of the developed methods and on the normal-mode solutions and variational principles for the vector wave motion and stability of general anisotropic media (with emphasis on magnetoplasmas) having tensor imperfect boundary conditions.