Convective and absolute instability

Abstract

Unstable systems characterized by linear partial differential equations with constant coefficients are considered. The question of convective or absolute instability has meaning only for a stated condition; then the Cauchy problem for the system is solved by Fourier and Laplace transforms. For one space dimension, we start from the solution which is written as a Laplace inversion integral, and evaluate it asymptotically (t→ ∞). For several space dimensions we start from a solution of the Cauchy problem that is written as a Fourier integral; this solution is evaluated asymptotically by the saddle point method. Six criteria distinguish between the two types of instabilities. The first is equivalent to the criterion found by Sturrock, and applies only to systems propagating in one space dimension. The fifth is more general because it applies to one or more space dimensions. The second and sixth give sufficient conditions for convective instability. The third and fourth give sufficient conditions for absolute instability. The “bulk” of the growing wave is defined to be the part of it having the highest growth rate. The velocity associated with the bulk of the wave is calculated for unstable systems satisfying the stated condition. This velocity may be zero for an absolutely unstable system.