The scattering of sound by a simple shear layer

Abstract

A fixed line source, oscillating harmonically in time, produces sound waves which fall on a two dimensional shear layer in which the velocity increases linearly over a finite distance and then remains constant. The linearized theory of sound allows a multiplicity of solutions. The ambiguity is resolved by an application of the principle of causality. As a result it is found that, for Strouhal numbers below a certain critical level, Helmholtz instability is evident but not if the Strouhal number is above critical. The instability wave fans out from a negligibly small region as the Strouhal number drops from critical until it occupies a wedge of 45° when the layer simplifies to a vortex sheet. The limit is the same as that derived by direct analysis of the vortex sheet but no ultra-distributions are necessary if the layer is not infinitesimally thin. Various other aspects of thin and thick layers are also discussed.