Energy paths in edge waves

Abstract

In this paper the energy streamlines, energy paths, and energy streak lines in a steady or unsteady inhomogeneous acoustic field next to an unstable oscillating boundary, such as a vortex sheet or shear layer, are determined. The theory in the paper applies also to an evanescent wave produced by total internal reflection, and to any other type of edge wave, e.g. a coastally or topographically trapped wave in geophysical fluid dynamics. The idea of the paper is that energy velocity, i.e. energy flux divided by energy density, is defined at every point in space and time, not merely when averaged over a cycle. Integration of the ordinary differential equation for energy velocity as a function of position and time gives the energy paths. These paths are calculated explicitly, and are found to have starting and finishing directions very different from those of cycle-averaged paths. The paper discusses the physical significance of averaged and non-averaged energy paths, especially in relation to causality. Many energy paths have cusps, at which the energy velocity is instantaneously zero. The domain of influence of an arbitrary point on the boundary of a steady acoustic edge wave is shown to lie within 45° of a certain direction, in agreement with a known result on shear-layer instability in compressible flow. The results are consistent with flow visualization photographs of near-field jet noise. The method of the paper determines domains of influence and causality in any wave problem with an explicit solution, for example as represented by a Fourier integral.