Influence of viscosity on acoustic diffraction by a half plane

Abstract

An approximate solution for the diffraction of sound by a half plane in a viscous fluid is given. The solution is derived from the linear equations of fluid dynamics, where the Euler equation is replaced by the Navier-Stokes equation. The resulting equations can be represented as the sum of acoustic and vorticity mode components. The acoustic mode is governed by the acoustic wave equation, while the vorticity mode is governed by a diffusion equation. The two modes each have nonzero velocity at the half plane, but the superposition of the two velocity fields is such that the sum of the tangential components and the sum of the normal components both are zero, in accord with the no-slip condition. Each mode can be approximately represented in terms of Fresnel integrals of complex argument. The solution implies that there is a boundary hemisphere in the vicinity of the edge in which there is no singularity and in which the vorticity mode is of equal importance to the acoustic mode. Outside the boundary hemisphere the acoustic mode dominates and the solution in that region is very nearly the same as originally predicted by Sommerfeld in 1896.