Multidimensional wave radiation from a source with Gaussian time variation and Gaussian-approximated distribution about a spherical sheet

Abstract

This paper concerns the wave field of a source with the title-indicated space–time function which, additionally, possesses an arbitrary directional variation. The multivariate solution obtained comprises an estimated error plus peak-induced spherical harmonics that are hyperconically confined, i.e., bounded by diverging and converging spherical fronts. Such fronts are not necessarily singular. Compliance with the radiation principle ensues, through contour integration, from Cauchy initial conditions. For an odd number of spatial dimensions, an inner zone created after a focusing phenomenon exhibits an analogy with a Petrowsky’s lacuna. Naturally, the wave field varies with direction, but only because its source does so. Spherically as well as axially symmetric cases constitute major corollaries. Asymptotic developments, evolving ultimately into steady limits, are also deducible. An indirect application is illustrated for magnetoacoustic flow parallel to a magnetic field; on induction by a cylindrical Gaussian-approximated current distribution, weak effects appear everywhere during the steady state and are superposed upon strong stationary wave effects bounded by cone sheets which project either (i) downstream for a supersonic–super-Alfvénic flow, or (ii) upstream for a restricted subsonic–sub-Alfvénic flow. Finally, the main results are directly applied to elastic wave propagation from a two-component Gaussian body force concentrated about a spherical base; a spherically symmetric radial component generates a strong irrotational wave field normally involving an instantaneous point singularity; an axisymmetric azimuthal component generates a strong solenoidal wave field.