Evolutionary isotropic radiation

Abstract

A directionally dependent source distribution activates within a multi-dimensional isotropic dispersive medium. It then develops into a purely pulsatory state. The radiation problem is approached via a convolution principle. Certain basic postulates are imposed to help secure integral convergence at various stages. The Sommerfeld radiation principle holds through an applied initial condition. One field quantity results from stationary phase approximations and represents a superposition of slowly transient spherical wavemodes over a continuously evolving set of wavenumbers and frequencies that must avoid the source frequency. Their radial group velocities vary coincidentally with a positive reception velocity parameter; the associated dispersion is basically a function of the medium, but admits amplitudes with some dependence on source frequency. Another field quantity accumulates relevant residues which contribute to a superposition of quasisteady spherical wavemodes over an intermittently growing set. This depends not only on the medium, but also on the source frequency, imparted to all such wavemodes, as well as an observation criterion, namely that any specific wavemode is observed after its enclosing energy front crosses the observer, in particular, with an invariant group velocity exceeding that common to all slowly transient wavemodes; amplitudes quickly lose their source-induced time dependence. It is this last quantity that survives in the long run and progresses into a non-trivial purely pulsatory steady state consistent with Lighthill’s radiation principle. Its ultimate survival is accomplished through a permanent flow of source-generated non-transient energy, permanency of the energy supply being guaranteed by an indefinitely sustained source amplitude; moreover, both medium and source never conspire to cancel the supply. On an energy basis, the fading of slowly transient modes may be due to the decreasing group velocity of their energy arrival. Spherically symmetric and axisymmetric cases are briefly examined. Finally, arguments and results are applied, with some modifications, to (i) an unsteadily vibrating elastic plate problem, and (ii) radiation of certain internal gravity waves in a Boussinesq fluid.