Violation of the inverse square law by the emissions of supersonically and superluminally moving volume sources

Abstract

The generally familiar notion that the conservation of energy requires the intensity of the radiation generated by a localized finite-duration source to decay like the inverse square of the distance from the source is not necessarily true. In this paper, we identify physically tenable sources of acoustic and electromagnetic radiations the amplitudes of whose emissions to particular distant zones decay cylindrically (like {R_P}^{-{1\over2}}) rather than spherically (like {R_P}^{-1}) as R_P tends to infinity (R_P denotes the distance of the observer from the source). These sources have moving distribution patterns which are in general three-dimensional and which propagate faster than the emitted waves. Their emission is characterized by a waveform of constant duration that consists of a continuous assemblage of cylindrically decaying subpulses. Each subpulse embodies a propagating caustic and is narrower the further away it is observed from the source. The change in the lifetime of the subpulses with range (\sim {R_P}^{-1}) is such that their energy---but not their intensity---follows the inverse square law and the Rayleigh distance associated with them is of the same order of magnitude as their distance from the source (R_P) for all values of this distance.\par We present our work in the context of the literature on the non-diffracting wave packets known as acoustic or electromagnetic missiles, and point out how these missiles allow the existing body of data on the emissions from supersonic jets and propellers and from pulsars to be seen in a different light. A supersonically convected aeroacoustic source of volumetric scale L^3 and lifetime T radiates conventional Mach waves whose mean square pressure-fluctuations level scales as ρ^2 U^4cTL /{R_P}^2 (ρand U are