  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 \rho^2 U^4cTL /{R_P}^2 (\rho and U are
