Asymptotic analysis of the radiation by volume sources in supersonic rotor acoustics

Abstract

The application of Lighthill's acoustic analogy to the generation of sound by rotating surfaces with supersonic speeds results in radiation integrals in which the stationary points of the phase function – that describes the sapce-time distance between each source point and a fixed observation point – have variable positions and coalesce at a caustic in the space of source points. Here, the leading term in the asymptotic expansion of the corresponding Green's function at this caustic is calculated, both in the time and the frequency domains, and it is shown that the radiation generated by volume sources, which are steady in the uniformly rotating blade-fixed frame, has an amplitude that does not obey the spherical spreading law, i.e. does not fall off like RP–1 with the radial distance RP away from the source. Within a finite solid angle, depending on the extent of the source distribution, the amplitude of this newly identified sound decays like RP–½, so that it is stronger in the far field than any previously studied element. That this is not incompatible with the conservation of energy is because the emission time intervals associated with the volume elements of the source distribution which contribute towards the non-spherically decaying component of the radiation are by a large (RP-dependent) factor greater than the time intervals during which the signals generated by these elements are received. The contributing source elements are those whose positions at the retarded time coincide witht the locus of singularities of the Green's function, i.e. with the one-dimensional locus of points, fixed in the rotating frame, which approach the observer with the wave speed and zero acceleration along the radiation direction. Because the signals received at two neighbouring instants in time arise from distinct, coherently radiating filamentary parts of the source which have both different extents and different strengths, the resulting overall waveform in the far zone consists of the superposition of a (continuous) set of narrow subpulses with uneven amplitudes. These subpulses are narrower the larger the distance from the source.The differences between the new results and those of the earlier works in the literature are shown to arise from the error terms in the far-field and high-frequency approximations, approximations that are inappropriate for volume sources moving supersonically.