The asymptotic theory of rapidly rotating sound fields

Abstract

A rapidly rotating sound field, such as that produced by a supersonic propeller, may contain several types of diffraction pattern, each in a different region of space. This paper determines these patterns and their locations for the field around a propeller of simple type ; the method used is stationary phase analysis of a certain double integral, and leads to asymptotic formulae valid when the number of blades is not too small or the high harmonics are being investigated. Physically, the results describe propagation along rays : each stationary phase point is a ‘ loud spot ’, producing a ray which points directly at the observer; most of the noise comes from these loud spots, because extensive cancellation takes place everywhere else. At most two interior and two boundary stationary points may be present: the number and type depend on the position of the observer in relation to a cusped torus and two hyperboloids of one sheet. As these surfaces are crossed, the acoustic field changes in character. For example, when two stationary points coalesce and annihilate each other, as they do at a caustic, an Airy function describes the transition from a loud zone of rapid oscillation to a quiet zone of exponential decay; and when an interior stationary point crosses the boundary of the disc the transition region is described either by a Fresnel integral or by a generalized Airy function. Separate analyses are given for regions close to and well away from the transition surfaces, and inner and outer limits are calculated for use in the method of matched asymptotic expansions. In all cases, an overlap region is found in which the leading term s agree. The results of the paper determine completely the geometry of the acoustic field, because the different regions have boundaries at known positions and cover the whole of space.