Derivation of Lighthill’s Eighth Power Law of an Aeroacoustic Quadrupole in Acoustic Spacetime

Abstract

Acoustic spacetime is a four-dimensional manifold analogue to the relativistic spacetime with the reference speed of light replaced by the speed of sound. It has been established primarily for the indirect studies of relativistic phenomena by means of their better understood acoustic analogues. More recently, it has also been used for the analytical treatment of sound propagation in various uniform and non-uniform flows of the background fluid. In this paper the analogy is extended and utilized to derive Lighthill’s eight power law for sound generation of an aeroacoustic quadrupole. Adding to the existing analogue theory, propagating sound waves are described in terms of a weak perturbation of the background acoustic spacetime metric. The obtained result proves that the acoustic analogy can be extended to cover both weak perturbation of the fluid due to the sound waves and certain sound generation mechanisms, at least in incompressible low Mach number flows.