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

Drasko Masovic,Ennes Sarradj

doi:10.3390/acoustics2030035

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.

Highlights

We show that Lighthill’s power law for aeroacoustic quadrupole in an inviscid flow can be derived from the acoustic spacetime analogy
The description of sound generation in acoustic spacetime appears to be more natural than the classical derivation based on the conservation laws
The entire stress–energy tensor is taken as the source of waves, without splitting it into the source and propagation terms or selecting an appropriate dynamic quantity for the aeroacoustic analogy

Summary

Introduction

We show that Lighthill’s power law for aeroacoustic quadrupole in an inviscid flow can be derived from the acoustic spacetime analogy. Unlike Lighthill’s aeroacoustic analogy, which is an exact reformulation of the governing equations of fluid dynamics, the eighth power law holds only for incompressible (low Mach number) flows in the source region and in the absence of a significant acoustic feedback. These conditions allow a purely kinematic analogy with general relativity to be established and applied for capturing sound generation in the following analysis. Several relevant outcomes which follow from the derivation of the eighth power law from the analogy are discussed in the concluding Section 4, which gives suggestions for future studies

Waves and Motion in Acoustic Spacetime

Aeroacoustic Sound Generation

Conclusions