Traffic Noise and the Hyperbolic Plane

TL;DR

This paper models sound propagation in a wind as charged particle trajectories in a magnetic field on the hyperbolic plane, providing geometric methods to estimate sound intensity and analyze source movement.

Contribution

It introduces a novel geometric framework linking sound rays in wind to magnetic trajectories on hyperbolic space, and relates the problem to null geodesics in a squashed anti-de Sitter spacetime.

Findings

Mapping sound rays to magnetic trajectories on hyperbolic plane

Developing a simple method to estimate sound intensity upwind/downwind

Connecting the problem to null geodesics in anti-de Sitter spacetime

Abstract

We consider the problem of sound propagation in a wind. We note that the rays, as in the absence of a wind, are given by Fermat's principle and show how to map them to the trajectories of a charged particle moving in a magnetic field on a curved space. For the specific case of sound propagating in a stratified atmosphere with a small wind speed we show that the corresponding particle moves in a constant magnetic field on the hyperbolic plane. In this way we give a simple `straightedge and compass' method to estimate the intensity of sound upwind and downwind. We construct Mach envelopes for moving sources. Finally, we relate the problem to that of finding null geodesics in a squashed anti-de Sitter spacetime and discuss the $SO(3,1)\times \mathbb{R}$ symmetry of the problem from this point of view.