The Nonlinear Theory of Sound

TL;DR

This paper establishes the existence of nonlinear, periodic sound waves across all frequencies in the compressible Euler equations, providing a rigorous foundation for classical acoustics and revealing shock-free solutions under certain conditions.

Contribution

It proves the existence of smooth, periodic nonlinear sound waves for all frequencies in the Euler equations, extending classical acoustic theory and analyzing shock formation.

Findings

Existence of smooth, periodic nonlinear sound waves for all frequencies.

Shock formation results consistent with Riemann's 1860 proof for specific flows.

Presence of shock-free solutions with nontrivial compressions and rarefactions.

Abstract

We prove the existence of ``pure tone'' nonlinear sound waves of all frequencies. These are smooth, space and time periodic, oscillatory solutions of the $3\times3$ compressible Euler equations in one space dimension. Being perturbations of solutions of a linear wave equation, they provide a rigorous justification for the centuries old theory of Acoustics. In particular, Riemann's celebrated 1860 proof that compressions always form shocks holds for isentropic and barotropic flows, but for generic entropy profiles, shock-free periodic solutions containing nontrivial compressions and rarefactions exist for every wavenumber $k$.