The Nonlinear Theory of Sound

Publication date: 24 May 2023

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 $3 i m e s 3$ 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 $k$ .

Mathematics Subject Classification ID

Hyperbolic conservation laws (35L65) Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics (76N10) Local and nonlocal bifurcation theory for dynamical systems (37G99)

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