Time-periodic solutions of the compressible Euler equations and the Nonlinear Theory of Sound

We prove the existence of ``pure tone’’ nonlinear sound waves of all frequencies. These are smooth, time periodic, oscillatory solutions of the $3\times 3$ compressible Euler equations satisfying periodic or acoustic boundary conditions in one space dimension. This resolves a centuries old problem in the theory of Acoustics, by establishing that the pure modes of the linearized equations are the small amplitude limits of solutions of the nonlinear equations. Riemann’s celebrated 1860 proof that compressions always form shocks is known to hold for isentropic and barotropic flows, but our proof shows that for generic entropy profiles, shock-free periodic solutions containing nontrivial compressions and rarefactions exist for every wavenumber $k$.