  We prove the existence of ``pure tone'' nonlinear sound waves of all
frequencies. These are smooth, time periodic, oscillatory solutions of the
$3\times3$ 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
$k$.
