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

The Nonlinear Theory of Sound