Plane, progressive, periodic sound waves of finite amplitude are considered. The well‐known solutions of Fay and Fubini are reviewed. At first glance, the two solutions seem contradictory, but, actually, each holds in a different region of the flow, the Fubini solution close to the source and the Fay solution rather far from the source. In the intermediate, or transition, region, neither solution is valid. A more general solution is obtained by using a method commonly employed for waves containing weak shocks. For distances up to the shock‐formation point, the general solution reduces exactly to the Fubini solution. For distances greater than about 3.5 shock‐formation lengths, the general solution is practically indistinguishable from the sawtooth solution, which, in turn, is the limiting form of Fay's solution for strong waves. The form of the general solution shows clearly how, in the transition region, the Fubini solution gives way to the sawtooth solution. The problem of an isolated cycle of an originally sinusoidal wave is also considered. Finally, some limitations on the weak‐shock method are discussed. In the periodic‐wave problem, the general solution is found to be inaccurate for distances greater than 1/α, approximately, where α is the small‐signal absorption coefficient. In Appendix A, a brief extension of the analysis to spherical and cylinrical waves is given.