The principal object of the analysis is to find the change in type of periodic plane waves of sound of finite amplitude propagated in free air.A solution of the exact equation of motion is obtained as a Fourier series. Due to the nonlinear relation between pressure and specific volume there is found to be a gradual transfer of energy from components of lower frequency to those of higher frequency. Since the effect of viscosity is to attenuate the higher frequency components more than the lower, there is always a wave form having the harmonic components in a stable relation such that the decrease in relative magnitude of any component due to viscosity is compensated by the relative increase due to nonlinearity. The conditions for stability vary with intensity. There is therefore no permanent wave form, but the stable wave will change its form more gradually than any other wave of the same intensity and wave length. The change in type of any wave is toward this stable form. There is a marked departure from the sinusoidal in the stable type even for waves of very moderate amplitude.