Nonlinear Dissipative Distortion of Progressive Sound Waves at Moderate Amplitudes

Abstract

The first part of this investigation deals with the case of a simple-harmonic wave which becomes eventually distorted into a relatively stable shape resembling the shape of a saw-tooth wave with rounded-off corners. A simplified-picture theory is developed in which the wave is idealized as consisting of a non-dissipative linear portion of length λ1 and a dissipative simple-harmonic portion of length λ2. The amplitude A of the distorted wave and its rate of absorption αA are computed (1) from the dissipative wavelength 2λ2 and (2) from the “build-up distance” L = Nλ at which stability of shape is achieved, both λ2 and N being regarded as experimental data. Tests of the theory are made by using shadowgrams of waves generated in air by a piezoelectric quartz at the frequency of 0.405 megacycle (see reference 2). The second part of the paper presents an approximation treatment of the classical wave equation in which either of two different transformations of variables lead to a quasi-linear differential equation of the type ∂2z/∂y2 = ∂z/∂x+∂(z2)/∂y. In the first transformation, more suitable for periodic waves, x is used for the distance from the source and y for the time, while the reverse is done in the second transformation, more suitable for shock waves. An exact solution is given for the case of the simple-harmonic source already discussed in the first part.