Convergence of the Keck‐Beyer Perturbation Solution for Plane Waves of Finite Amplitude in a Viscous Fluid

Keck and Beyer have given a perturbation solution of the equation for plane sound waves of finite amplitude in a viscous fluid. The solution may be represented as a power series in the parameter N, which characterizes the importance of nonlinearity relative to dissipation. Convergence properties with respect to N have not heretofore been established. In this Letter, a global definition of the function represented by the Keck‐Beyer solution is obtained by showing that certain approximations used by Keck and Beyer are equivalent to reducing the differential equation to Burgers' equation, which has an exact solution. The exact solution is then expanded to obtain the Keck‐Beyer results. Since the expansion is valid for all values of N, it is concluded that the Keck‐Beyer series converges for all values of N.