Nonlinear resonant oscillations in closed tubes—An application of the averaging method

The application of the averaging method to the one-dimensional inhomogeneous, nonlinear acoustic wave equation with dissipative term makes it possible to give asymptotic solutions for any kind of external resonance excitation. It shows that the lowest-order solution consists of the superposition of two modulated counterpropagating waves, where the amplitude of each is a solution of Burgers equation. The method is extended to the treatment of oscillating boundaries; in that case it also leads to Burgers equations. Explicit stationary solutions are given for the particularly important forms of the external excitation, harmonic distributed forces, and harmonic oscillating boundaries. The application of several other computational methods to this problem leads to the same results.