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A new analytical approach is developed for the description of standing waves caused by arbitrary periodic vibration of a boundary. The approach is based on the nonlinear evolution equation written for an auxiliary function. This equation offers the possibility to study not only the steady-state acoustic field, but also its evolution in time. One can take into account the dissipative properties of the medium and the difference between one of the resonant frequencies and the fundamental frequency of the driving motion of the wall. An exact nonsteady-state solution is derived corresponding to the sawtooth-like periodic vibration of the boundary. The maximal “amplitude” values of the particle velocity and the energy of a standing wave are calculated. The temporal profiles of standing waves at different points of the layer are presented. A new possibility of pumping a high acoustic energy into a resonator is indicated for the case of a special type of the wall motion having the form of an “inverse saw.” Theoretically, such a vibration leads to an “explosive instability” and an unlimited growth of the standing wave. For a harmonic excitation, the exact non-steady-state solution is derived as well. The standing wave profiles are described by Mathieu functions, and the energy characteristics by their eigenvalues.

Nonlinear standing waves in a layer excited by the periodic motion of its boundary