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

Abstract

-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. © 2001 MAIK "Nauka/Interperiodica".

Finite-amplitude standing acoustic waves in a cubically nonlinear medium

“…A detailed explanation and the physical foundation of this approach can be found in [2,12]. A similar approach can be used for standing waves in a cubically nonlinear resonator.…”

“…Such a nonlinearity is known to be dominant for acoustic waves in fluids and for longitudinal waves in solids. Reviews of the results obtained for quadratically nonlinear resonators can be found in [1][2][3].…”

Standing Waves In Quadratic And Cubic Nonlinear Resonators: Q-Factor And Frequency Response

“…This approach is described in detail in [1] and used by several authors before. For quadratic nonlinearity and harmonic vibration of one wall the functional equation is derived [1]:…”

“…Parameter D is the small ratio of the length of resonator to the characteristic absorption distance. The absorption coefficient b and corresponding dissipative terms are not presented, for simplicity, in the initial functional equation (1), but this modifying can be easily performed in the differential equation (2). Exact analytical solutions to equation (2) for travelling waves were obtained in Refs.…”

“…Equation (1) can describe not only steady-state vibrations, but transient processes as well [1], including unlimited linear amplitude growth at resonance (with 0 = ε ). If the following requirements are fulfilled -namely, the length L of resonator is small in comparison with the shock formation distance, frequency ω of driving force differs slightly from the resonant frequency and the Q-factor is large, Q>>1 -the general equation (1) can be reduced [1] to the inhomogeneous Burgers equation:…”