Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response

Abstract

Simplified nonlinear evolution equations describing non-steady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach based on a nonlinear functional equation is used. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. The process of development of a standing wave is described analytically on the base of exact nonlinear solutions for different laws of periodic motion of the wall. For harmonic excitation the wave profiles are described by Mathieu functions, and their mean energy characteristics by the corresponding eigenvalues. The sawtooth-shaped motion of the boundary leads to a similar process of evolution of the profile, but the solution has a very simple form. Some possibilities to enhance the Q-factor of a nonlinear system by suppression of nonlinear energy losses are discussed.

Dissipative and hysteresis loops as images of irreversible processes in nonlinear acoustic fields

“…Loops (25) are shown in Fig. 3 for three different values of xs : 0.2 (dotted line), 1 (solid line), and 3 (dashed line).…”

“…24 Their solution describing nonlinear standing waves can be derived by the asymptotic methodology developed before for highpower resonant vibrations. 25,26 V. DISCUSSION For clarification of the results above, a discussion is desirable. In accordance with a classification given in Ref.…”

Standing and propagating waves in cubically nonlinear media

“…In analogy with the solution attempted by the present authors for a quadratic nonlinearity [7] we attempt a solution to Eq. (14) in the form…”

“…The equation with a quadratic nonlinearity corresponding to the cubic nonlinear equation (14) is studied for a resonator by the present authors [7]. The resonator boundary conditions are V x…”

“…Two examples of cubically nonlinear wave theory are studied. In section 3 standing waves in a cubically nonlinear resonator are studied and the frequency response function is calculated and compared with the corresponding function in the quadratically nonlinear case [7]. In section 4 cubically nonlinear propagation of N-waves is studied using methods developed for quadratically nonlinear waves [8].…”