Nonlinear standing waves in an acoustical resonator

Abstract

A one-dimensional model is developed to analyze nonlinear standing waves in an acoustical resonator. The time domain model equation is derived from the fundamental gasdynamics equations for an ideal gas. Attenuation associated with viscosity is included. The resonator is assumed to be of an axisymmetric, but otherwise arbitrary shape. In the model the entire resonator is driven harmonically with an acceleration of constant amplitude. The nonlinear spectral equations are integrated numerically. Results are presented for three geometries: a cylinder, a cone, and a bulb. Theoretical predictions describe the amplitude related resonance frequency shift, hysteresis effects, and waveform distortion. Both resonance hardening and softening behavior are observed and reveal dependence on resonator geometry. Waveform distortion depends on the amplitude of oscillation and the resonator shape. A comparison of measured and calculated wave shapes shows good agreement.