Abstract:

A theoretical investigation of nonlinear standing waves in an acoustical resonator is presented. The motivation for this research stems from the new technology of resonant macrosonic synthesis (RMS) developed at MacroSonix. RMS creates high amplitude standing waves, e.g., overpressures in excess of 300% of ambient pressure. The analysis is based on a one-dimensional model equation for the velocity potential that is derived from the fundamental gas dynamics equations for an ideal gas. Nonlinearity, gas viscosity, and entire resonator driving are included. The resonator is assumed to be of an axisymmetric, but otherwise arbitrary, shape. Nonlinear spectral equations are integrated numerically for a two-point boundary-value problem. The harmonic amplitudes and phases of the velocity potential wave are obtained directly from the solution of the frequency-domain equations. The pressure wave shape, the harmonic amplitudes of the pressure wave, and harmonic amplitude distribution along the resonator axis are then calculated. Results are presented for three resonator geometries: a cylinder, a cone, and a bulb. Both hardening and softening behaviors are observed and shown to be geometry dependent. At high amplitude, hysteresis effects are present in the frequency-response curves. Comparisons between measured and calculated waveforms show good agreement.