Effect of resonator dimensions on nonlinear standing waves

An investigation of the effect of resonator dimensions on nonlinear standing waves in shaped resonators is conducted. Simple forms of the shear viscosity term in the momentum equations are developed for an axisymmetric (2D) resonator and a low aspect ratio rectangular (3D) resonator. The cross sections of the resonators are exponentially expanded and the one-dimensional wave equations are solved by using the Galerkin's method. The quality factors, pressure waveforms, compression ratios, and resonance frequencies are calculated for different dimensionless cross sections and lengths of the resonators. The results show that, apart from the resonator length, the ratio of the cross-section dimension to the length of the resonator is an important parameter. If the ratio is greater than 0.04, the characteristics of the shaped resonator are not affected significantly. However, when the ratio is less than 0.01, the resonance becomes weak, the compression ratio drops substantially, and the frequency response changes as well.