Linear and nonlinear frequency shifts in acoustical resonators with varying cross sections

The frequency response of a nonlinear acoustical resonator is investigated analytically and numerically. The cross-sectional area is assumed to vary s lowly but otherwise arbitrarily along the axis of the resonator, such that the Webster horn equation provides a reasonable one-dimensional model in th e linear approximation. First, perturbation theory is used to derive an asy mptotic formula for the natural frequencies as a function of resonator shap e. The solution shows that each natural frequency can be shifted independen tly via appropriate spatial modulation of the resonator wall. Numerical cal culations for resonators of different shapes establish the limits of the as ymptotic formula. Second, the nonlinear interactions of modes in the resona tor are investigated with Lagrangian mechanics. An analytical result is obt ained for the amplitude-frequency response curve and nonlinear resonance fr equency shift for the fundamental mode. For a resonator driven at its lowes t natural frequency, it is found that whether hardening or softening behavi or occurs depends primarily on whether the nonlinearly generated second-har monic frequency is greater or less than the second natural frequency of the resonator. A fully nonlinear one-dimensional numerical code is used to ver ify the analytical result. (C) 2001 Acoustical Society of America.