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

Abstract

The frequency response of a nonlinear acoustical resonator is investigated analytically and numerically. The cross-sectional area is assumed to vary slowly but otherwise arbitrarily along the axis of the resonator, such that the Webster horn equation provides a reasonable one-dimensional model in the linear approximation. First, perturbation theory is used to derive an asymptotic formula for the natural frequencies as a function of resonator shape. The solution shows that each natural frequency can be shifted independently via appropriate spatial modulation of the resonator wall. Numerical calculations for resonators of different shapes establish the limits of the asymptotic formula. Second, the nonlinear interactions of modes in the resonator are investigated with Lagrangian mechanics. An analytical result is obtained for the amplitude-frequency response curve and nonlinear resonance frequency shift for the fundamental mode. For a resonator driven at its lowest natural frequency, it is found that whether hardening or softening behavior occurs depends primarily on whether the nonlinearly generated second-harmonic frequency is greater or less than the second natural frequency of the resonator. A fully nonlinear one-dimensional numerical code is used to verify the analytical result.