Perturbation (normal-mode) theory predicts the amplitude-dependent resonance-frequency shift and saturation of gas-filled acoustic resonators whose cross section varies along the axis, including the cone and bulb shapes proposed for acoustic compressors. Because the expansion is ordered in small parameters and is intended for approximate but rapid parameter estimation, its accuracy at finite drive amplitudes cannot be guaranteed a priori. The open problem is a validated quantitative account of the finite-amplitude frequency response of cone- and bulb-shaped resonators — the resonance-frequency shift as a function of drive amplitude and the saturation of the pressure amplitude — together with the resulting validity range of the asymptotic predictions.

Background

When a gas-filled closed resonator is driven strongly at one of its acoustic resonance frequencies, the standing wave reaches finite amplitude and the nonlinearity of the gas changes the response in two characteristic ways: the resonance frequency shifts by an amount that grows with amplitude (a "hardening" response shifts it upward, a "softening" response downward), and the pressure amplitude saturates instead of growing linearly with the drive. Experiments on resonant macrosonic synthesis (Lawrenson et al., 1998) showed that shaping the resonator — letting its cross-sectional area S(x)S(x) vary along the axis xx — can suppress shock formation and produce shock-free standing waves of extremely high amplitude, which motivated acoustic compressors and other no-moving-part devices. In the long-wavelength limit, where the cross-section dimensions are small compared with the acoustic wavelength, the linear modes of such a cavity are described by the Webster horn equation, a one-dimensional wave equation for the mean pressure with a geometry-dependent term S′(x)/S(x)S'(x)/S(x). Ilinskii, Lipkens, Lucas, Van Doren and Zabolotskaya (1998) derived the corresponding nonlinear one-dimensional model equation, including thermoviscous boundary-layer losses (the dominant dissipation mechanism at the resonator walls), and solved it numerically in the frequency domain with a finite number of harmonics; this remains the standard "direct numerical" reference. Hamilton, Ilinskii and Zabolotskaya (2001) developed a Lagrangian perturbation theory in the normal modes for shapes close to cylindrical, and Hamilton, Ilinskii and Zabolotskaya (2009) extended it to any shape for which the Webster horn equation is a valid linear model, including the cone and the bulb. The theory gives rapid approximate estimates of the frequency response and the nonlinear resonance-frequency shift; physically, the direction of the shift is set by how efficiently the fundamental generates its second harmonic into modes lying below versus above the second harmonic frequency, compared with the cubic nonlinearity and the static (mean) deformation of the gas. The asymptotic expansion is ordered in small parameters (acoustic Mach number and the strength of the geometric variation), and its authors present it as appropriate for approximate but rapid parameter estimations, so its accuracy at finite drive amplitudes cannot be guaranteed a priori; the 2009 paper compares its cone and bulb results with losses against the direct numerical solution of the one-dimensional model. An independent asymptotic school (Mortell and Seymour, 2004) treats the same class of variable-cross-section resonators with different techniques, and its authors publicly disputed aspects of the Hamilton–Ilinskii–Zabolotskaya criteria for hardening versus softening in a 2008 Comment, to which the theory's authors replied — evidence that the quantitative accuracy question for shaped resonators is a recognized open issue in the community.

Problem Statement

For gas-filled acoustic resonators whose cross section varies along the axis in shapes described by the Webster horn equation (the long-wavelength one-dimensional wave equation with geometry-dependent term S′(x)/S(x)S'(x)/S(x), where S(x)S(x) is the cross-sectional area) — specifically the cone and the bulb proposed for acoustic compressors — with thermoviscous boundary-layer losses, determine the nonlinear frequency response at finite and large drive amplitudes with quantified accuracy: the resonance-frequency shift as a function of drive amplitude and the saturation of the pressure amplitude. The concrete cone and bulb geometries, loss model and drive-amplitude scales are those of the cited source papers (Hamilton, Ilinskii and Zabolotskaya, 2009, DOI 10.1121/1.3075585; Ilinskii et al., 1998, DOI 10.1121/1.423850). The scientific objective is the quantitative validation of the reduced asymptotic theory for these shapes: frequency-response curves for cone and bulb resonators at finite and large drive amplitudes, and the range of drive amplitudes over which the lowest-order normal-mode predictions of the shift (both sign and magnitude) and of saturation are trustworthy.

References

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