Nonlinear resonance of a gas between coaxial cylinders and concentric spheres: the shock-forming, finite-dissipation, and nonaxisymmetric regimes beyond the shock-free modulation theory

When a gas confined between two coaxial cylinders or two concentric spheres is driven at resonance by harmonic oscillation of the outer boundary, the curved geometry makes the mode spectrum dissonant, so the weakly nonlinear response is shock-free and its amplitude grows to $O(M^{1/3})$, where $M$ is the acoustic Mach number. Kurihara and Yano (2006) derived, by the method of multiple scales with $\epsilon=M^{1/3}$, a cubic amplitude equation whose undamped dynamics produces a periodic modulation of amplitude and phase with period of order $M^{-2/3}$; their analysis explicitly excluded the shock-forming regime above the numerically observed critical Mach number, the nearly plane-wave limit $\alpha\to 1$, and nonaxisymmetric geometries, and treated dissipation only briefly. Seymour, Mortell and Amundsen (2011, 2012) later constructed continuous-versus-shocked criteria and steady shocked asymptotic solutions via nonlinear geometric acoustics valid in the small-gap limit $L\gg 1$, recovering the plane-wave limit, which resolves the steady-state part of the first two exclusions. The surviving open core is: a reduced analytic description of the shock-forming regime and its transient at order-one gap ratio, including the breakdown of the modulation at the critical Mach number; the continuous-to-shocked transition map over the $(M,\alpha)$ plane for this resonator family; the quantitative effect of finite dissipation on the modulation and its bifurcation structure; and the nonaxisymmetric (eccentric-cylinder) problem.

Background

Acoustic resonance in a closed cavity arises when a boundary oscillates harmonically at a frequency near a linear eigenfrequency of the gas, and the response amplitude is then limited only by nonlinearity and dissipation. The strength of the excitation is measured by the acoustic Mach number $M$, the ratio of the boundary velocity amplitude to the sound speed, typically $M\ll 1$. The classical case is the plane standing wave in a closed tube studied by Chester (1964): the eigenfrequencies are integer multiples of the fundamental, so the nonlinearly generated higher harmonics are themselves resonant, periodic shock waves form, and the quasi-steady amplitude saturates at $O(\sqrt{M})$.

Curved radial geometries behave differently. For radially symmetric standing waves between concentric spheres or in a cylinder, the eigenfrequency of the $n$-th mode is not an integer multiple of the fundamental; the spectrum is called dissonant. The higher harmonics generated by nonlinearity are then detuned from resonance, shock formation is impeded, and the resonant amplitude can reach $O(M^{1/3})$, much larger than in the plane case at the same $M$. This dissonance mechanism underlies macrosonics, the exploitation of shock-free high-amplitude operation in shaped cavities (measured by Lawrenson et al., 1998). Steady-state theories of these dissonant resonances were given by Chester (1991) for a spherical shell and by Ellermeier (1994) for an infinitely long cylinder, the latter extended to a thermoviscous gas (Ellermeier, 1997); because these analyses assumed the steady state from the outset, the temporal evolution of the wave motion was not clarified.

The geometry is parameterized by the radius ratio $\alpha$, the radius of the inner boundary divided by that of the outer boundary. As $\alpha\to 0$ the inner boundary vanishes and cylindrical or spherical focusing raises the amplitude near the center; as $\alpha\to 1$ with fixed spacing the wave motion approaches that of a plane standing wave, for which shocks necessarily form when dissipation is negligible, which is why a special, shock-resolving treatment of $\alpha\to 1$ is needed (the nearly-plane regime was discussed by Ockendon et al., 1993). A numerical study of the coaxial-cylinder problem (Kurihara, Inoue and Yano, 2005), solving the Euler equations for an inviscid gas, found that no shock forms when $M$ lies below a critical value depending on $\alpha$ (written here $M_{\rm crit}(\alpha)$), that the amplitude then reaches $O(M^{1/3})$ and is periodically modulated with a period of order $M^{-2/3}$, and that above criticality the waveform contains shocks accompanied by shock heating near the inner cylinder.

The source paper of the present problem (Kurihara and Yano, 2006) supplied the reduced theory of that shock-free regime for two coaxial cylinders and two concentric spheres, with the wave motion excited by radial oscillation of the outer boundary. Using the method of multiple scales, an asymptotic technique that introduces a slow time $T=\epsilon^2 t$ with $\epsilon=M^{1/3}$ on which weak nonlinearity, detuning from exact resonance, and damping all act, they derived a cubic nonlinear evolution equation for the complex amplitude of the fundamental mode, with coefficients depending explicitly on $\alpha$ and on whether the geometry is cylindrical or spherical. For negligible dissipation this amplitude equation possesses a first integral, a conserved quantity constraining the motion in the phase plane of amplitude and phase, and almost every trajectory in that plane is periodic: the resonant oscillation is accompanied by a periodic modulation of amplitude and phase on the slow time $M^{2/3}t$, with an explicit modulation period for an initially quiescent gas. The steady states correspond to fixed points of the amplitude equation, for zero detuning typically three (one saddle and two centers), which organize the frequency response and constitute the bifurcation structure of the resonance; small damping slowly destroys the modulation so that the oscillation approaches a steady state matching the earlier steady-state theories.

The analysis is thereby confined to the weakly nonlinear, shock-free, weakly dissipative, axisymmetric corner of the parameter space, and the paper itself flags what lies outside: the regime $M>M_{\rm crit}(\alpha)$, where the numerical work shows shocks appear but no reduced analytic description exists; the limit $\alpha\to 1$, where the plane-wave tendency to form shocks invalidates the shock-free expansion and a matched asymptotic or shock-resolving theory is required; dissipation of moderate or large magnitude, relevant when the driving frequency is very high, whose effect on the modulation amplitude, decay rate, and bifurcation structure is not systematically characterized (the source gives only a brief discussion, with example phase planes and steady-response curves at one moderate damping value, all within the amplitude equation); and nonaxisymmetric geometries such as two eccentric cylinders, where asymmetric standing-wave modes are excited, a problem the authors state was then under study.

Subsequent work by Seymour, Mortell and Amundsen substantially advanced the first two of these regimes for the concentric family. They first analyzed resonant oscillations between concentric spheres (2011), showing how the dissonant spectrum, stratification of the gas, the geometric parameter, and the input Mach number each determine whether the motion is continuous or shocked, and recovering limiting geometries (full sphere, cone, straight tube). They then constructed asymptotic solutions for the shocked resonant oscillations between concentric spheres and coaxial cylinders (2012) using nonlinear geometric acoustics, a high-frequency approximation that transports the nonlinearly distorted waveform along radial rays, valid for $L\gg 1$, where $L$ is the ratio of the inner radius to the gap between the boundaries (so $L\to\infty$ is precisely the $\alpha\to 1$ nearly-plane limit); they compared shocked profiles and shock strengths with exact numerical solutions and recovered the plane-wave (Chester) limit as $L\to\infty$, noting that these shocked solutions complement the continuous solutions previously derived from a dominant first-mode approximation. The same group later mapped the transition between continuous and shocked solutions for the frustum of a closed cone (2015; 2022), but the analogous transition map for the concentric-spheres/coaxial-cylinders family, the shocked regime at $L=O(1)$, the shock-forming transient, finite-dissipation effects on the modulation, and nonaxisymmetric annuli all remain without a quantitative theory.

Problem Statement

Establish a quantitative reduced description of the nonlinear resonant oscillations of a gas driven between two coaxial cylinders and two concentric spheres, and in nonaxisymmetric annular variants of this resonator, valid outside the weakly nonlinear, shock-free, weakly dissipative, axisymmetric corner completed by Kurihara and Yano (2006) and by the steady shocked asymptotics of Seymour, Mortell and Amundsen (2012). The single overarching objective is to account for how nonlinearity (acoustic Mach number $M$), geometry (radius ratio $\alpha$, and eccentricity in nonaxisymmetric cases), and dissipation jointly determine whether the resonant response is a modulated, steady, or shocked oscillation. The jointly supporting targets, each currently without a quantitative theory, are: (1) the shock-forming regime at order-one gap ratio, $L=O(1)$, where $L$ is the ratio of the inner radius to the gap between the boundaries (equivalently, the radius ratio $\alpha$ away from 1), for $M>M_{\rm crit}(\alpha)$: a reduced analytic description of the shocked oscillation and of the shock-forming transient, in particular how the periodic amplitude-phase modulation of the shock-free regime breaks down and is replaced by shocked oscillation as $M$ crosses $M_{\rm crit}(\alpha)$, and a quantitative characterization of $M_{\rm crit}(\alpha)$ itself, for which an analytic formula is the preferred form but an explicit numerical determination is acceptable; (2) the transition map between continuous and shocked solutions over the $(M,\alpha)$ plane for this resonator family, connecting the dominant-first-mode modulation theory valid at $L=O(1)$ with the nonlinear geometric acoustics valid for $L\gg 1$, including the $\alpha\to 1$ nearly-plane limit where both meet the classical plane-wave shocked resonance; (3) the effect of finite (moderate to large) dissipation, relevant at high driving frequency, on the modulation amplitude and decay rate, on the approach to steady state, and on the bifurcation structure of the amplitude equation (the fixed-point organization and separatrix structure of the phase plane as damping grows from order $\epsilon^2$, with $\epsilon=M^{1/3}$, to magnitudes comparable with the nonlinear term), of which the source gives only a brief, example-level discussion; and (4) in nonaxisymmetric annuli, where asymmetric standing-wave modes are excited: the resonant mode content, the coupling among those modes, and the slow modulation of their amplitudes and phases, taken for the required instance of two eccentric cylinders with eccentricity-to-radius ratio up to $O(1)$. All four targets are mandatory for completion, and each requires a quantitative result: an analytic closed-form result is preferred where one is available, but a quantitative result obtained by a stated, reproducible method, numerical or asymptotic, with its regime of validity, satisfies a target where no analytic route is available. Partial coverage of the stated regimes, or a qualitative survey without a quantitative result, does not complete the task.

References

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