For a one-dimensional resonator filled with a medium whose nonlinearity is purely cubic (odd, with no quadratic term — the situation realized by shear waves in isotropic solids and soft media), Rudenko, Hedberg, and Enflo (2007) constructed an analytical theory of finite-amplitude standing waves: representing the field as two strongly distorted counterpropagating waves coupled through their period-averaged intensities, they derived resonance curves, discontinuous shock-containing profiles, a nonlinear saturation of the stored intensity, and predicted hysteresis and bistability by analogy with the Duffing oscillator. The theory rests on idealizations — a two-wave reduction and a small-dissipation algebraic approximation — and its distinctive predictions were never checked against direct numerical integration of the original driven dissipative equation or against dynamical branch selection. This problem asks for that completion: compute the resonance curves and steady discontinuous solutions of the full driven system with finite small dissipation and exact sinusoidal forcing; establish the stability and dynamical selection of the solution branches, verifying or refuting the predicted hysteresis and bistability; and quantify the validity domains of the two-wave and algebraic approximations. Partial progress exists for realistic shear-resonator models (Andreev et al. 2011; Cormack and Hamilton 2015–2018), but the exact source problem remains open.

Background

When a fluid or solid is driven hard enough that the wave amplitude is no longer infinitesimal, the small departure of the medium's constitutive law from linearity reshapes the wave itself: harmonics grow, the waveform steepens, and in a resonator the response near resonance changes qualitatively. For ordinary fluids the leading nonlinearity is quadratic (the leading term in the equation of state beyond the linear one), and nonlinear standing waves in fluid-filled resonators have been studied since Chester's work on resonant tube oscillations; Rudenko, Hedberg, and Enflo developed the corresponding layer theory for a quadratically nonlinear medium driven by a periodically moving boundary (Acoustical Physics 47, 452, 2001; JASA 117, 601, 2005). A different and less explored case is a medium whose nonlinearity is purely cubic — the leading correction is of odd (third) order, with the quadratic term absent. This is not exotic: for shear waves in isotropic elastic media the shear stress is an odd function of shear strain, so the leading nonlinearity is exactly of this cubic type, which is why rubber-like polymers, biological tissue, and geomaterials at large shear strain are the natural realizations; the same odd nonlinearity also arises for longitudinal waves in special media and for certain signals where quadratic contributions cancel.

Rudenko, Hedberg, and Enflo (Acoustical Physics 53, 455, 2007) analyzed the one-dimensional resonator filled with such a cubically nonlinear, weakly dissipative medium and driven by harmonic oscillation of its boundary. Their key structural finding is that unlike the quadratic case — where two counterpropagating waves making up the standing field interact directly — in a cubic medium the two waves are coupled only through their mean (period-averaged) intensities. Working in this two-wave representation, they constructed discontinuous steady profiles containing compression and expansion shock fronts (a shock front is the near-discontinuous jump the waveform develops when nonlinear steepening overtakes dissipative smoothing), computed resonance curves — the dependence of the mean field intensity on the detuning between the driving frequency and a resonator eigenfrequency — and showed that the governing nonlinear integro-differential equation degenerates, when dissipation is small, into a third-degree algebraic equation with an undetermined coefficient fixed by a supplementary condition. From this they extracted two distinctive predictions: a nonlinear saturation, meaning that beyond a certain drive amplitude the stored intensity stops growing and cannot exceed a limiting value set by nonlinear (shock-front) attenuation; and a hysteresis with bistability of the resonance curve, argued by analogy with the Duffing oscillator — the standard lumped model of a mass in a cubic force field, whose frequency response bends over, supports two stable oscillation levels in a range of detunings, and jumps between them, and which is also the classical route diagram for period-doubling and chaos in driven nonlinear oscillators.

The analytical route, however, idealizes the problem twice: the exact field is replaced by the two-wave ansatz, and the dissipative dynamics by a steady algebraic balance that neglects the dissipation parameter everywhere except in narrow boundary layers. The authors themselves left the completion open. Later work has advanced the neighboring, physically equivalent shear-wave realization: Andreev, Krit, and Sapozhnikov (Acoustical Physics 57, 779, 2011) computed finite-difference resonance curves for a plastisol shear resonator with a monorelaxing cubic constitutive model and observed the predicted asymmetry and a bistability region experimentally, and Cormack and Hamilton (JASA 143, 1035, 2018) built an augmented Duffing model for a relaxing shear-wave resonator and validated it against finite-difference solutions of their full wave equation. These studies, however, use their own viscoelastic model equations rather than the source system, focus on the lowest mode, and do not address the saturation limit, the global branch-selection dynamics, or a quantitative test of the source theory's distinctive predictions — leaving the completion of the 2007 program an open problem.

Problem Statement

For the one-dimensional resonator filled with a cubically nonlinear (odd, with no quadratic term), weakly dissipative medium and driven by harmonic oscillation of its boundary, exactly as formulated by Rudenko, Hedberg, and Enflo (2007), establish the quantitative nonlinear resonance response beyond the idealized analytical description. The starting point is the original driven dissipative wave equation with cubic nonlinearity and resonator boundary conditions from which the authors derive their two-wave representation and nonlinear integro-differential equation; the governing equations, boundary driving condition, dissipation parameter, and definition of the mean field intensity are those of the cited paper (DOI 10.1134/S1063771007040057). The tasks are:

  1. Direct numerical integration of the original driven dissipative wave equation (finite small dissipation, exact sinusoidal boundary forcing) to compute the resonance curves — mean field intensity versus detuning between drive frequency and resonator eigenfrequency — and the steady discontinuous (shock-containing) solutions, including the saturated limiting intensity and its dependence on drive amplitude.

  2. The dynamical selection and stability of the solution branches: determine, from the time-dependent evolution, which branches are attracting, how jumps between branches occur under frequency sweeps, and verify or refute the predicted hysteresis and bistability (argued in the source by analogy with a concentrated Duffing-type system at nonlinear resonance). Should chaotic or quasi-periodic regimes be observed in parameter ranges where multistability is predicted, documenting them is of interest, but this is not a completion condition.

  3. Quantified validity domains of the analytical approximations: assess by comparison with the full solution when the two-wave representation and the small-dissipation algebraic approximation (with its undetermined coefficient) are accurate, in particular the effect of finite but small dissipation on the selection and stability of the periodic solutions.

These three tasks jointly serve one objective: replace the source paper's idealized analytical description of the driven cubically nonlinear resonator with a quantitatively verified account of its resonance response. The problem is complete when that objective is met for the source's own model equation system as defined in Rudenko, Hedberg, and Enflo (2007); substituting a refined or alternative model equation system for the source system does not meet the completion criterion. Experimental realization of the predictions in a medium with cubic-dominant nonlinearity (such as the shear-resonator experiments of Andreev et al.) would strengthen the verification, but a new laboratory experiment is not a completion condition.

References

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