When a strongly nonlinear solitary wave strikes an interface between two hexagonal granular lattices in the sonic-vacuum regime, a quasiparticle collision model based on energy and momentum conservation predicts the amplitudes of transmitted solitary-wave trains and yields a granular analogue of Snell's law in which the amplitude-dependent solitary-wave speed replaces the vanishing sound speed (Tichler et al., Phys. Rev. Lett. 111, 048001 (2013)). Four limitations of that model remain unaddressed in the subsequent literature: the amplitudes and timing of the delayed secondary solitary-wave trains are not predicted; the domain of validity of the quasi-one-dimensional reduction and the fixed effective mass meff≈1.4mm_{\mathrm{eff}}\approx1.4m is unestablished across mass ratios and incidence angles within the source's hexagonal mass-contrast setting; the dynamics at and beyond the critical incidence angle θc\theta_c, including delayed reflection, has no theory; and the empirically observed near-independence of the Snell-like relations on the incident amplitude lacks a first-principles derivation. The problem is to construct a theory of strongly nonlinear solitary-wave scattering at granular interfaces that repairs these four gaps.

Background

A granular crystal is an ordered packing of macroscopic elastic particles that interact only through compressive contact forces. For two touching spherical beads, the Hertz contact law states that the repulsive force grows with the overlap δ\delta as F∝δ3/2F\propto\delta^{3/2}; the interaction is one-sided, so a contact can open (contact breaking) as soon as the grains separate. A lattice of such beads with zero precompression is called a sonic vacuum, because the speed of small-amplitude linear sound vanishes there; finite disturbances propagate instead as strongly nonlinear solitary waves — localized, supersonic pulses whose speed VsV_s grows with amplitude (for Hertzian spheres, Vs∝u1/5V_s\propto u^{1/5} for striker speed uu). Because the speed depends on amplitude and the governing equations are nonintegrable, no exact scattering theory exists. A device validated in one-dimensional chains is to treat the solitary wave as a quasiparticle carrying momentum PP and energy E=P2/(2meff)E=P^2/(2m_{\mathrm{eff}}), with an effective mass meff≈1.4mm_{\mathrm{eff}}\approx1.4m for beads of mass mm (the pulse spans roughly five bead diameters). In one dimension, when such a wave crosses an interface between chains of different bead masses, a train of transmitted solitary waves is generated, together with delayed secondary pulses associated with gap opening and closing at the interface (Vergara, Phys. Rev. E 73, 066623 (2006)); global conservation laws alone do not fix the energy partition among the pulses.

Tichler et al. (2013) extended the question to two dimensions: an incident solitary-wave front, parallel to an interface separating two hexagonal (close-packed triangular) granular lattices differing only in bead mass, with mass ratio A≡m2/m1A\equiv m_2/m_1, was studied by molecular-dynamics simulation of about 10410^4 frictionless Hertzian grains. Modeling the dynamics as successive elastic collisions between the heavy interfacial bead and the solitary-wave quasiparticle of the light lattice, they predicted the momentum of the nn-th transmitted solitary wave, P2,n/P2,1=(B−1B+1)n−1P_{2,n}/P_{2,1}=\left(\frac{B-1}{B+1}\right)^{n-1} with B=m1/m2,effB=m_1/m_{2,\mathrm{eff}}, in agreement with simulations. For oblique interfaces they found that, below a critical incidence angle θc\theta_c at which the refracted front would run parallel to the interface, the angles of refraction and reflection obey an analogue of Snell's law, V0sin⁡θrefr=V2,1sin⁡θiV_0\sin\theta_{\mathrm{refr}}=V_{2,1}\sin\theta_i, where V0V_0 and V2,1V_{2,1} are the speeds of the incident and of the leading transmitted solitary wave, respectively — the amplitude-dependent solitary-wave speeds replacing the sound speeds that appear in the linear optical or acoustic case, which are zero in the sonic vacuum. Remarkably, the ratio sin⁡θrefr/sin⁡θi\sin\theta_{\mathrm{refr}}/\sin\theta_i is nearly independent of the incident amplitude even though the wave speeds themselves depend on amplitude, an observation the authors called seemingly puzzling and rationalized only qualitatively by quasiparticle scattering with proportional speeds.

Problem Statement

Construct a theory of strongly nonlinear solitary-wave scattering at interfaces between granular lattices in the sonic-vacuum regime that repairs the four acknowledged gaps of the quasiparticle collision model of Tichler et al. (2013), for the setting defined there — two uncompressed hexagonal granular lattices whose beads differ only in mass (A≡m2/m1A\equiv m_2/m_1), interacting via the Hertz law and treated as frictionless — and establish where that setting's approximations extend:

(1) Delayed secondary trains: predict the number, amplitudes, and emission times of the delayed secondary sequences of solitary waves generated by the repeated collisions of the interfacial ('dancing') heavy bead, which the existing model deliberately neglects.

(2) Validity of the quasi-one-dimensional reduction: determine the domain of accuracy of treating the two-dimensional front as a one-dimensional quasiparticle with fixed effective mass meff≈1.4mm_{\mathrm{eff}}\approx1.4m, within the hexagonal mass-contrast setting defined above, as a function of the mass ratio AA and the incidence angle, and quantify how violations affect the predicted transmission and reflection amplitudes.

(3) Critical and beyond-critical incidence: provide a mechanistic theory of the dynamics at and beyond the critical angle θc\theta_c, at which the Snell-like law would make the transmitted front propagate parallel to the interface — including the delayed reflection observed when θi>θc\theta_i>\theta_c and the dynamics when the incoming and reflected waves cross each other at the interface.

(4) Amplitude independence: derive from first principles — beyond the qualitative quasiparticle argument — why the Snell-like ratios sin⁡θrefr/sin⁡θi\sin\theta_{\mathrm{refr}}/\sin\theta_i and sin⁡θrefl/sin⁡θi\sin\theta_{\mathrm{refl}}/\sin\theta_i are nearly independent of the amplitude of the incident solitary wave, and state the limits of that independence.

The common objective is a predictive, physically grounded account of how strongly nonlinear solitary-wave fronts are transmitted, reflected, and refracted at granular interfaces; the four items are facets of that single objective and are jointly required, and the conclusions of each hold only within the restricted regimes in which its approximations apply.

References

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