In a granular packing, ultrasound recorded by a time-reversal mirror (TRM) and re-emitted time-reversed refocuses at the source through multiply scattered coda waves at small driving amplitude, but the refocusing degrades at large amplitude because the vibration itself rearranges the inter-grain contact network between the forward and backward propagation passes. The source experiment (Harazi et al., EPJ Spec. Top. 226, 1487 (2017)) left two linked issues open, neither of which is known to have been answered in the literature to date: (i) how the loss of time-reversal fidelity depends on the arrival time — equivalently the path length and scattering order — of the multiply scattered waves in the coda of a rearranged packing, i.e. whether late-arriving, deeply scattered components are more or less sensitive to contact-network rearrangement than early coda; and (ii) how to focus time-reversed ultrasound inside a granular medium so as to controllably generate a localized high-amplitude excitation at a chosen interior position, sufficient to trigger, probe, and potentially control rearrangements of the contact network.

Background

A time-reversal (TR) experiment records the wavefield emitted by a source with an array of transducers (a time-reversal mirror, TRM), flips the recorded signals in time, and re-emits them; by reciprocity of the wave equation the re-emitted field retraces its scattering history and refocuses at the original source. In a strongly multiply scattering medium the late, multiply scattered tail of the recorded signal (the coda) acts as a lens: time-reversing the coda yields a refocused spot finer than what the ballistic (direct, unscattered) arrival provides. Granular packings — disordered assemblies of elastic grains compressed into contact — are an unusual host for such waves. Under static compression the grains form a heterogeneous contact network of force chains (chains of grains carrying disproportionately large load), and elastic waves propagate through the contacts with wavelengths comparable to a few grain diameters, scattering strongly. Experiments and simulations (Harazi, Yang, Fink, Tourin, and Jia, EPJ Spec. Top. 226, 1487–1497, 2017) showed that in this regime TR of the multiply scattered coda refocuses well at small driving amplitude, implying that continuum-like effective elastic moduli hold down to the scale of a few grains, whereas at large driving amplitude the contacts slip and the grains undergo nonaffine rearrangements (motions not captured by a smooth, affine deformation of the packing), changing the contact network itself. The propagation medium is then different between the forward and backward passes, the impulse responses are modified, and the recompression peak at the source — the TR fidelity — drops: TR invariance breaks down. The authors note that granular acoustics sits between the extreme sensitivity of chaotic particle motion to perturbations and the wavelength-coarse-grained stability of wave propagation, because waves in a packing travel along preferred discrete contact paths (see also Snieder and Scales, Phys. Rev. E 58, 5668, 1998, who used time-reversed imaging as a diagnostic of wave versus particle chaos). Related sensitivity questions are treated in seismology by coda wave interferometry, which uses changes in the coda to detect small perturbations of the medium (Snieder, Pure Appl. Geophys. 163, 455–473, 2006), but that theory addresses weak perturbations of a fixed scatterer configuration rather than the strong, dissipative change of a contact network. The same laboratory program also established ultrasound as an actuator: nanometer-amplitude ultrasound can trigger granular avalanches through shear acoustic lubrication of the contacts (Léopoldès, Jia, Tourin, and Mangeney, Phys. Rev. E 102, 042901, 2020), a mechanism reviewed in Jia and Tourin (J. Acoust. Soc. Am. 153, A112, 2023). Within this context, the source paper's conclusion poses two unresolved experimental and practical questions that define the present problem: how TR fidelity loss quantitatively depends on coda arrival time when the packing has rearranged, and how to focus high-amplitude ultrasound inside the medium at a chosen position to trigger rearrangements deliberately.

Problem Statement

The problem, as posed by the concluding remarks of Harazi et al. (2017), has two linked deliverables that together define one objective — quantifying and exploiting the (breakdown of) time-reversal invariance of multiply scattered ultrasound in granular packings whose contact networks can rearrange. (1) Diagnostic leg: quantify how the loss of TR fidelity — the decrease of the time-reversed recompression signal at the source when the contact network has rearranged between forward and backward propagation — depends on the arrival time of the multiply scattered waves within the coda, i.e. on the path length and scattering order. Determine whether late-arriving (deeply scattered) components are more or less sensitive to the rearrangement than early coda, and what this implies for the sensitivity hierarchy between particle-like and wave-like propagation in granular media. (2) Actuation leg: develop operational protocols for time-reversal focusing inside a granular medium that generate a localized high-amplitude ultrasonic field at a chosen interior position — focusing through the disordered contact network, at amplitudes large enough to trigger rearrangements of that network — and characterize the triggered rearrangement dynamics (and the degree to which they can be controlled). The two legs are connected: the arrival-time dependence of fidelity under rearrangement bounds which coda windows and depths remain usable for refocusing, and therefore how much energy can be deposited, and how deep, by interior TR focusing in a medium that evolves during the experiment. A complete answer establishes the fidelity-versus-arrival-time dependence for a characterized rearranged packing and demonstrates (or delimits) targeted interior high-amplitude triggering, with the protocol, metrology, and interpretation made explicit. The dependence may be established analytically, numerically, or experimentally; what the answer must supply is a quantitative relation (or an explicit delimitation of the regime in which it holds) rather than a qualitative statement alone, and a rearranged packing is characterized when its preparation and confinement, the driving that rearranges it, and the resulting change of the contact network are specified.

References

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