Experimental evidence of solitary wave interaction in Hertzian chains

Abstract

We study experimentally the interaction between two solitary waves that approach one another in a linear chain of spheres interacting via the Hertz potential. When these counterpropagating waves collide, they cross each other and a phase shift in respect to the noninteracting waves is introduced as a result of the nonlinear interaction potential. This observation is well reproduced by our numerical simulations and is shown to be independent of viscoelastic dissipation at the bead contact. In addition, when the collision of equal amplitude and synchronized counterpropagating waves takes place, we observe that two secondary solitary waves emerge from the interacting region. The amplitude of the secondary solitary waves is proportional to the amplitude of incident waves. However, secondary solitary waves are stronger when the collision occurs at the middle contact in chains with an even number of beads. Although numerical simulations correctly predict the existence of these waves, experiments show that their respective amplitudes are significantly larger than predicted. We attribute this discrepancy to the rolling friction at the bead contact during solitary wave propagation.

Characterizing traveling-wave collisions in granular chains starting from integrable limits: The case of the Korteweg–de Vries equation and the Toda lattice

“…The latter provided a highquality description especially of counter-propagating wave collisions, while its 2-soliton solutions can also be used for capturing co-propagating cases, at least qualitatively. It is relevant to note here that at the level of δ 0 = 0 (i.e., when the precompression is absent) such collisions have already been studied in [34,61,62]. It thus seems that an extension of that work to experimentally consider collisions in the more theoretically and analytically tractable case of finite precompression would be possible, as much as it would be desirable.…”

Investigation of co-travelling solitary wave collisions in a granular chain

“…Numerical simulations by Manciu, et al 9 have noted that the head-on collision of two equal waves results in the formation of secondary solitary waves (SSWs) due to the lack of tensional forces and the discreteness of the system: the spheres break away from each other and as they regain contact their collisions generate new waves. Santibanez, et al 10 have experimentally confirmed the creation of these waves in an equal head-on collision, as well as noting that the waves undergo a phase advance that can be described as the interaction of two quasiparticles. Simulations by Zhen-Ying, et al 11 have shown that head-on and co-travelling collisions of solitary waves of unequal magnitude result in the exchange of energy between the solitary waves as well as the formation of secondary waves.…”

Attenuation of short strongly nonlinear stress pulses in dissipative granular chains

“…This large differences between velocities of neighboring particles in one mass chain result in larger viscous dissipative losses (Eqs. [8][9][10][11][12] in comparison with two mass chain were localized pulses (not solitary waves) have a longer dimensions and thus a smaller gradients of velocity between neighboring particles. This difference may explain the reversal in impact mitigation effectiveness of these systems with increased damping coefficients.…”

“…The behavior of these waves was investigated by different groups of researchers numerically, for example [2,6,7], and experimentally using gauges embedded in the particles made from different materials [3,4,5,[8][9][10][11] and also high speed photography allowing to measure particles displacements with a micrometer-scale resolution [12].…”