Transmission and Reflection of Strongly Nonlinear Solitary Waves at Granular Interfaces

The interaction of a solitary wave with an interface formed by two strongly nonlinear noncohesive granular lattices displays rich behavior, characterized by the breakdown of continuum equations of motion in the vicinity of the interface. By treating the solitary wave as a quasiparticle with an effective mass, we construct an intuitive (energy- and linear-momentum-conserving) discrete model to predict the amplitudes of the transmitted solitary waves generated when an incident solitary-wave front, parallel to the interface, moves from a denser to a lighter granular hexagonal lattice. Our findings are corroborated with simulations. We then successfully extend this model to oblique interfaces, where we find that the angle of refraction and reflection of a solitary wave follows, below a critical value, an analogue of Snell’s law in which the solitary-wave speed replaces the speed of sound, which is zero in the sonic vacuum.

Figure 1

Time sequence leading to the generation of a solitary-wave train in simulations. The (red) beads on the left of the interface constitute the heavier medium with mass $m_1$, and the (yellow) beads on the right of the interface constitute the lighter medium with mass $m_2$. The mass ratio $A\equiv m_2/m_1=0.125$. The velocity field, overlaid in green, denotes the instantaneous speeds of the beads. An incident solitary wave (a) collides with the interface of two sonic vacuums (b)–(d) leading to the formation of a train of transmitted solitary waves (e).

Figure 2

(a) Schematic illustration of our model for the formation of a solitary-wave train, side view. (b) Momentum ratios $P_{2,n}/P_{2,1}$ between the $n$th solitary wave and the leading one in the train for $A=m_2/m_1=0.125$. Red circles are the theoretical predictions, while the black squares are the numerical values from the simulations of Fig. 1.

Figure 3

Snapshot of the simulations showing a solitary wave incident upon an interface separating two hexagonal lattices in a sonic vacuum. (a) For $A\equiv m_2/m_1=0.125$, the transmitted disturbance propagates in the form of a nonlinear oscillatory wave analogous to the train of solitary waves shown for granular chains in Fig. 1 for $A<1$. (b) For $A>1$, we find both a reflected and a transmitted solitary wave as shown in Fig. 1 for $A=8$.

Figure 4

(a) Angle of refraction ${\theta }_{\mathrm{refr}}$ vs angle of incidence ${\theta }_i$ for the hexagonal lattice when $A<1$. The square (circle) symbols correspond to the case when $A=0.125$ ($A=0.9$) and compares the numerically obtained ratio of the speed of the leading transmitted solitary wave to the incident solitary wave against the analytical estimates given by solid curves. (b) Comparison of numerical data (symbols) with the analytical estimate (solid curves) for the angle of refraction (black data) and angle of reflection (blue data) vs the angle of incidence for the hexagonal lattice when $A>1$. The top panels of (a) and (b) describe the relevant angles, where the interface is shown as the dashed (red) line and arrows represent the direction of propagation of the solitary-wave front (thick dark region).

Figure 5

Delayed reflection of a solitary wave for the case ${\theta }_i>{\theta }_c$, when $A=0.125$ and ${\theta }_i=1.8{\theta }_c$.