Effective Rankine-Hugoniot conditions on two-dimensional layered media

It is well known that nonlinear waves traveling on homogeneous media may develop shocks even if they start being smooth. If, however, they travel on periodic media the shock formation may be avoided.

In [Santosa&Symes1991] Bloch expansions are used to show that dispersive effects are introduced in periodic medium. Homogenization theory has also been widely applied to wave propagation on one-dimensional periodic medium [Chen&Fish2001, Conca&Vanninathan1997, Fish&Chen2001, Yong&Kevorkian2002]. In one dimension, the dispersion is an effective consequence of reflections at the material interfaces. When the material parameters changes along the different media such that the material impedance remains constant, there are no reflections and, as a consequence, there is no dispersion. Therefore, having a change of impedance between the different materials is crucial to obtain effective dispersion.

In one-dimensional periodic medium with large impedance contrast the effective dispersion introduced may avoid shock formation. In [LeVeque&Yong2003] homogenization and numerical simulations via finite volume are used to demonstrate that this effective dispersion may interact with the nonlinear effects to create solitary waves. If, on the other hand, the dispersion is not large enough shocks may form. In XX, an effective condition is proposed to predict the moment when shocks form; in addition, if shocks form, they propose a speed for the shock propagation.

In [Quezada&Ketcheson2014] finite volume methods are used to obtain cylindrical solitary waves on two dimensional periodic media arranged in a checkerboard pattern. Later in [Quezada&Ketcheson], homogenization and finite volumes are used to demonstrate that effective dispersion is also introduced in two dimensional layered medium even if the impedance is constant. This dispersion is an effective consequence of diffraction created by changes in the sound speed. Just as in the one-dimensional setting, the effective dispersion may avoid shock generation and even create solitary waves [Ketcheson&Quezada]. If, however, the effective dispersion is not large enough, shocks appear.

In two dimensional layered media, we can consider a more general setting to introduce effective dispersion via reflections due to changes in impedance and via diffraction due to changes in sound speed. This depends not just on the changes in impedance and sound speed but also on the direction of propagation. Again, this dispersion may be large to avoid shock formation and even to create solitary waves. Otherwise, shocks form; in this work we consider this situation and study the speed at which shocks propagate. We propose an effective speed of propagation of shocks on two dimensional layered media. This estimation is based obtained by applying the Rankine-Hugoniot conditions to the leading order homogenized equations.