Interface waves in almost incompressible elastic materials

We study the problem of two elastic half-planes in contact and the Stoneley interface wave that may exist at the interface between two different elastic materials, emphasis being put on the case when the half-planes are almost incompressible. We show that numerical simulations involving interface waves require an unexpectedly high number of grid points per wavelength as the materials become more incompressible. Let λ, μ, ρ and λ′, μ′, ρ′ be the Lamé parameters and densities of the first and second half-plane, respectively. A theoretical study shows that if K is a real constant, λ′=Kλ, μ′=Kμ, ρ′=Kρ and μ→0, then for an accurate solution the required number of grid points per wavelength scales as (μ/λ)−1/p, where p is the order of accuracy of the numerical method. This requirement becomes very restrictive close to the incompressible limit μ≪λ, especially for lower order methods i.e., a small p. The theoretical findings are supported by numerical experiments that illustrate the demanding resolution requirement as well as the superiority of higher order methods. The scaling is also seen to hold for a more general choice of Lamé parameters. Numerical experiments when one of the half-planes is a vacuum are also presented, where the higher resolution requirement is illustrated in a numerical solution of Lamb's problem.