Surface elastic waves in granular media under gravity and their relation to booming avalanches

Due to the nonlinearity of Hertzian contacts, the speed of sound c in granular matter is expected to increase with pressure as P-1/6. A static layer of grains under gravity is thus stratified so that the bulk waves are refracted toward the surface. The reflection at the surface being total, there is a discrete number of modes (both in the sagittal plane and transverse to it) localized close to the free surface. The shape of these modes and the corresponding dispersion relation are investigated in the framework of an elastic description taking into account the main features of granular matter: Nonlinearity between stress and strain and the existence of a yield transition. We show in this context that the surface modes localized at the free surface exhibit a waveguide effect related to the nonlinear Hertz contact. Recent results about the song of dunes are reinterpreted in light of the theoretical results. The predicted propagation speed is compared with measurements performed in the field. Taking into account the finite depth effects, we show that the booming instability threshold can be explained quantitatively by a waveguide cutoff frequency below which no sound can propagate. Therefore, we propose another look at a recent controversy, confirming that the song of dunes can well originate from a coupling between avalanching grains and surface elastic waves once the specificity of surface waves (we baptized Rayleigh-Hertz) is correctly taken into account.