Wavelength selection of ripples in a vertically vibrating dynamically thick granular layer due to density-wave refraction

A numerical study was made on the wavelength selection mechanism of the ripples observed on the surface of a granular layer that is oscillated vertically. Multiple collisions of the one-dimensional array of beads show a time-dependent particle distribution, which induces a density wave. The magnitude of the wave velocity is estimated by the theory of elasticity, which reveals the refraction of the density wave in a quasi-two-dimensional granular layer. Our theory explains how the vertical excitation of particles determines the horizontal characteristic scale on the surface, or the wavelength of the ripples, which is applicable even to the saturation regime of a dynamically thick granular layer, e.g., the one that is thick enough to allow an immobilized region in the lower part.