Dispersion Relation of Standing Waves on a Vertically Oscillated Thin Granular Layer

摘要

An experimental study of standing waves on a thin granular layer, which oscillated vertically under atmospheric pressure, was made. We controlled frequencies f and amplitudes of oscillations as well as the layer heights h, and wave lengths λ were observed by means of a high-speed video camera. At first and second critical acceleration, the granular layer forms standing wave patterns oscillating at half and quarter, respectively, of the excitation frequency. Glass beads of a different diameter were tested, and the data normalized by means of three different scales are compared. Scaling by particle diameter d is relevant near the onset of waves, whereas scaling of either by h or by g/f~2 (g: acceleration of gravity) is appropriate for fully developed waves. At higher frequencies, wave length has a tendency to saturate, reflecting the fact that the effective depth pertinent to the wave formation is confined to a region of a few layers near the upper surface. Dispersion relation is characterized by as power-law whose exponent depends on layer height and acceleration amplitude, which seems to be explained by a shallow water gravity wave of viscous fluid.