Wavelength scaling and square/stripe and grain mobility transitions in vertically oscillated granular layers

Laboratory experiments are conducted to examine granular wave patterns near onset as a function of the container oscillation frequency f and amplitude A, layer depth H, and grain diameter D. The primary transition from a flat grain layer to standing waves occurs when the layer remains dilated after making contact with the container. With a flat layer and increasing dimensionless peak container acceleration Γ=4π2f2A/g(g is the acceleration due to gravity), the wave transition occurs for Γ≈2.6, but with decreasing Γ the waves persist to Γ=2.2. For 2.2<Γ<3.8, patterns are squares for f<fss and stripes for f>fss; H determines the square/stripe transition frequency fss=0.33g/H. The dispersion relations for layers with varying H collapse onto the curve λ/H=1.0+1.1(fH/g)−1.32±0.03 when the peak container velocity v=2πAf exceeds a critical value, vgm≈3Dg. Local collision pressure measurements suggest that vgm is associated with a transition in the horizontal grain mobility: for v>vgm, there is a hydrodynamic-like horizontal sloshing motion, while for v<vgm, the grains are essentially immobile and the stripe pattern apparently arises from a bending of the granular layer. For f at vgm less than fss and v<vgm, patterns are tenuous and disordered.