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<f_ss and stripes for f>f_ss; H determines the square/stripe transition frequency f_ss = 0.33√g/H. The dispersion relations for layers with varying H collapse onto the curve λ/H = 1.0+1.1(f√H/g)^-1.32±0.03 when the peak container velocity v = 2πAf exceeds a critical value, v_gm≈3D̄ḡ. Local collision pressure measurements suggest that v_gm is associated with a transition in the horizontal grain mobility: for v>v_gm, there is a hydrodynamic-like horizontal sloshing motion, while for v<v_gm, the grains are essentially immobile and the stripe pattern apparently arises from a bending of the granular layer. For f at v_gm less than f_ss and v<v_gm, patterns are tenuous and disordered.