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 G = 4 pi^2 f^2 A/g (g is the acceleration due to gravity), the wave transition occurs for G=2.6, but with decreasing G the waves persist to G=2.2. For 2.2<G<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)^0.5. The dispersion relations for layers with varying H collapse onto the curve L/H=1.0+1.1[f(H/g)^0.5]^(-1.32 +/- 0.03) (L is the wavelength) when the peak container velocity v exceeds a critical value v_gm of approximately 3 (Dg)^0.5. 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. 
Paul B. Umbanhowar and Harry L. Swinney