Aspect ratio of undulation in a vertically vibrated granular layer

The aspect ratio of the height δ to the wavelength λ of the undulation generated by a vertical vibration of the granular layer was investigated experimentally, and its dependence on the frequency f and amplitude a   is disclosed. We found that δ/λδ/λ is well described by an almost linear function of fa   rather than by that of Γ≡(2πf)2a/gΓ≡(2πf)2a/g, irrespective of the horizontal size of the container, where g is the acceleration of gravity. Appearance of sub-arches to maintain the main eigenmode and the transitions between eigenmodes of undulation are also elucidated.