Density wave propagation of momentum and the formation of arches in the undulation of a vertically vibrated thin granular layer

Experimental and theoretical investigations are made on the aspect ratio of the height <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> to wavelength <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> of the undulations generated in a dry granular layer confined between two parallel vertical planes under vertical vibration of frequency <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> and amplitude <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> </mml:math> . We elucidated the propagation of the density wave along the layer, which suggests one or two order smaller macroscopic elastic constant ascribed to a configuration change of the constituent particles. Our experiment also suggests the relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∝</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> irrespective of the size of the container, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∝</mml:mo> <mml:mi>f</mml:mi> <mml:mi>a</mml:mi> </mml:mrow> </mml:math> ) is the momentum given by the external forcing, which is reminiscent of the de Broglie wave known in quantum mechanics. We further performed a numerical simulation on the onset and development of undulations using our simplified model, in which horizontal dilatation generated by mutual intrusion of particles near the bottom of the container develops to an arch formation of the granular layer. Influence of the friction of the granular material is examined.