Travelling waves on the surface of a falling liquid film past a permeable bed

The paper presents the existence of gravity capillary waves travelling down the surface of a falling liquid film past a permeable bed. The bed is characterized by a parameterβ = (α +σ)/ασ 2, where α is a property of the porous material and\(\sigma = h_0 /\sqrt {K,} h_0 \) the depth of the liquid film andK the absolute permeability of the porous medium. It is shown that the range of allowed dimensionless wave celerity widens as β increases. The celerity depends less and less on the Weber numberW and also on the dimensionless wave numberN, as β increases. When β=0 we recover the gravity capillary waves on liquid films predicted byKapitza.