Heavy hydroelastic travelling waves

This is a mathematical study of steady two-dimensional waves of prescribed period which propagate without change of form on the surface of an infinitely deep expanse of fluid that is moving under gravity and bounded above by heavy, frictionless, thin elastic sheet. The flow, which is supposed irrotational, is at rest at infinite depth and its velocity is stationary relative to a frame moving with the wave. In that frame, the elastic sheet coincides with the zero streamline and its material points move according to the equations of hyperelasticity.From the mechanics of the surface material, the pressure on a point of the sheet in the moving frame is shown to be a function of its height, its slope and derivatives with respect to arc length of these quantities. Independently, according to classical fluid mechanics, the pressure in the fluid at the same point is determined by its height and the fluid velocity tangent to the zero streamline.Therefore, this is a free-boundary problem: to find a non-self-intersecting curve in the plane which is the zero contour of a harmonic function (the stream function) and at which the normal derivative of the same harmonic function is a prescribed function of the shape of the curve.With the wavelength fixed, the parameters are the density ρ of the undeformed sheet, the wave velocity c₀, the mean velocity c of the surface sheet relative to the wave and the gravitational acceleration g. The case of a weightless sheet, in which ρ=0 and c does not appear, is a special case. Under quite general hypotheses on the elastic response of the sheet, the existence of rapidly propagating or very slow (depending on parameter values) steady waves is established by finding a critical point of the Lagrangian. This is a saddle-point problem in the calculus of variations.