Steady periodic hydroelastic waves on the water surface of finite depth with constant vorticity

This study analyzes steady periodic hydroelastic waves propagating on the water surface of finite depth beneath nonlinear elastic membranes. Unlike previous work \cite{BaldiT,BaldiT1,Toland,Toland1}, our formulation accommodates rotational flows in finite-depth water. We employ a conformal mapping technique to transform the free-boundary problem into a quasilinear pseudodifferential equation, resulting in a periodic function of a single variable. This reduction allows the existence question for such waves to be addressed within the framework of bifurcation theory. With the wavelength normalized to 2$π$, the problem features two free parameters: the wave speed and the constant vorticity. Under the assumption of the local convexity of undeformed membrane's stored energy, it is observed that the problem, when linearized about uniform horizontal flow, has at most two independent solutions for any values of the parameters. Fixing the vorticity and treating the wave speed as the bifurcation parameter, the linearized problem possesses a single solution. We demonstrate that the full nonlinear problem exhibits a sheet of solutions comprising a family of curves bifurcating from simple eigenvalues. Taking both the wave speed and vorticity as parameters, when the constant vorticity approaches critical values, the linearized problem exhibits a two-dimensional kernel. Near these critical points, a secondary bifurcation curve emerges from the primary solution branch. This secondary branch consists of ripple solutions on the surface.