Steady periodic hydroelastic waves and Wilton ripples with constant vorticity

Abstract

We study two-dimensional steady periodic hydroelastic waves in a finite-depth fluid with constant vorticity, with the free surface modeled as an extensible nonlinear hyperelastic membrane whose stored energy is $E(\nu ,\mu )$. A key point is that the material coordinate of the membrane and the conformal boundary coordinate are not identified. We introduce the material reparametrization explicitly, use the tangential membrane equilibrium together with the period constraint to eliminate it locally by the implicit function theorem, and thereby obtain a fourth-order quasilinear pseudodifferential equation for the free-surface profile alone. We prove a converse reconstruction, including the material parametrization, so that the reduced fixed-domain problem is locally equivalent to the full hydroelastic free-boundary system. For fixed vorticity, isolated simple roots of the resulting dispersion relation generate local Crandall--Rabinowitz branches. Our main result concerns isolated $1:2$ resonances. We identify the appropriate nonlinear invariant subspace, derive the exact double-kernel conditions, and compute the quadratic resonant coefficient for the material-relaxed membrane. Under the nonvanishing of this coefficient, a Lyapunov--Schmidt secondary-bifurcation lemma adapted from Shearer yields a period-doubling secondary branch from the $2n$-mode primary family; the secondary profiles contain both the $n$- and $2n$-harmonics and have minimal period $2\pi /n$. For the quadratic constitutive law $E(\nu ,\mu )=\frac{A}{2}(\nu -1{)}^2+\frac{B}{2}{\mu }^2$, $A,B>0$, the resonant coefficient is given in closed form and is strictly positive under an explicit depth condition. We finally record the exact stagnation-line criterion for the underlying laminar flows, without imposing unsupported conclusions on the topology of nonlinear critical layers.