标题： 具有恒定旋度的有限深度水面上的稳定周期性水弹性波 显示英文标题

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

摘要： 本研究分析了在有限深度水体表面下非线性弹性膜下方传播的稳定周期性水弹性波。 与之前的工作\cite{BaldiT,BaldiT1,Toland,Toland1}不同，我们的公式考虑了有限深度水中的旋转流。 我们采用共形映射技术将自由边界问题转化为一个拟线性伪微分方程，从而得到一个单变量的周期函数。 这种简化使得在分岔理论框架内解决此类波的存在性问题成为可能。 将波长归一化为 2$\pi$，该问题有两个自由参数：波速和常数涡量。 在假设未变形膜的存储能量局部凸性的前提下，观察到当关于均匀水平流动线性化时，对于任何参数值，该问题最多有两个独立解。 固定涡量并将波速作为分岔参数，线性化问题具有一解。 我们证明了完整的非线性问题表现出一个由简单特征值分岔出的曲线族组成的解面。 将波速和涡量都作为参数，当常数涡量接近临界值时，线性化问题表现出二维核。 在这些临界点附近，从主解分支中出现了一个次级分岔曲线。 该次级分支包含表面上的波纹解。 显示英文摘要

摘要： 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$\pi$, 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.