A uniqueness result for periodic traveling waves in water of finite depth (Q1883096)

The author shows that in a perfect fluid of finite depth, the surface profile \(\eta\) of a periodic traveling wave uniquely determines the corresponding flow in the body of the fluid. This holds also for rotational flows if the vorticity \(\gamma\) is given by a function \(\gamma(\psi)\), where \(\psi\) is the stream function, and satisfies the condition \(\gamma'(s)\cdot\max_{x\in\mathbb R}\eta^2(x)< \pi^2\), where \(0< s< m\), and \(m\) is the stream function value on the bottom. This condition is also shown to be sharp. The proof is based on a reformulation of the original problem in terms of the stream function, and on an application of the strong maximum principle for second-order elliptic equations.