Abstract
We prove that no two-dimensional Stokes and solitary waves exist when the vorticity function is negative and the Bernoulli constant is greater than a certain critical value given explicitly. In particular, we obtain an upper bound F le sqrt{2} + epsilon for the Froude number of solitary waves with a negative constant vorticity, sufficiently large in absolute value.
Highlights
We consider the classical water wave problem for two-dimensional steady waves with vorticity on water of finite depth
The flow force constants corresponding to flows with d = d± are denoted by S±(r). It was recently proved in [13] that all solitary waves are supported by supercritical depths d−(r) and the corresponding flow force constant equals to S−(r); here r is the Bernoulli constant of a solitary wave
First we prove the claim about solitary waves
Summary
We consider the classical water wave problem for two-dimensional steady waves with vorticity on water of finite depth. (u, v) are components of the velocity field, y = η(x) is the surface profile, c is the wave speed, P is the pressure and g is the gravitational constant. It is often assumed in the literature that the flow is irrotational, that is vx − uy is zero everywhere in the fluid domain. Under this assumption the components of the velocity field are harmonic functions, which allows to apply methods of complex analysis. In the present paper we will consider rotational flows, where the vorticity function is defined by ω = vx − uy.
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