On the existence theory for irrotational water waves

Abstract

This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and ⅙π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, andwhere g is the acceleration due to gravity. Krasovskii's set of solutions is included in the set that we obtain.

On periodic water-waves and their convergence to solitary waves in the long-wave limit

“…It is well-known that both periodic waves [9], [12] and solitary waves [1] of large amplitude may occur in these circumstances.…”

“…water-waves (though the physical interpretation of its solutions there is different from ours). Equation (1.31), which is equivalent to the usual integral equations for periodic waves ( [9], [12], (20], (281), is introduced because it has distinct advantages for our purposes in section 3. The most important of these is its striking resemblance to the approximation used in [; section 3.2]…”

Steady Water Waves

“…In the course of his work Krasovskii made two further conjectures about Stokes waves, namely that θ ν β tends to an extreme wave as β → π/6 and that there is no solution (ν, θ ν ) with sup s∈[0,π] θ ν > π/6. This result was later improved by Keady & Norbury [39] using the modern global bifurcation theory becoming available at the time. Keady & Norbury showed that the local branch of solutions bifurcating from the trivial solution at ν = 3 extends to a global branch parameterised by ν ∈ (3, ∞).…”

On the recovery of traveling water waves with vorticity from the pressure at the bed

“…In the case of zero vorticity, the rigorous existence theory of Stokes waves goes back to constructions in [Nek51,LC25] and [Str26] of small amplitude waves, and it includes global bifurcation results in [Kra61,KN78], for instance, and the resolution in [AFT82] of Stokes' conjecture about the wave of greatest height. All these works strongly use the assumption that the flow in the bulk is irrotational, whereby one may reformulate the problem in terms of quantities defined at the fluid surface.…”