Gravity perturbed Crapper waves

Abstract

Crapper waves are a family of exact periodic travelling wave solutions of the free-surface irrotational incompressible Euler equations; these are pure capillary waves, meaning that surface tension is accounted for, but gravity is neglected. For certain parameter values, Crapper waves are known to have multi-valued height. Using the implicit function theorem, we prove that any of the Crapper waves can be perturbed by the effect of gravity, yielding the existence of gravity-capillary waves nearby to the Crapper waves. This result implies the existence of travelling gravity-capillary waves with multi-valued height. The solutions we prove to exist include waves with both positive and negative values of the gravity coefficient. We also compute these gravity perturbed Crapper waves by means of a quasi-Newton iterative scheme (again, using both positive and negative values of the gravity coefficient). A phase diagram is generated, which depicts the existence of singlevalued and multi-valued travelling waves in the gravity-amplitude plane. A new largest water wave is computed, which is composed of a string of bubbles at the interface.

On the existence of stationary splash singularities for the Euler equations

“…The structure of Γ is already known as well. In fact, we have the following result, which we borrow from [1] and whose proof we sketch here for completeness: Lemma 4. The operator Γ : H 1 odd → L 2 even is injective, but not surjective.…”

“…Recall that q represents the surface tension, g is the gravity and ǫ indicates the presence of the upper fluid. Setting ǫ to zero, the equations decouple and we recover the capillary-gravity wave problem as studied in [1]. Once a solution θ of the Bernoulli equation is known, the vorticity can be recovered from the second equation as long as the underlying interface is non self-intersecting (cf.…”

“…Since D θ,ω G is not surjective, we cannot use the implicit function theorem directly to prove Theorem 1 (the remaining hypothesis of the implicit function theorem are easy to check). Therefore, following [1], we use an adaptation of the Lyapunov-Schmidt reduction argument. So let Π denote the L 2 projector onto the linear span of cos θ A , i.e., Πu := cos θ A , u cos θ A cos θ A 2 2 .…”

“…Let us fix the density of the second fluid ρ 2 > 0 and consider any η > 0. For any small enough ρ 1 ≥ 0 and g there is some positive surface tension coefficient σ such that there is a periodic solution to the two-fluid problem (1) for which the interface S is an η-splash curve.…”

Periodic traveling interfacial hydroelastic waves with or without mass

“…The particular results in [2] are that the formulation for a traveling parameterized curve was introduced and was used to prove a local bifurcation theorem, and families of waves were computed, showing that curves of traveling waves typically ended when a self-intersecting wave was reached. Subsequently, Akers, Ambrose, and Wright showed that Crapper waves, a family of exact pure capillary traveling water waves, could be perturbed by including the effect of gravity, and the formulation was again used to compute these waves [5]. Further numerical results were demonstrated in [4], where the non-density-matched vortex sheet was considered.…”