For the problem describing steady, gravity waves with vorticity on a twodimensional, unidirectional flow of finite depth the following results are obtained. (i) Bounds for the free-surface profile and for Bernoulli's constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli's constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.
We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an incompressible fluid (say, water). The flow is assumed to be unidirectional of finite depth and the water motion is supposed to be rotational. Our aim is to verify the Benjamin-Lighthill conjecture for flows whose total head (Bernoulli's constant) is close to the critical one; the latter is determined by the vorticity distribution so that no horizontal shear flows exist for smaller values of the total head.Originally, the conjecture was made about irrotational wave trains in order to describe them in terms of the parameters Q (rate of flow), R (total head/Bernoulli's constant) and S (flow force). Let r and s be dimensionless versions of R and S, respectively, for fixed Q, and let C be the region in the (r, s)-plane whose cusped boundary ∂C represents all possible uniform streams; moreover, the part of ∂C corresponding to supercritical streams is included into C, whereas the other part not. The Benjamin-Lighthill conjecture says that (a) each wave train is represented by a point of C and (b) every point of C corresponds to some wave train. In 2010-11, this form of the conjecture was proved by Kozlov and Kuznetsov for irrotational waves corresponding to nearcritical values of Bernoulli's constant.Here, we modify the Benjamin-Lighthill conjecture to adapt it for rotational waves on unidirectional flows. Let ω be a vorticity distribution, then the corresponding cusped region Cω (its boundary represents all possible horizontal shear flows) must be truncated by the line r = r0, where the constant r0 defined by ω is finite for some vorticity distributions. Under the assumptions that ω is Lipschitz continuous and the problem's parameter r attains nearcritical values, we prove the following extended version of the conjecture. Namely, along with the assertions (a) and (b) formulated above we show that the correspondence between wave trains
The problem for two-dimensional steady water waves with vorticity is considered. Using methods of spatial dynamics, we reduce the problem to a finite dimensional Hamiltonian system. The reduced system describes all smallamplitude solutions of the problem and, as an application, we give a proof of the existence of non-symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one.
We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to “half-solitary” waves (e.g. bores) which decay in only one direction.
Abstract. Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with an arbitrary number of crests per period. The role of bifurcation parameters is played by the roots of the dispersion equation.
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