Three-dimensional Water-wave Problem Research Articles

A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $\beta$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<\beta<\frac{1}{3}$. A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey-Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.

Motion trapping structures in the three-dimensional water-wave problem

Trapped modes in the linearized water-wave problem are free oscillations of finite energy in an unbounded fluid with a free surface. It has been known for some time that such modes are supported by certain structures when held fixed, but recently it has been demonstrated that in two dimensions trapped modes are also possible for freely floating structures that are able to respond to the hydrodynamic forces acting upon them. For a freely floating structure such a mode is a coupled oscillation of the fluid and the structure that, in the absence of viscosity, persists for all time. Here previous work on the two-dimensional problem is extended to give motion trapping structures in the three-dimensional water-wave problem that have a vertical axis of symmetry.

Trapping structures in the three-dimensional water-wave problem

Trapped modes in the linearized water-wave problem are free oscillations of the fluid which have finite energy. They are known to exist at isolated frequencies in the presence of certain special structures. The existence of a trapped mode implies the non-uniqueness, or non-existence, of the solution to physically relevant radiation and diffraction problems for such a structure.Previous work on the three-dimensional problem has established the existence of vertically axisymmetric structures that support trapped modes with either a single interior free surface, or two concentric interior free surfaces. In the present work the existence of several new types of trapping structures is established. These include non-axisymmetric structures with a single interior free surface and various structures with multiple interior free surfaces. The method used is an indirect one in which flow fields without wave radiation are specified, and corresponding structures are found by constructing suitable stream surfaces. Computations of the added-mass coefficients for these structures provide independent support for the existence of a trapping mode and illustrate their hydrodynamic characteristics at other wavenumbers.

Nonlinear Free-Surface Effects on a Submerged Sphere

Nonlinear effects on the wave-induced forces acting on a submerged sphere in a steady translation are investigated. A recent endeavor by Tuck indicates that in the case of a submerged cylinder the forces due to the nonlinearity of the free surface are quite discernible in comparison with the second-order forces derived from the linearized homogeneous freesurface condition. The purpose of the present work is to assess the influence of the nonlinearity upon a three-dimensional water-wave problem.