It was pointed out in the Preface that methods of investigation of the uniqueness and solvability for the water-wave problem depend essentially on the type of obstacle in respect to its intersection with the free surface. Among various possibilities, the simplest one is the case in which the free surface coincides with the whole horizontal plane (and so rigid boundaries of the water domain are represented by totally submerged bodies and the bottom of variable topography); we restrict our attention to this case in the present chapter.

We begin with the method of integral equations (Section 2.1), which not only provides information about the unique solvability of the water-wave problem but also serves as one of the most frequently used tools for a numerical solution of the problem. In Section 2.2, various geometric criteria of uniqueness are obtained with the help of auxiliary integral identities. The uniqueness theorem established allows us to prove the solvability of the problem for various geometries of submerged obstacles in Section 2.3. The last section, Section 2.4, contains bibliographical notes.

Method of Integral Equations and Kochin's Theorem

When Green's function is constructed it is natural to solve the water-wave problem by applying integral equation techniques, which is a standard approach to boundary value problems. In doing so, a proof of the solvability theorem for an integral equation is usually based on Fredholm's alternative and the uniqueness of the solution to the boundary value problem.

The most well-known trapped mode in the theory of water waves is the simple exponential solution derived by Stokes [313]. It describes a wave over a uniformly sloping beach that can travel unchanged in the direction of the shoreline, and that decays exponentially to zero in the seaward direction. The Stokes edge wave is characterized by being confined to or trapped by the boundary, in this case the beach, despite the fact that the fluid region is unbounded. Little interest was shown in this solution for over a 100 years. Thus Lamb [179], p. 447, states “it does not appear that the type of motion here referred to is very important.” Only in the last forty years or so have further examples of such trapped modes been discovered, and it was recognized that “there is now considerable evidence which indicates that edge waves are common in occurrence and practical use” (see Le Blond and Mysak's book [183], p. 227). Thus the aim of the present chapter is to review many papers representing a considerable progress in the investigation of edge waves and other types of trapped modes that have occurred during the past twenty-five years. The material here is to a great extent borrowed from a survey paper by Evans and Kuznetsov [75], but we also cover new results published since 1995, when that paper was written.

The aim of the present book is to give a self-contained and up-to-date account of mathematical results in the linear theory of water waves. The study of different kinds of waves is of importance for various applications. For example, it is required for predicting the behavior of floating structures (immersed totally or partially) such as ships, submarines, and tension-leg platforms and for describing flows over bottom topography. Furthermore, the investigation of wave patterns of ships and other vehicles in forward motion is closely related to the calculation of the wave-making resistance and other hydrodynamic characteristics that are used in marine design. Another area of application is the mathematical modeling of unsteady waves resulting from such phenomena as underwater earthquakes, blasts, and the like.

The history of water wave theory is almost as old as that of partial differential equations. Their founding fathers are the same: Euler, Lagrange, Cauchy, Poisson. Further contributions were made by Stokes, Lord Kelvin, Kirchhoff, and Lamb, who constructed a number of explicit solutions.

As for the water-wave problem investigated in Part 1, it is natural to solve the Neumann–Kelvin problem by applying integral equation techniques, since Green's function is constructed. However, in the theory of ship waves this approach is less straightforward than in the theory of time-harmonic waves. First of all, well-posed statements of the two-dimensional Neumann–Kelvin problem are different for totally submerged and surface-piercing bodies because certain supplementary conditions should be imposed in the latter case. Another essential point distinguishes the Neumann–Kelvin problem for a subcritical flow from the water-wave problem. In fact, any solution to the homogeneous water-wave problem has a finite energy, but for solutions of the homogeneous Neumann–Kelvin problem the unconditional validity of this property is still an open question.

So, using integral equations, we have to rely on the method that does not involve an a priori knowledge of uniqueness in the boundary value problem. Such a method was applied to the water-wave problem in Chapters 2 and 3. Its main features are related to the analyticity of integral operators as functions of the parameter ν and to the properties of these operators in limiting cases.

As in Part 1, we treat the simplest problem first, and this is the two-dimensional problem of a body totally submerged in water of infinite depth (see Section 7.1).

This chapter is concerned with various statements of the two-dimensional Neumann-Kelvin problem for a surface-piercing body. We speak about various statements because the Neumann–Kelvin problem as it is formulated in Chapter 7 for totally submerged bodies proves to be under definite when a body is surface piercing. It took several decades to realize that this under-definiteness occurs and to develop several well-posed formulations of the problem (see a brief consideration of the question's history in Section 8.6).

The plan of this chapter is as follows. The problem augmented by general linear supplementary conditions is considered in Section 8.1. The question of total resistance to the forward motion for a surface-piercing cylinder is considered in a short Section 8.2, where we present formulae generalizing those in Section 7.3. A number of other statements of the Neumann–Kelvin problem are reviewed in Section 8.3. Among them, there are statements leading to the so-called least singular and wave-free solutions. Also, a statement of the Neumann–Kelvin problem for a tandem of surface-piercing cylinders is considered. This statement involves a set of four supplementary conditions canceling both the wave resistance and the spray resistance and providing a well-posed statement of the problem. This means that a unique solution exists for all values of the forward speed U except for a sequence tending to zero.

At the same time, for the exceptional values of U, examples of non-uniqueness are constructed in Section 8.4.

Here we give a brief account of physical assumptions (first section) and the mathematical approximation (second section) used for developing a mathematical model of water waves. The resulting linear boundary value problems are formulated in the third and fourth sections for the wave-body interaction and ship waves, respectively.

Mathematical Formulation

Conventions

Water waves (the terms surface waves and gravity waves are also in use) are created normally by a gravitational force in the presence of a free surface along which the pressure is constant. There are two ways to describe these waves mathematically. It is possible to trace the paths of individual particles (a Lagrangian description), but in this book an alternative form of equations (usually referred to as Eulerian) is adopted. The motion is determined by the velocity field in the domain occupied by water at every moment of the time t.

Water is assumed to occupy a certain domain W bounded by one or more moving or fixed surfaces that separate water from some other medium. Actually we consider boundaries of two types: the above-mentioned free surface separating water from the atmosphere, and rigid surfaces including the bottom and surfaces of bodies floating in and/or beneath the free surface.

It is convenient to use rectangular coordinates (x1, x2, y) with origin in the free surface at rest (which usually coincides with the mean free surface), and with the y axis directed opposite to the acceleration caused by gravity.

As in the case of time-harmonic waves (see Chapter 1), we begin with the simplest model, replacing a ship by a point source in the uniform forward motion in calm water. The corresponding velocity potential is sometimes referred to as the Kelvin source, but to keep the terminology unified we call it the Green's function in what follows. Similar to the theory of time-harmonic waves developed in Part 1, the theory of ship waves presented here relies essentially on Green's functions. They are of importance not only for proving solvability theorems (see Chapters 7 and 8) but also for constructing examples of trapped waves (nontrivial solutions to homogeneous boundary value problems) in Section 8.4.

The three-dimensional Green's function of a point source in deep water is considered in detail in Sections 6.1 and 6.2. General facts about the three-dimensional Green's function are considered in Section 6.1 and the far-field expansions for Green's function and the corresponding elevation of the free surface are obtained in Section 6.2. Two-dimensional Green's functions are treated in Section 6.3, which we begin with the simpler case of deep water (Subsection 6.3.1). For water of finite depth, which will be referred to as shallow water, we consider Green's function in Subsection 6.3.2.