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.

In the present chapter, the first of two chapters dealing with surface-piercing bodies, we impose an essential restriction that no bounded part of the free surface is separated from infinity. For the three-dimentional problem, this means that the free surface is a connected two-dimensional region (possibly multiply connected). In two dimensions, the assumption requires that there is only one surface-piercing body. However, a finite number of totally submerged bodies might be present in both cases. Supplementing this general restriction by one condition of technical nature or another, a method was developed (essentially by John) for proving the uniqueness theorem for various geometries and all values of ν > 0 (see Section 3.2). Provided the uniqueness is established, the machinery of integral equations developed in Section 3.1 leads to the unique solvability of the water-wave problem. Without the assumption about uniqueness, the integral equations method possibly does not guaranee the solvability for a certain sequence of values tending to infinity. Moreover, application of integral equations is rather tricky for semisubmerged bodies even when the uniqueness holds because of so-called irregular frequencies, which are also investigated in Section 3.1.

Integral Equations for Surface-Piercing Bodies

The essential point in application of the integral equation techniques to the case of a surface-piercing body is that the wetted boundary S is not a closed surface (contour) in three (two) dimensions, and it is bounded by a curve (a finite set of points) along the body's intersection with the free surface.

Results presented in Chapter 9 provide no details of the transient behavior of flows and do not yield direct hydrodynamic corollaries. However, there are situations in which information about developing waves in time can be extracted so that it leads to specific properties of hydrodynamic characteristics. In particular, an asymptotic analysis allows us to do this at least for two classes of disturbances. One of these classes constitutes rapidly stabilizing disturbances (this class includes brief disturbances as an important subclass), and the second class is formed by high-frequency disturbances. Both of these classes can be treated by using the same technique of two-scale asymptotic expansions for velocity potentials. The latter allows us to derive principal terms in asymptotics of some hydrodynamic characteristics.

Rapidly Stabilizing Surface Disturbances

In this section we are concerned with the effect of rapidly stabilizing disturbances on magnitudes characterizing unsteady water waves. For this purpose we consider several initial-boundary value problems describing waves caused by surface and underwater disturbances. The main example of the first kind is given by a pressure system applied to the free surface at the initial moment and rapidly stabilizing to a given distribution (a particular case is an impulsive pressure system). Underwater disturbances are presented by a source having a strength rapidly stabilizing in time to a constant value, and a rapidly stabilizing bottom movement. Complete asymptotic expansions in powers of a nondimensional small duration of disturbance are constructed for velocity potentials.

It was demonstrated in Section 3.1 that in the presence of a surface-piercing obstacle the water-wave problem is solvable for an arbitrary right-hand-side term in the Neumann condition on the obstacle's surface. However, there is an uncertainty about the set of frequencies providing the solvability. According to the proof given in Subsection 3.1.1, a sequence νn → ∞ (n = 1, 2, …) possibly exists such that for these exceptional values the solvability could be violated for some data given on the obstacle's surface. In particular, this must occur for values νn that are point eigenvalues of the water-wave problem embedded in the continuous spectrum (the latter is known to be the whole positive half-axis as is shown in the Examples section of the Introduction). If a value of the spectral parameter ν belongs to the point spectrum, then the homogeneous problem possesses a nontrivial solution with finite energy, or in other words, there is no uniqueness of solution for the nonhomogeneous problem.

In this chapter (see Section 4.1), we give examples of such non-uniqueness for the two-dimensional and axisymmetric problems, and so the exceptional values of ν do exist at least for some obstacle geometries. Moreover, for every ν > 0 a certain family of obstacles exhibiting the non-uniqueness property can be obtained. An essential point in all these examples is the presence of an isolated portion of the free surface inside the obstacle where the eigenmode waves are trapped.

The simplest “obstacle” to be placed into water is a point source. The corresponding velocity potential (up to a time-periodic factor) is usually referred to as the Green's function. This notion is crucial for the theory we are going to present in this book, since a wide class of time-harmonic velocity potentials (in particular, solutions to the water-wave problem) admit representations based on Green's function (see Section 1.3).

Potentials constructed by using Green's functions form the basis for such different topics as proving solvability theorems (see Chapters 2 and 3) and constructing examples of trapped waves (nontrivial solutions to homogeneous boundary value problems given in Chapter 4).

The plan of this chapter is as follows. Beginning with Green's functions of point sources in water of infinite (Subsection 1.1.1) and finite (Subsection 1.1.2) depths, we proceed with straight line sources and ring sources (Section 1.2) arising in two-dimensional problems and problems with axial symmetry, respectively. Green's representation of velocity potentials and related questions are given in Section 1.3. Bibliographical notes (Section 1.4) contain references to original papers treating the material of this chapter as well as other related works.

Three-Dimensional Problems of Point Sources

Point Source in Deep Water

In the present subsection, we consider in detail Green's function describing the point source in deep water. In Subsection 1.1.1.1, we define it as a solution to the water-wave problem having Dirac's measure as the right-hand-side term in the equation.

In the first two chapters of this book we described many properties of the Euler and the Navier–Stokes equations, including some exact solutions. A natural question to ask is the following: Given a general smooth initial velocity field v(x, 0), does there exist a solution to either the Euler or the Navier–Stokes equation on some time interval [0, T)? Can the solution be continued for all time? Is it unique? If the solution has a finite-time singularity, so that it cannot be continued smoothly past some critical time, in what way does the solution becomes singular? This chapter and Chap. 4 introduce two different methods for proving existence and uniqueness theory for smooth solutions to the Euler and the Navier–Stokes equations. In this chapter we introduce classical energy methods to study both the Euler and the Navier–Stokes equations. The starting point for these methods is the physical fact that the kinetic energy of a solution of the homogeneous Navier–Stokes equations decreases in time in the absence of external forcing. The next chapter introduces a particle method for proving existence and uniqueness of solutions to the inviscid Euler equation. As is true for all partial differential evolution equations, the challenge in proving that the evolution is well posed lies in understanding the effect of the unbounded spatial differential operators. The particle method exploits the fact that, without viscosity, the vorticity is transported (and stretched in three dimensions) along particle paths.