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 time-harmonic water waves. The study of these waves is important for prediction of behaviour of floating structures (immersed totally or partially), such as ships, submarines, tension-leg platforms etc., and in describing flows over bottom topography.
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 had been made by Stokes, Lord Kelvin, Kirchhoff, Lamb, who had found a number of explicit solutions. In the 20th century Havelock, Kochin, Sretenskiy, Stoker, John et al. had applied the Fredholm theory of boundary integral equations to the field of water waves.
Although the last decades have brought a renewed interest in both mathematical and applied aspects of the theory, some fundamental questions still remain open. A list of (at the time) unsolved problems is given by Ursell in 1992, where the problem of uniqueness of a time-harmonic velocity potential is placed first. The reason for this is that there exists a number of conditions providing uniqueness for some geometries and frequencies, and, at the same time, some examples of non-uniqueness (so-called trapped modes) have been constructed recently.
This topic forms the core of our book which distinguishes it from other monographs on water waves. Among them there are general expositions of the theory by Lamb, Lighthill, Stoker, Wehausen & Laitone ; works focussed on the applied aspect of the wave-body interaction such as Haskind, Mei, Newman, Wehausen ; surveys on ship waves and wave resistance by Kostyukov, Timman et al., Wehausen, and recent books concerned with non-linear waves by Debnath and Ovsyannikov et al. . The reader interested in the theory of transient waves might consult the following works: Beale, Euvrard , Finkelstein, Friedman & Shinbrot, Garipov, Maskell & Ursell, Ursell , Vullierme-Ledard . However, there is no monograph concerning the mathematical aspect of time-harmonic water waves, and the present work is aimed to fill in this gap, at least partially. The book is arranged in five parts. Part I contains an auxiliary material, and consists of two chapters. The derivation of the linearized water-wave problem, that is, the boundary value problem describing the time-harmonic water waves in the open sea, is outlined in ch. 1 on the basis of general dynamics of an inviscid incompressible fluid (water is the standard example of such a fluid). In ch. 2 we give an account of Green's functions in three and two dimensions. This chapter is crucial for understanding of several following chapters. First, Green's function gives a key to proving solvability theorem by reducing the water-wave problem to an integral equation on the wetted surface (contour) of an immersed body, or of a bottom obstruction (see chs. 3, 5 and 6). Secondly, Green's function is a tool used in ch. 8 for construction of examples of trapped waves, that is, examples of non-uniqueness in the water-wave problem. Part II is concerned with the cases when the free surface coincides with the whole horizontal plane. The application of the integral equation technique to the problem of submerged body is developed in ch. 3. It provides the solvability of the water-wave problem for all frequencies except possibly for a finite number of values. In the next ch. 4 sufficient conditions on the body shape are established which guarantee the unique solvability for all frequencies. Moreover, the so-called auxiliary integral identity is derived for proving the uniqueness theorem. This identity finds further applications in chs. 5, 7 and 9. In ch. 5 we pass to the case of bottom obstructions in three and two dimensions. The obtained results are similar to those in ch. 4, but the derivation of the existence theorem requires a more advanced operator theory technique. In Part III semisubmerged bodies are allowed in the way leaving no bounded components of the free surface. As in Part II we apply first the method of integral equations (ch. 6). However, the integral equation based on the source distribution over the wetted rigid surface gives rise to the so-called irregular frequencies, that is, the frequencies at which the integral equation is not solvable for an arbitrary right-hand side term. These values are not related to the water-wave problem, and arise from the fact that a certain boundary value problem in the domain between the body surface and the free surface plane has these values as eigenvalues. There are several ways leading to other integral equations without irregular frequencies. We consider one of them in detail, and give a survey of the others in ch. 6. In ch. 7 we present uniqueness theorems related to geometries under consideration. We begin with John's theorem in sect. 7.1, consider extensions of John's method in sect. 7.2, and give further uniqueness results in sect. 7.3. Besides, it should be mentioned that the question whether there are examples of non-uniqueness is still open for geometries without bounded components of the free surface. Part IV deals with the case when isolated portions of the free surface are present. It distinguishes from the situations described in Parts II and III, since examples of trapped waves (that is, of non-uniqueness in the water-wave problem) involving such geometries have been constructed. In ch. 8 we give two-dimensional (sect. 8.1) and axisymmetric (sect. 8.2) examples. They show that the exceptional values of frequency, when the water-wave problem is not uniquely solvable, do exist, at least for special geometries obtained by means of the so-called inverse procedure. In ch. 9 we give a geometrical condition providing uniqueness theorems in the two-dimensional problem with two bodies symmetric about a vertical axis (sect. 9.1), and in the axisymmetric problem for a toroidal body (sect. 9.2). According to these theorems it occurs that intervals of uniqueness alternate with intervals of possible non-uniqueness on the frequency half-axis. There are cases when the latter does take place. However, if more restrictions are imposed on the geometry, then it is possible to prove that some intervals of possible non-uniqueness are free of it. These results are given in sect. 9.3 for the two-dimensional problem and in sect. 9.4 in three dimensions. A survey of results obtained in the extensive field of trapped waves periodic in a horizontal direction is given in Part V. A classification of such trapped waves is given in short ch. 10. Edge waves are treated in ch. 11. We present results on trapped modes above submerged cylinders and bottom protrusions in ch. 12. Modes trapped by surface-piercing structures are considered in ch. 13. The last ch. 14 is concerned with trapped modes near vertical cylinders in channels. To complete the description of the book structure we mention that parts are divided into chapters consisting of sections, which are mostly divided into subsections. The sections and subsections are numbered by two and three numbers respectively (for example, 4.2 is sect. 2 in ch. 4, and 2.4.2 is subsect. 2 in sect. 2.4). Every chapter has independent through numbering of formulae (for example, (2.36) denotes the 36th formula in ch. 2). Most of the references are collected in the last section of each chapter entitled Bibliographical Notes. This does not apply to chapters or sections presenting surveys. It should be mentioned that the book has a flavour of authors' preferences in the chosen field, since its substantial part is based on their own contributions. We add that the presentation of the material is mathematically rigorous, despite we avoid the lemma-theorem style. Instead, we adopt more or less informal style, formulating, nevertheless, all proved assertions italicized. The prerequisite for reading the book is a knowledge of Mathematical Analysis including a familiarity with Bessel functions and a little with the Fourier transform. We assume that the reader is aware of the elements of Functional Analysis (for example, the Fredholm alternative is widely used in the book). The book is supposed to be a research monograph in Applied Mathematics. Some its aspects might be of interest to mathematicians specialized in Partial Differential Equations and in Spectral Operator Theory. Nevertheless, the authors hope it could be used as an advanced text for Applied and Engineering Mathematics students.