**Preface**

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.