Green function technique for water waves

The monograph is devoted to the subject of dynamics of water waves. The technique of Green Function (GF) is taken as the principal tool to describe the wave processes. The classical interpretation of this method is used and some generalisations are proposed. The basic purposes of the book are the following: construction of representations for the GF, reduction the problems into integral equations and derivation of asymptotic representations. The Green Function can be treated as a key function of the wave theory. Physically it can be interpreted as a source in a fluid domain with free surface; mathematically the GF is solution of a special boundary value problem, which contains delta-function. The boundary value problem for Green Function involves the delta-function either in equation or in boundary conditions. In this accordance we introduce two types of the GF: the space GF and the boundary GF. This separation allows better perform a systematisation of different cases of the wave problems and reduce these problems into integral equations. There are at least two steps in the Green Function method: construction of GF and application of this function to water waves problem by means of integral relations either in terms of Green identity or in terms of potential layers. In order to construct representations for the GF we use integral transform method. Generally there are two possibilities to apply integral transforms by construction of the GF: the transform in horizontal direction (Fourier, Hankel, Fourier-Bessel) and the transform with respect to vertical variable (Havelock transforms). The first method gives one-term representation for GF in the form of contour integral with singular integrand; the second method gives at least two-component representation for GF by means of regular integrals. Further the hydrodynamic problems can be reduced into integral equations of the first or the second kind by means of potential layers. Asymptotic representations for wave fields can be constructed by means of Green identity and expressed in terms of Kochin functions. The following hydrodynamic problems are investigated in this monograph: radiation-diffraction, ship waves, non-stationary problems, containers, wave-current systems, two-layer fluid systems and gravity-capillary wave systems. Basic results of this work are consolidated in the form of a Table of Green Functions, which contains greater than hundred different representations for various geometrical and hydrodynamic situations. Expressions for Green Functions are represented in different integral and series forms: Fourier integrals and series, Hilbert integrals, Hankel integrals, Fourier-Bessel series, Havelock integrals and series, fractional integrals, convolution integrals, principal value integrals, contour integral in complex plane. The Appendix contains some mathematical relations, appearing in the context of the book.