Diffraction of Water Waves : Integral Transforms and Special Functions

This monograph is devoted to subject of diffraction of water waves. Methods of integral transforms and special functions are taken as principal tool to describe diffraction of water waves. The classical interpretation of these methods is used and some generalisations are proposed. The objects of this book are either to construct an exact solutions of diffraction problems in integral form or to reduce the problems into integral equations and to obtain their asymptotic solutions. Diffraction problems are reduced into Fredholm integral equations of the second kind. Asymptotic solutions of these equations are obtained.Two groups of bodies are considered: submerged and floating bodies with non-zero displacement (vertical cylindrical bodies, horizontal cylindrical bodies, vertical axisymmetrical bodies) and thin-walled structures with sharp edge (horizontal plates on free surface, submerged horizontal plates, vertical ducts and barriers). Adequate mathematical technique is applied and developed to investigate corresponding diffraction problems: sequential integral transforms, double integral transforms, operator triplets (combinations of two transforms with regular kernels and fractional integrals), generalized integral transforms.The work is constructed as follows. Part 1 is concerned with foundations of diffraction theory, it consists of four chapters: mathematical models of hydrodynamic processes, integral transforms and fractional integrals as technique for investigation of these processes, examples of diffraction problems. The second part is devoted to problems, which can be analysed by means of sequential application of integral transforms: radiation problems for vertical cylindrical bodies and waves on a sloping beach. In the third part method of double integral transforms is given, which intends for investigation of boundary value problems in two-connected domains. Based on this method the diffraction problems for submerged bodies with non-zero displacement are investigated. These are bodies of revolution with vertical axis and cylindrical bodies with horizontal axis. In the fourth part of the work the diffraction problems for thin-walled structures with free edge (plates on free surface, immersed plates, vertical ducts and barriers) are discussed. Combinations of integral transforms and fractional integrals (triplets of transforms) are used to obtain solutions of these diffraction problems. Interaction of wave motion and current are investigated in the fifth part. Mathematical technique of this part is based on generalized Havelock transforms. Expressions for corresponding Green Functions are constructed. The Appendix contains some mathematical relations, appearing in context of the monograph. The Appendix consists of the following components: description of curvilinear coordinates, definitions and properties of special functions and separation of variables in partial differential equations.