This is a book on the theoretical studies of nonlinear water waves as they appear in the literature over the last three decades. The book is aimed at the reader interested in the wider theory of nonlinear wave phenomena, their applications, and other nonlinear aspects of fluid flows.
The first chapter deals with the fundamental equations governing the motion of inviscid and viscous fluids with boundary conditions. The energy equation and the energy flux in an inviscid incompressible fluid are discussed. This is followed by a variational principle for water waves that enable us to derive the basic equations for water waves. The final section is devoted to equations for energy dissipation in an incompressible viscous fluid and their physical significance.
In Chapter 2, we discuss the propagation of surface gravity waves and surface capillary-gravity waves on a flat water-air interface. The steady-state wave problem and its solution using a radiation condition or an equivalent device are discussed in some detail. This is followed by an introduction to group velocity and its physical interpretation.
Chapter 3 is concerned with initial value problems for water waves that are generated by an oscillatory disturbance acting at the free surface of still water and of a running stream. The classical two-dimensional and the axisymmetric Cauchy-Poisson problems are studied in detail by the methods of integral transforms and of stationary phase approximation.
In Chapter 4 we study the theory of nonlinear shallow water waves and solitons, with emphasis on methods and solutions of several evolution equations that are originated in the theory of water waves. Special attention is given to the Boussinesq equation, the Korteweg-de Vries equation and the inverse scattering transform, conservation laws and nonlinear transformations, Bäcklund transformations and nonlinear superposition principle.
Chapter 5 brings the classical theory of ship waves and wave resistance. Several formulae for the wave resistance are derived explicitly in terms of wave amplitude or pressure distribution. The theory of linear and nonlinear diffraction of water waves is the main focus of Chapter 6. Special attention is paid to nonlinear wave forces on offshore solid structures and to nonlinear modification of the linear Morison equation for the hydrodynamic drag force on the structure.
Chapter 7 reviews the developments of the theory of nonlinear dispersive waves in both nondissipative and dissipative media, with major emphasis on those that contribute to the unification and allow extension to nonlinear dispersive problems. The final Chapter 8 gives a review of nonlinear instability of dispersive waves with applications to water waves.
We have not considered numerical and experimental aspects of the subject because of the limited size of the book and the lack of experience of the author. For readers wishing to learn more about the numerical and experimental investigations, a large and reasonably complete bibliography is included.