Green's Functions

Abstract

As in the case of time-harmonic waves (see Chapter 1), we begin with the simplest model, replacing a ship by a point source in the uniform forward motion in calm water. The corresponding velocity potential is sometimes referred to as the Kelvin source, but to keep the terminology unified we call it the Green's function in what follows. Similar to the theory of time-harmonic waves developed in Part 1, the theory of ship waves presented here relies essentially on Green's functions. They are of importance not only for proving solvability theorems (see Chapters 7 and 8) but also for constructing examples of trapped waves (nontrivial solutions to homogeneous boundary value problems) in Section 8.4. The three-dimensional Green's function of a point source in deep water is considered in detail in Sections 6.1 and 6.2. General facts about the three-dimensional Green's function are considered in Section 6.1 and the far-field expansions for Green's function and the corresponding elevation of the free surface are obtained in Section 6.2. Two-dimensional Green's functions are treated in Section 6.3, which we begin with the simpler case of deep water (Subsection 6.3.1). For water of finite depth, which will be referred to as shallow water, we consider Green's function in Subsection 6.3.2.