As shown in the previous chapters, efficient methods for two-dimensional free-surface flows can be derived by using the theory of analytic functions. In particular, free streamline problems, series truncation methods and boundary integral equation methods based on the Cauchy integral formula can be used to obtain highly accurate solutions. Unfortunately such techniques are not available for three-dimensional free-surface flows. However, as we shall see in this chapter, boundary integral equation methods can still be derived using Green's theorem.

Boundary integral equation methods based on Green's theorem can also be used for two-dimensional free-surface flows as an alternative to methods based on the Cauchy integral formula. We first show this for twodimensional free-surface flows generated by moving disturbances in water of infinite depth. Gravity is included in the dynamic boundary condition but surface tension is neglected.

Green's function formulation for two-dimensional problems

We describe the numerical method based on Green's functions by considering the free-surface flows generated by a moving pressure distribution (see Figure 4.4) or by a moving surface-piercing object (see Figure 4.3). We will assume that the water is of infinite depth. The corresponding method based on the Cauchy theorem was described in Chapter 7 for a moving pressure distribution.

Pressure distribution

We consider the two-dimensional free-surface flow generated by a pressure distribution moving at a constant velocity U at the surface of a fluid of infinite depth.