We now return to the free-surface flow generated by a moving disturbance and extend the results of Chapter 4 to the nonlinear regime. We shall see that the wave trains in the far field (if they exist) are then described by the nonlinear theories of Chapters 5 and 6. Furthermore we will show that the nonuniformities of Figures 4.9 and 4.8 are removed when a nonlinear theory is used. Some nonlinear solutions described in this chapter approach the linear solutions of Chapter 4 as the size of the disturbance approaches zero, while others approach solitary waves.

We have organised the results in the following way. In Section 7.1 we present pure gravity free-surface flows (i.e. g ≠ 0, T = 0) in water of finite depth and show in Section 7.1.1 that the nonuniformity of the linear supercritical solutions near F = 1 (see Figure 4.9) is removed when a nonlinear theory is used. Subcritical flows are considered in Section 7.1.2. In Section 7.2 we consider gravity–capillary free-surface flows. Solutions in water of finite depth are described in Section 7.2.1. We show in Section 7.2.2 that the nonuniformity of the linear theory near α = 0.25 (see Figure 4.8) is removed when a nonlinear theory is used. We examine in Section 7.3 the implications of the existence of multiple branches of periodic gravity–capillary waves (see Sections 5.1.2 and 6.5.3) for free-surface flows generated by moving disturbances in water of infinite depth.

Free-surface problems occur in many aspects of science and everyday life. They can be defined as problems whose mathematical formulation involves surfaces that have to be found as part of the solution. Such surfaces are called free surfaces. Examples of free-surface problems are waves on a beach, bubbles rising in a glass of champagne, melting ice, flows pouring from a container and sails blowing in the wind. In these examples the free surface is the surface of the sea, the interface between the gas and the champagne, the surface of the ice, the boundary of the pouring flow and the surface of the sail.

In this book we concentrate on applications arising in fluid mechanics. We hope to convince the reader of the beauty of such problems and to present the challenges faced when one attempts to describe these flows mathematically. Many of these challenges are resolved in the book but others are still open questions. We will always attempt to present fully nonlinear solutions without restricting assumptions on the smallness of some parameters. Our techniques are often numerical. However, it is the belief of the author that a deep understanding of the structure of the solutions cannot be gained by brute-force numerical approaches. It is crucial to combine numerical methods with analytical techniques, especially when singularities are present.

Introduction

The first ten chapters of this book were devoted to steady free-surface flows. An equally important topic is that of time-dependent free-surface flows. Boundary integral equation methods can still be used to investigate these problems. The idea is to ‘march in time’ and to solve at each time step a linear integral equation similar to those derived in the previous chapters, by using Cauchy integral equation formula or Green's theorem. Such methods have been developed and used by many authors (see for example and the references cited in these papers). In particular, results have been obtained for breaking waves. An obvious use of time-dependent codes is to study the stability of steady solutions.

In this chapter we will confine our attention to one type of time-dependent free-surface flow, namely gravity–capillary standing waves. We will solve the problem by a series expansion similar to that used in Section 5.1 to study periodic travelling waves. The analysis follows Vanden-Broeck closely. The choice of this problem is motivated by the fact that gravity–capillary standing waves have properties similar to those of Wilton ripples (see Section 6.5.3.1).

We note that a proof of the existence of nonlinear gravity standing waves was provided only recently.

Nonlinear gravity–capillary standing waves

The concept of linear standing waves was introduced in Section 2.4.3. Here we extend the theory of standing waves to the nonlinear regime.

Two fundamental approaches have been used in the previous chapters to calculate free-surface flows. The first involves perturbing known exact solutions. Often these exact solutions are trivial, e.g. a uniform stream. To leading order this approach gives a linear theory (see for example the calculations of Chapter 4) and at higher order a weakly nonlinear theory (see for example the small-amplitude expansions and the Korteweg–de Vries equation of Chapter 5).

In the second approach fully nonlinear solutions are computed. This approach involves a discretisation leading to a system of nonlinear algebraic equations, which is then solved by iteration (e.g. using Newton's method). Iteration requires the choice of an initial guess. These initial guesses are often trivial solutions or asymptotic solutions derived in the first approach. After convergence of the iterations, the solution obtained is then used as an initial guess to compute a new solution for slightly different values of the parameters. For example, the linear solutions of Section 2.4 were used as an initial guess in Chapter 6 to compute a nonlinear solution of small amplitude. This solution was then used as an initial guess to compute a solution of larger amplitude and so on. This method of ‘continuation’ leads to families of solutions; an application is the ‘continuation in ∈’ used in Section 7.1.1. We can then investigate whether other solution branches bifurcate from these branches (see Section 6.5.2.1 for an example).

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.