9 - Waves with constant vorticity

Summary

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).