So far we have studied linear deep-water and shallow-water waves. Now we shall turn our attention to nonlinear shallow-water waves, particularly solitary waves. We begin in the following section by re-formulating the shallow-water wave equations to include effects of nonlinearity and dispersion that were ignored in the previous chapter. Then in § 13.2 these equations are scaled and non-dimensionlized. This procedure introduces two dimensionless parameters representing nonlinearity and dispersion. Four limiting cases are identified and studied in the follow four sections, the most interesting of which is the nonlinear dispersive case investigated in § 13.6, involving solitary waves.

Re-Formulation

In order to develop the equations governing nonlinear shallow-water waves, let's begin with the full nonlinear formulation for surface-water waves in a body of water having depth D. The governing equation, valid for −D < z < h remains Laplace's equation:

where ϕ is the velocity potential. The nonlinear kinematic and dynamic conditions at the free surface are

The bottom boundary condition is simply

The surface boundary conditions are a pair of coupled nonlinear partial differential equations, applied at an unknown position z = h(x, t). Nonlinear equations are even more difficult to solve than linear equations with variable coefficients, so we have our work cut out for us in attempting to analyze and solve these equations. This provides us with a chance to use some of the tools of the trade, including

Scaling and Non-Dimensionalization

Let's begin to use our tools by scaling the dependent and independent variables h, ϕ, x, z and t. To begin with, it is obvious that we should scale x with the length of the wave _, z with the water depth D and h with the maximum wave height, hm.