Nonlinear Effects in Wave Scattering and Generation

Abstract

When a fluid flow interacts with a topographic feature, and the fluid can support wave propagation, there is the potential for waves to be generated upstream and/or downstream. In many cases when the topographic feature has a small amplitude the situation can be successfully described using a linearised theory, and any nonlinear effects are determined as a small perturbation on the linear theory. However, when the flow is critical, that is, the system supports a long wave whose group velocity is zero in the reference frame of the topographic feature, then typically the linear theory fails and it is necessary to develop an intrinsically nonlinear theory. It is now known that in many cases such a transcriticai, weakly nonlinear and weakly dispersive theory leads to a forced Korteweg-de Vries (fKdV) equation. In canonical form, this is

$$ - {u_t} - \Delta {u_x} + 6u{u_x} + {u_{xxx}} + {f_x} = 0,$$

where u(x, t) is the amplitude of the critical mode, t is the time coordinate, x is the spatial coordinate, Δ is the phase speed of the critical mode, and f(x) is a representation of the topographic feature.

In this article we shall sketch the contexts where the fKdV equation is applicable, and describe some of the most relevant solutions. There are two main classes of solutions. In the first, the initial condition for the fKdV equation is u(x, 0) = 0 so that the waves are generated directly by the flow interaction with the topography. In this case the solutions are characterised by the generation of upstream solitary waves and an oscillatory downstream wavetrain, with the detailed structure being determined by A and the polarity of the topographic forcing term f(x). In the second class a solitary wave is incident on the topography, and depending on the system parameters may be repelled with a significant amplitude change, trapped with a change in amplitude, or allowed to pass by the topography with only a small change in amplitude.