The effective behaviour of linear and nonlinear waves in irregular channels
wave propagation | asymptotic theory | scientific computing
A fluid mechanics model is built in several stages. The passage from a simplified, prototypical model to a more sophisticated one leads to a natural question: will given properties be preserved by the more complex model? Linear potential theory is taken to be our reduced/prototypical model for water wave propagation. The hydrostatic Navier--Stokes equation is chosen as the nonlinear, more sophisticated, model. It is shown to be a consistent limiting model for reflection--transmission problems in the presence of rapidly varying topographies. The persistence of linear potential theory results are verified for surface pulse propagation in the presence of both periodic and disordered topographies. Numerical experiments show that effective behaviour is preserved. In particular, waves propagating over large amplitude, rapidly varying topographies do not break. Stochastic theory results from linear potential theory are validated with the nonlinear numerical model. This corroborates linear theory. Additional wave--topography experiments are performed beyond the regime of validity of linear potential theory. Nonlinear waves, bores and viscous flows are considered. These new results include the delay of bores by rapidly varying topographies. Evidence is given that apparent diffusion, due to disordered topographies, can dominate viscous diffusion.