Resonant reflection of shallow-water waves due to corrugated boundaries

One-dimensional, nonlinear, shallow-water wave equations are derived from two-dimensional Boussinesq equations to investigate resonant reflections due to corrugated boundaries. Small, but short-wave undulations are introduced through water depth and channel width. Coupled nonlinear equations for the transmitted and reflected wave fields are derived and solved numerically. In the simple case where undulations are zero (the reflection is also zero), the governing equations are used to study the propagation of permanent shallow-water waves (cnoidal waves) and to examine the generation of higher harmonics in shallow-water waves. The present numerical results show that the nonlinear effects are very important in considering the resonant reflection of cnoidal waves from a rippled bed.