Nonlinear free surface flows past a semi- infinite flat plate in water of finite depth

We consider the steady free surface two-dimensional flow past a semi-infinite flat plate in water of a constant finite depth. The fluid is assumed to be inviscid, incompressible and the flow is irrotational; surface tension at the free surface is neglected. Our concern is with the periodic waves generated downstream of the plate edge. These can be characterized by a depth-based Froude number, F , and the depth d (draft) of the depressed plate. For small d and subcritical flows, we may use the linearized problem, combined with conservation of momentum, to obtain some analytical results. These linear results are valid when F is not close to 0 or 1. As F approaches 1, we use a weakly nonlinear longwave analysis, and in particular show that the results can be extended to supercritical flows. For larger d nonlinear effects need to be taken account, and so we solve the fully nonlinear problem numerically using a boundary integral equation method. Here the predicted wavelength from the linear and weakly nonlinear results is used to set the mean depth condition for the nonlinear problem. The results by these three approaches are in good agreement when d is relatively small. For larger d our numerical results are compared with known results for the highest wave. We also find some wave-free solutions, which when compared with the weakly nonlinear results are essentially just one-half of a solitary wave solution.