Free surface flow under gravity and surface tension due to an Applied Pressure Distribution II bond number less then one-third

We consider steady free surface two-dimensional flow due to a localized applied pressure distribution under the effects of both gravity and surface tension in water of a constant depth, and in the presence of a uniform stream. The fluid is assumed to be inviscid and incompressible, and the flow is irrotational. The behaviour of the forced nonlinear waves is characterized by three parameters: the Froude number, F, the Bond number, τ < 1/3, and the magnitude and sign of the pressure forcing term ǫ. The fully nonlinear wave problem is solved numerically by using a boundary integral method. For small amplitude waves and F < Fm < 1 where Fm is a certain critical value where the phase and group velocities for linearized waves coincide, linear theory gives a good prediction for the numerical solution of the nonlinear problem in the case of a bifurcation from the uniform flow. As F approaches Fm, however, some nonlinear terms need to be taken in the problem. In this case the forced nonlinear Schr¨odinger equation is found to be an appropriate model to describe bifurcations from an unforced envelope solitary wave. In general, it is found that for given values of F < Fm and τ < 1/3, there exist both elevation and depression waves.