Free Surface Flow under Gravity and Surface Tension due to an Applied Pressure Distribution II Bond number less than 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 Schrödinger 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.