Flexural gravity wave generation by initial disturbances in the presence of current

The effect of uniform current on the generation of flexural gravity waves resulting from initial disturbances at a point was analyzed in two dimensions. The problem was formulated as an initial boundary value problem under the assumptions of the linearized theory of water waves. By direct application of the Laplace transform and then the Fourier transform, explicit expressions for the velocity potential and free surface elevation were obtained in integral forms; these were evaluated asymptotically for large distances and times by the application of the method of the stationary phase to obtain the far field behavior of the surface elevations in specific cases. Simple numerical computations were performed to illustrate the effect of uniform current on the surface elevation, wavelength, phase velocity, and group velocity of the flexural gravity waves and on the far field behavior of the progressive waves in two different cases, namely, when there is an initial depression concentrated at the origin and an initial impulse concentrated at the origin.