Time dependent flexural gravity waves in the presence of current

In the present study, a combined effect of current and compressive force on time dependent flexural gravity wave motion in both the cases of single and two-layer fluids is analyzed in finite and infinite water depths in two dimensions. The roots of the dispersion relation associated with the plane flexural gravity waves are analyzed via contour plots and by plotting various terms of the dispersion relation separately. The characteristic of plane flexural gravity waves is studied by analyzing the phase and group velocities along with the law of conservation of energy flux to understand the combined effect of current and compressive force on the wave motion. The integral form of the time dependent Green's function in the presence of current is obtained using the Laplace transform method and used in Green's identity to derive the time dependent velocity potential for the flexural gravity wavemaker problem. The time harmonic Green's function and velocity potentials are obtained as a special case from the time dependent problems. Numerical results are computed and analyzed in particular cases using the method of stationary phase to obtain the asymptotic results for Green's function and the deflection of ice sheet. The integral form of Green's function derived here will be suitable to deal with physical problem when the roots of the dispersion relation for the flexural gravity wave problem coalesces which were otherwise not possible in the eigenfunction expansion method used for time harmonic problems.