Analytical solutions for the flexural–gravity and capillary–gravity wave resistances due to a steadily moving concentrated load

The interfacial flexural–gravity waves due to a concentrated load steadily moving on the thin elastic plate floating on the interface between two immiscible fluids of different densities are investigated analytically. The two inviscid fluids are assumed to be incompressible and homogeneous, and the motion be irrotational. The equation of motion for the plate, including the elastic force, the compressive force and the inertial force, is imbedded in the dynamic boundary conditions on the interface. Based on the linear potential theory for small-amplitude waves, the integral solutions for the interfacial wave profiles and wave resistances are derived by the Fourier transform. The dispersion relation and the exact solutions for the wave numbers are explicitly deduced. It is found the wave dynamical behaviors depend on the relation between the moving speed and the minimal phase speed. The maximal allowable value for the compressive force is obtained, at which the minimal phase speed is zero and the wave motion can be stimulated with any nonzero moving speed of the load due to the presence of compressive stress in the plate. The effects of the flexural rigidity and the fluid density ratio are also explored. The solutions for the capillary–gravity wave can readily be recovered by discarding the elastic force in the present formulation.