Generation of water waves by an inertial surface disturbance in the presence of currents

In this paper, we investigate the two-dimensional problem of wave generation by an initial disturbance at the upper surface of infinite-depth water, which is covered by an inertial surface in the presence of a uniform current. The inertial surface is modeled as a thin, uniformly distributed layer of non-interacting floating material, such as a floating mat or broken ice. Two types of initial disturbances are considered: (i) a predefined depression of the inertial surface and (ii) an impulsive force applied at the inertial surface. Using Laplace and Fourier transforms, the surface depression is expressed as an infinite integral, which is subsequently approximated for large spatial and temporal scales via the method of stationary phase. The effects of the inertial surface and the uniform current on wave generation are analyzed at different times and distances. The results show that the presence of an inertial surface reduces the amplitude of the surface depression, whereas increasing the current speed leads to a larger amplitude of the depression.