In the previous chapters, all the physical problems are considered for solution based on the assumption that the motion is simple harmonic in time, which has reduced the physical problems to boundary value problems. However, in the presence of a sudden disturbance such as an impulse or in the case of a vehicle moving on the surface of a floating elastic plate or during landing and taking off of an aircraft on a floating plate-/ice-covered free surface, this simple harmonic motion is not a valid assumption. Further, often there are situations where the disturbances grow with the increase in time assuming that there was no disturbance at time t = 0. The significance of the associated wave structure interaction problem is that t is not an active variable of the governing differential equation. However, the time derivative appears on the plate-covered mean free surface and elsewhere with higher order derivative terms arising on the structural boundaries. To deal with such time-dependent problems, often the Laplace transform is employed to reduce the initial boundary value problem to a boundary value problem of the type discussed in the previous chapters. However, in many situations for problems in the unbounded domain, an appropriate integral transform method is employed (Hankel transform is used for axi-symmetric problems) to obtain integral representation of the boundary value problems incorporating all the boundary conditions, and asymptotic results are obtained using the method of stationary phase. In many other situations, Green’s integral theorem is employed along with a suitable Green’s function to deal with the time domain problem in a direct manner. There has been significant progress on initial value problems for transient water waves since Finkelstein [35]. Recently, there is a growing interest in dealing with time domain unsteady wave structure interaction problems due to their wide applications in various areas of marine science and technology including polar science (see [67], [124] and other cited papers). Significant progress on timedependent wave structure interaction problems has been based on the mode expansion method, the transform method and the spectral method (as in [36], [65], [96], [98], [132], [135], and the cited literature). There are several studies for determining the source singularities associated with unsteady water wave

wave interaction with flexible

motion in single-and double-layer fluid having free surface and interface (see [108] and the literature cited therein). Due to the growing interest in the wave structure interaction problem, attempts are being made to generalise several results available in the literature for gravity wave problems to apply to flexural gravity wave motion problem as in [11] and [83].