Time-domain motion of a floating or obliquely submerged non-uniform elastic plate

We consider the motion of a thin elastic plate with non-uniform thickness. The plate is either submerged and has some inclination with the vertical or is floating on the upper surface of the water. Green's function arising from the fourth-order boundary condition for the non-uniform plate (which we refer to as plate Green's function) is determined using two different methods in terms of the vibrating modes of the plate. These, in turn, are derived from the modes of a plate with constant thickness. The problem is finally reduced to a boundary integral equation involving the plate Green's function and the fundamental Green's function. This equation is hypersingular in the case of a submerged plate. A numerical solution to the integral equation is used to find results for elastic plates with variable thicknesses. The results are validated by comparing them with those of an elastic plate with uniform thickness. We also present simulations of the time-domain motion when the plate-fluid system is subject to an incident wave pulse using Fourier transform.