ar X iv : 0 81 2 . 16 60 v 3 [ m at h . A P ] 9 A pr 2 00 9 A Novel Method of Solution For The Fluid Loaded Plate

We study the equations governing a fluid loaded plate. We first reformu-late these equations as a system of two equations, one of which is an explicit non-local equation for the wave height and the velocity potential on the free surface. We then concentrate on the linearised equations and show that the problems formulated either on the full or the half line can be solved by employing the unified approach to boundary value problems introduced by on of the authors in the late 1990's. The problem on the full line was analysed by Crighton et. al. using a combination of Laplace and Fourier transforms. The new approach avoids the technical difficulty of the a priori assumption that the amplitude of the plate is in L 1 dt (R +) and furthermore yields a simpler solution representation which immediately implies that the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, which however, involves two unknown functions. The main difficulty with the half-line problem is the characterisation of these two functions. By employing the so-called global relation, we show that the two functions can be obtained via the solution of a complex valued integral equation of the convolution type. This equation can be solved in closed form using the Laplace transform. By prescribing the initial data η0 to be in H 3 3 (R +), or equivalently twice differentiable with sufficient decay at infinite, we show that the solution depends continuously on the initial data, and hence, the problem is well-posed.