A Novel Method of Solution for the Fluid Loaded Plate

TL;DR

This paper introduces a new unified approach to solve the fluid loaded plate problem on the line and half-line, simplifying previous methods and establishing well-posedness through integral equations and Laplace transforms.

Contribution

It presents a novel unified boundary value problem method that avoids technical assumptions and provides explicit solution representations for fluid loaded plates.

Findings

The approach simplifies the solution process compared to previous methods.

The problem on the full line is shown to be well-posed.

The half-line problem's solution involves solving a convolution-type integral equation.

Abstract

We study the Cauchy problem associated with the equations governing a fluid loaded plate formulated on either the line or the half-line. We show that in both cases the problem can be solved by employing the unified approach to boundary value problems introduced by on of the authors in the late 1990s. 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 the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, but this formula 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 $\eta_0$ to be in $H^3(R^+)$, we show that the solution depends continuously on the initial data, and hence, the problem is well-posed.