Green Function Methods for Floating Elastic Plates

Introduction

The problem of a two-dimensional Floating Elastic Plate was solved using a Free-Surface Green Function by Newman 1994 and Meylan and Squire 1994. We describe here both methods (which are closely related). A related paper was given by Hermans 2003 and we extended to multiple plates in Hermans 2004.

Equations of Motion

We begin with the equations of motion in non-dimensional form for a single Floating Elastic Plate which occupies the region [math]\displaystyle{ -b\leq x\leq b }[/math]. The full derivation of these equation is presented in Eigenfunction Matching Method for Floating Elastic Plates. We assume that the plate is infinite in the [math]\displaystyle{ y }[/math] direction, but we allow the wave to be incident at an angle which we do by introducing a wavenumber [math]\displaystyle{ k_y }[/math]. These means that the total potential is given by

[math]\displaystyle{ \Phi(x,y,z,y) = \Re\left(\phi(x,z)e^{i\omega t}e^{i k_y y}\right). }[/math]

The free-surface is at [math]\displaystyle{ z=0 }[/math] and the sea floor is at [math]\displaystyle{ z=-h }[/math]

[math]\displaystyle{ \begin{matrix} \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} - k_y^2\right) \phi = 0 \;\;\;\; \mbox{ for } -h \lt z \leq 0, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \frac{\partial \phi}{\partial z} = 0 \;\;\;\; \mbox{ at } z = - h, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \left( \beta \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2 - \gamma\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha\phi = 0 \;\;\;\; \mbox{ at } z = 0, \;\;\; -b \leq x \leq b, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \frac{\partial \phi}{\partial z} - \alpha\phi = 0 \;\;\;\; \mbox{ at } z = 0, \;\;\; x\lt -b \,\,\mathrm{or}\,\, b\lt x, \end{matrix} }[/math]

where [math]\displaystyle{ \alpha = \omega^2 }[/math] and

[math]\displaystyle{ \begin{matrix} \left(\frac{\partial^3}{\partial x^3} - (2 - \nu)k^2_y\frac{\partial}{\partial x}\right) \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = \pm b, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = \pm b. \end{matrix} }[/math]

Transformation Using the Green function

We use the Free-Surface Green Function for two-dimensional waves incident at an angle which we denote by [math]\displaystyle{ G(x,\zeta) }[/math] since we are only interested in its value at [math]\displaystyle{ z=0. }[/math] Using this we can transform the system of equations to

[math]\displaystyle{ \phi(x) = \phi^{i}(x) + \int_{-b}^{b}G(x,\xi) \left( \alpha\phi(\xi) - \phi_z(\xi) \right)d \xi }[/math] [math]\displaystyle{ \begin{matrix} \left( \beta \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2 - \gamma\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha\phi = 0 \;\;\;\; \mbox{ at } z = 0, \;\;\; -b \leq x \leq b, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \left(\frac{\partial^3}{\partial x^3} - (2 - \nu)k^2_y\frac{\partial}{\partial x}\right) \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = \pm b, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; \mbox{ for } x = \pm b. \end{matrix} }[/math]

We will consider now the case where [math]\displaystyle{ k_y=0 }[/math], although the solutions presented here can be generalised to the case when [math]\displaystyle{ k_y\neq 0 }[/math]. Under this assumption the equations reduce to

[math]\displaystyle{ \phi(x) = \phi^{i}(x) + \int_{-b}^{b}G(x,\xi) \left( \alpha\phi(\xi) - \phi_z(\xi) \right)d \xi }[/math] [math]\displaystyle{ \begin{matrix} \left( \beta \frac{\partial^4}{\partial x^4} - \gamma\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha\phi = 0 \;\;\;\; \mbox{ at } z = 0, \;\;\; -b \leq x \leq b, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm b, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm b. \end{matrix} }[/math]

Transformation using Eigenfuctions for a Beam

We can find a the eigenfunction which satisfy

[math]\displaystyle{ \partial_x^4 w_n = \lambda_n^4 w_n }[/math]

plus the edge conditions.

[math]\displaystyle{ \begin{matrix} \frac{\partial^3}{\partial x^3} \frac{\partial\phi}{\partial z}= 0 \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm b, \end{matrix} }[/math] [math]\displaystyle{ \begin{matrix} \frac{\partial^2}{\partial x^2} \frac{\partial\phi}{\partial z} = 0\mbox{ for } \;\;\;\; \mbox{ at } z = 0 \;\;\; x = \pm b. \end{matrix} }[/math]

This solution is discussed further in Eigenfunctions for a Beam.

Expanding

[math]\displaystyle{ \frac{\partial \phi}{\partial z} = \sum a_n w_n }[/math]

we obtain

[math]\displaystyle{ \alpha \phi = \sum \left(\beta\lambda_n^4 - \gamma\alpha + 1\right)a_n w_n }[/math]

This leads to the following equation

[math]\displaystyle{ \phi(x) = \frac{1}{\alpha} \int_{-b}^{b} \frac{w_n(x)w_n(\xi)}{\beta\lambda_n^4 - \gamma\alpha + 1}\phi_z(\xi)d\xi }[/math]

or

[math]\displaystyle{ \phi(x) = \frac{1}{\alpha} \int_{-b}^{b} g(x,\xi)\phi_z(\xi)d\xi }[/math]

where

[math]\displaystyle{ g(x,\xi) = \frac{w_n(x)w_n}{(\xi\beta\lambda_n^4 - \gamma\alpha + 1} }[/math]