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.

We present here the solution for a floating elastic plate using dry modes. We begin with the equations. The solution can also be found using Eigenfunction Matching for a Finite Floating Elastic Plate using Symmetry.

Equations for a Finite Plate in Frequency Domain

We consider the problem of small-amplitude waves which are incident on finite floating elastic plate occupying water surface for [math]\displaystyle{ -L\lt x\lt L }[/math]. These equations are derived in Floating Elastic Plate The submergence of the plate is considered negligible. We assume that the problem is invariant in the [math]\displaystyle{ y }[/math] direction. We also assume that the plate edges are free to move at each boundary, although other boundary conditions could easily be considered using the methods of solution presented here. We begin with the Frequency Domain Problem for a semi-infinite Floating Elastic Plates in the non-dimensional form of Tayler 1986 (Dispersion Relation for a Floating Elastic Plate). We also assume that the waves are normally incident (incidence at an angle will be discussed later).

[math]\displaystyle{ \Delta \phi = 0, \;\;\; -h \lt z \leq 0, }[/math] [math]\displaystyle{ \partial_z \phi = 0, \;\;\; z = - h, }[/math] [math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt -L,\,\,{\rm or}\,\,x\gt L }[/math] [math]\displaystyle{ \partial_x^2\left\{\beta(x) \partial_x^2\partial_z \phi\right\} - \left( \gamma(x)\alpha - 1 \right) \partial_z \phi - \alpha\phi = 0, \;\; z = 0, \;\;\; -L \leq x \leq L, }[/math]

where [math]\displaystyle{ \alpha = \omega^2 }[/math], [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \gamma }[/math] are the stiffness and mass constant for the plate respectively. The free edge conditions at the edge of the plate imply

[math]\displaystyle{ \partial_x^3 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L, }[/math] [math]\displaystyle{ \partial_x^2 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L, }[/math]

Transformation using Eigenfunctions 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 L, \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 L. \end{matrix} }[/math]

This solution is discussed further in Eigenfunctions for a Beam.

Expanding

[math]\displaystyle{ \frac{\partial \phi}{\partial z} = i\omega \sum_{n=0}^{\infty} a_n w_n }[/math]

Equation in Terms of the Modes of the Plate

Under these assumptions, the equations become

[math]\displaystyle{ \Delta\phi =0,\,\,-h\lt z\lt 0, }[/math] [math]\displaystyle{ \partial_{z}\phi =0,\,\,z=-h, }[/math] [math]\displaystyle{ \omega^{2}\phi =\partial_{z}\phi,\,\,x\notin(-b,b),\ \ z=0, }[/math] [math]\displaystyle{ i\omega\sum_{n=0}^{\infty}\alpha_{n}w_{n} =\partial_{z}\phi,\,\,x\in (-b,b),\,\, z=0, }[/math] [math]\displaystyle{ \sum_{n=0}^{\infty}\alpha_{n}\left( 1+\beta\lambda_{n}^{4}\right) w_{n}-\omega^{2}\gamma\sum_{n=0}^{\infty}\alpha_{n}w_{n} = -i\omega \phi,\,\,x\in(-b,b),\,\, z=0. }[/math]

We solve for the potential (and displacement) as the sum of the diffracted and radiation potentials in the standard way, as for a rigid body. We begin with the diffraction potential [math]\displaystyle{ \phi^{(d)} }[/math] which satisfies the following equations

[math]\displaystyle{ \begin{matrix} \Delta\phi^{(d)} & =0,\,\,-h\lt z\lt 0,\\ \partial_{z}\phi^{(d)} & =0,\,\,z=-h,\\ \partial_{z}\phi^{(d)} & =\omega^{2}\phi^{(d)},\,\,x\notin(-b,b),\,\, z=0,\\ \partial_{z}\phi^{(d)} & =0,\,\,x\in(-b,b),\,\,z=0. \end{matrix} }[/math]

Furthermore, [math]\displaystyle{ \phi^{(d)} }[/math] satisfies the radiation condition

[math]\displaystyle{ \frac{\partial}{\partial x} \left(\phi^{(d)}-\phi_{\kappa}^{\rm In} \right) \pm ik\left( \phi^{(d)}-\phi^{\rm In}_{\kappa}\right) = 0 ,\,\,\mathrm{as} \,\,x\rightarrow\pm\infty, }[/math]

where [math]\displaystyle{ k }[/math] is the wavenumber, which is the positive real solution of the dispersion equation

[math]\displaystyle{ k\tanh(kh)=\omega^{2}, }[/math]

and [math]\displaystyle{ \phi^{\rm In}_{\kappa} }[/math] is the incident wave given by

[math]\displaystyle{ \phi^{\rm In}_\kappa = \frac{i\omega}{k\sinh kh} \cosh k(z+h) e^{i\kappa kx} }[/math]

(which has unit amplitude in displacement) and [math]\displaystyle{ \kappa }[/math] is either [math]\displaystyle{ 1 }[/math] for a wave travelling towards negative infinity or [math]\displaystyle{ -1 }[/math] for a wave travelling towards positive infinity (we will need both these solutions). We now consider the radiation potentials [math]\displaystyle{ \phi^{(n)} }[/math], which satisfy the following equations

[math]\displaystyle{ \begin{matrix} (11) \Delta\phi^{(n)} & =0,\,\,-h\lt z\lt 0,\\ \partial_{z}\phi^{(n)} & =0,\,\,z=-h,\\ \partial_{z}\phi^{(n)} & =\omega^{2}\phi^{(n)},\,\,x\notin(-b,b),\, \, z=0\\ (12) \partial_{z}\phi^{(n)} & = i\omega w_{n},\,\,x\in(-b,b),\,\,z=0. \end{matrix} }[/math]

The radiation condition for the radiation potential is

[math]\displaystyle{ (13) \frac{\partial\phi^{(n)}}{\partial x}\pm ik\phi^{(n)}=0,\,\,\mathrm{as} \,\,x\rightarrow\pm\infty. }[/math]

The method used to solve these equations for deep water is described in sturova06, and for finite depth in newman94 . Therefore we find the potential as

[math]\displaystyle{ \phi=\phi^{(d)}_\kappa +\sum_{n=0}^{\infty}\alpha_{n,\kappa}\phi^{(n)}, }[/math]

so that

[math]\displaystyle{ \sum_{n=0}^{\infty}\left( 1+\beta\lambda_{n}^{4} - \omega^{2}\gamma\right) \alpha_{n,\kappa}w_{n}=-i\omega\phi ^{(d)}_\kappa-i\omega\sum_{n=0}^{\infty}\alpha_{n,\kappa}\phi^{(n)}. }[/math]

If we multiply by [math]\displaystyle{ w_m }[/math] and take an inner product over the plate we obtain

[math]\displaystyle{ (14) \left( 1+\beta\lambda_{n}^{4}+\omega^{2}\gamma\right) \alpha_{n,\kappa }=i\omega \int_{-b}^{b}\phi^{(d)}_\kappa w_{n}dx + \sum_{m=0}^{\infty}\left(-\omega^2 a_{mn}(\omega)+ i\omega b_{mn}(\omega)\right) \alpha_{m,\kappa}, }[/math]

where the functions [math]\displaystyle{ a_{mn}(\omega) }[/math] and [math]\displaystyle{ b_{mn}(\omega) }[/math] are given by

[math]\displaystyle{ a_{mn}(\omega) + b_{mn}(\omega)/i\omega=\int_{-b}^{b}\phi^{(m)}w_{n}dx, }[/math]

and they are referred to as the added mass and damping coefficients respectively. Equation \eqref{single_freq_numeric} is solved by truncating the number of modes. The expression for the displacement is

[math]\displaystyle{ (15) \zeta_{\kappa}(x,\omega) = \left\{ \begin{matrix}[c]{ll} \sum_{n=0}^{\infty}\alpha_{n,\kappa}(\omega)w_{n}(x), \,\,x\in(-b,b), \\ -i\omega \left(\phi^{(d)}_\kappa(x,0) + \sum_{n=0}^{\infty}\alpha_{n,\kappa}\phi^{(n)}(x,0) \right),\,\,x\notin(-b,b). \end{matrix} \right. }[/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 Eigenfunctions 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]