Eigenfunction Matching for a Semi-Infinite Floating Elastic Plate

Introduction

We show here a solution for a semi-infinite Floating Elastic Plate on Finite Depth. The problem was solved by Fox and Squire 1994 but the solution method here is slightly different. The simpler theory for a Dock describes many of the ideas here in more detail.

Equations

We consider the problem of small-amplitude waves which are incident on a semi-infinite floating elastic plate occupying water surface for [math]\displaystyle{ x\gt 0 }[/math]. The submergence of the plate is considered negligible. We assume that the problem is invariant in the [math]\displaystyle{ y }[/math] direction, although we allow the waves to be incident from an angle. 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{ \begin{matrix} \Delta \phi = 0, \;\;\; -h \lt z \leq 0, \end{matrix} }[/math] [math]\displaystyle{ \partial_z \phi = 0, \;\;\; z = - h, }[/math] [math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt 0, }[/math] [math]\displaystyle{ \beta \partial_x^4\partial_z \phi - \left( \gamma\alpha - 1 \right) \partial_z \phi + \alpha\phi = 0, \;\; z = 0, \;\;\; x \geq 0, }[/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 = 0, }[/math] [math]\displaystyle{ \partial_x^2 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = 0, }[/math]

Method of solution

Eigenfunction expansion

We will solve the system of equations using an Eigenfunction Matching Method. The method was developed by Fox and Squire 1994. The separation of variables for the left hand region where there is open water, and the incident wave, is described in Eigenfunction Matching for a Semi-Infinite Dock and we consider here only the separation of variables in the plate covered region.

Separation of variables under the Plate

The potential velocity can be written in terms of an infinite series of separated eigenfunctions under each elastic plate, of the form

[math]\displaystyle{ \psi = \frac{\cos(\kappa (z+h))}{\cos\kappa h}.\, }[/math]

If we apply the boundary conditions given we obtain the Dispersion Relation for a Floating Elastic Plate

[math]\displaystyle{ \kappa \tan{(\kappa h)}= -\frac{\alpha}{\beta \kappa^{4} + 1 - \alpha\gamma} }[/math]

Solving for [math]\displaystyle{ \kappa }[/math] gives a pure imaginary root with positive imaginary part, two complex roots (two complex conjugate paired roots with positive imaginary part in most physical situations), an infinite number of positive real roots which approach [math]\displaystyle{ {n\pi}/{h} }[/math] as [math]\displaystyle{ n }[/math] approaches infinity, and also the negative of all these roots (Dispersion Relation for a Floating Elastic Plate) . We denote the two complex roots with positive imaginary part by [math]\displaystyle{ \kappa_(-2) }[/math] and [math]\displaystyle{ \kappa_(-1) }[/math], the purely imaginary root with positive imaginary part by [math]\displaystyle{ \kappa_{0} }[/math] and the real roots with positive imaginary part by [math]\displaystyle{ \kappa_(n) }[/math] for [math]\displaystyle{ n }[/math] a positive integer. The imaginary root with positive imaginary part corresponds to a reflected travelling mode propagating along the [math]\displaystyle{ x }[/math] axis. The complex roots with positive imaginary parts correspond to damped reflected travelling modes and the real roots correspond to reflected evanescent modes.

Expressions for the potential velocity

We now expand the potential in the two regions using the separation of variables solution. We define

[math]\displaystyle{ \phi_{m}\left( z\right) =\frac{\cos k_{m}(z+H)}{\cos k_{m}H},\quad m\geq0 }[/math]

as the vertical eigenfunction of the potential in the open water region and

[math]\displaystyle{ \psi_{m}\left( z\right) =\frac{\cos\kappa_{m}(z+H)}{\cos\kappa_{m}H},\quad m\geq-2 }[/math]

as the vertical eigenfunction of the potential in the plate covered region. The potential [math]\displaystyle{ \phi }[/math] can now be expressed as the following sum of eigenfunctions: be expanded as

[math]\displaystyle{ \phi(x,z)= e^{-k_{0}x}\phi_{0}(z) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x\lt 0 }[/math]

and

[math]\displaystyle{ \phi(x,z)=\sum_{m=-2}^{\infty}b_{m} e^{-\kappa_{m}x}\psi_{m}(z), \;\;x\gt 0 }[/math]

where [math]\displaystyle{ a_{m} }[/math] and [math]\displaystyle{ b_{m} }[/math] are the coefficients to be determined.

For later reference, we note that:

[math]\displaystyle{ \int\nolimits_{-H}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn} }[/math]

where

[math]\displaystyle{ A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}H\sin k_{m}H+k_{m}H}{k_{m}\cos ^{2}k_{m}H}\right) }[/math]

and

[math]\displaystyle{ \int\nolimits_{-H}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn} }[/math]

where

[math]\displaystyle{ B_{mn}=\frac{k_{n}\sin k_{n}H\cos\kappa_{m}H-\kappa_{m}\cos k_{n}H\sin \kappa_{m}H}{\left( \cos k_{n}H\cos\kappa_{m}H\right) \left( k_{n} ^{2}-\kappa_{m}^{2}\right) } }[/math]

An infinite dimensional system of equations

The potential and its derivative must be continuous across the transition from open water to the plate covered region. Therefore, the potentials and their derivatives at [math]\displaystyle{ x=0 }[/math] have to be equal. We also truncate the sum at [math]\displaystyle{ N }[/math] being careful that we have two extra modes on the plate covered region to satisfy the edge conditions. We obtain

[math]\displaystyle{ \phi_{0}\left( z\right) + \sum_{m=0}^{N} a_{m} \phi_{m}\left( z\right) =\sum_{m=-2}^{N}b_{m}\psi_{m}(z) }[/math]

and

[math]\displaystyle{ -k_{0}\phi_{0}\left( z\right) +\sum _{m=0}^{N} a_{m}k_{m}\phi_{m}\left( z\right) =-\sum_{m=-2}^{N}b_{m}\kappa_{m}\psi _{m}(z) }[/math]

for each [math]\displaystyle{ N }[/math]. We solve these equations by multiplying both equations by [math]\displaystyle{ \phi_{l}(z)\, }[/math] and integrating from [math]\displaystyle{ -h }[/math] to [math]\displaystyle{ 0 }[/math] to obtain:

[math]\displaystyle{ A_{0}\delta_{0l}+a_{l}A_{l} =\sum_{m=0}^{N}b_{m}B_{ml}+b_{-1}B_{-1l}+b_{-2}B_{-2l} }[/math]

and

[math]\displaystyle{ -k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l =-\sum_{m=0}^{N}b_{m}\kappa_{m}B_{ml}+b_{-1}\kappa_{-1}B_{-1l}+b_{-2}\kappa_{-2}B_{-2l} }[/math]

We separate the terms for [math]\displaystyle{ m=-1 }[/math] and [math]\displaystyle{ m=-2 }[/math] because [math]\displaystyle{ l \in [0,N] }[/math]. If we multiply the first equation by [math]\displaystyle{ k_l }[/math] and subtract the second equation we obtain

[math]\displaystyle{ 2k_{0}A_{0}\delta_{0l} =\sum_{m=0}^{N}b_{m}(k_l + \kappa_{m})B_{ml}+b_{-1}(k_l+\kappa_{-1})B_{-1l}+b_{-2}(k_l+\kappa_{-2})B_{-2l} }[/math]

Finally, we need to apply the conditions at the edge of the plate to give us two further equations,

[math]\displaystyle{ \partial_x^2\partial_z\phi = - \sum_{m=-2}^{N}b_{m} \kappa_m^3 \tan\kappa_m h = 0 }[/math]

and

[math]\displaystyle{ \partial_x^3\partial_z\phi = \sum_{m=-2}^{N}b_{m} \kappa_m^4 \tan\kappa_m h = 0 }[/math]

These equations will permit us to obtain [math]\displaystyle{ 2N+4 }[/math] equations in order to calculate [math]\displaystyle{ b_{-1} }[/math], [math]\displaystyle{ b_{-2} }[/math] and [math]\displaystyle{ a_m }[/math] and [math]\displaystyle{ b_m }[/math], [math]\displaystyle{ m \in [0,N] }[/math].

Numerical Solution

To solve the system of equations previously defined we set the upper limit of [math]\displaystyle{ l }[/math] to be [math]\displaystyle{ N }[/math], as we already said before. In terms of matrix, we obtain

[math]\displaystyle{ \begin{bmatrix} \begin{bmatrix} A_0&0 \quad \cdots&0\\ 0&&\\ \vdots&A_l&\vdots\\ &&0\\ 0&\cdots \quad 0 &A_N \end{bmatrix} & \begin{bmatrix} -B_{-20}&-B_{-10}\\ &\\ \vdots&\vdots\\ &\\ -B_{-2N}&-B_{-1N} \end{bmatrix} & \begin{bmatrix} -B_{00}&\cdots&-B_{0N}\\ &&\\ \vdots&-B_{ml}&\vdots\\ &&\\ -B_{N0}&\cdots&-B_{NN} \end{bmatrix} \\ \begin{bmatrix} 0&&\cdots&&0\\ 0&&\cdots&&0 \end{bmatrix} & \begin{bmatrix} \kappa_{-2}^3\tan\kappa_{-2}h&\kappa_{-1}^3\tan\kappa_{-1}h\\ \kappa_{-2}^4\tan\kappa_{-2}h&\kappa_{-1}^4\tan\kappa_{-1}h \end{bmatrix} & \begin{bmatrix} \kappa_0^3\tan\kappa_0h&\cdots&\kappa_N^3\tan\kappa_Nh\\ \kappa_0^4\tan\kappa_0h&\cdots&\kappa_N^4\tan\kappa_Nh \end{bmatrix} \\ \begin{bmatrix} 0&&\cdots&&0\\ &&&&\\ \vdots&&\ddots&&\vdots\\ &&&&\\ 0&&\cdots&&0 \end{bmatrix} & \begin{bmatrix} (k_0+\kappa_{-2})B_{-20}&(k_0+\kappa_{-1})B_{-10}\\ &\\ \vdots&\vdots\\ &\\ (k_N+\kappa_{-2})B_{-2N}&(k_N+\kappa_{-1})B_{-1N} \end{bmatrix} & \begin{bmatrix} (k_0 + \kappa_0) \, B_{00}&\cdots&(k_N + \kappa_{0}) \, B_{0N}\\ &&\\ \vdots&(k_l + \kappa_{m}) \, B_{ml}&\vdots\\ &&\\ (k_0 + \kappa_N) \, B_{N0}&\cdots&(k_N + \kappa_{N}) \, B_{NN}\\ \end{bmatrix} \end{bmatrix} \begin{bmatrix} a_{0} \\ \\ \vdots \\ \\ a_N \\ b_{-2}\\ b_{-1}\\ b_{0}\\ \\ \vdots \\ \\ b_N \end{bmatrix} = \begin{bmatrix} - A_{0} \\ 0 \\ \vdots \\ \\ 0 \\ 0 \\ 0 \\ 2k_{0}A_{0} \\ 0 \\ \vdots \\ \\ 0 \end{bmatrix} }[/math]

We then simply need to solve the linear system of equations.

Waves Incident at an Angle

We can consider the problem when the waves are incident at an angle [math]\displaystyle{ \theta }[/math]. In this case we have the wavenumber in the [math]\displaystyle{ y }[/math] direction is [math]\displaystyle{ k_y = \sin\theta k_0 }[/math] where [math]\displaystyle{ k_0 }[/math] is as defined previously (note that [math]\displaystyle{ k_y }[/math] is imaginary).

This means that the potential is now of the form [math]\displaystyle{ \phi(x,y,z)=e^{k_y y}\phi(x,z) }[/math] so that when we separate variables we obtain

Therefore the potential can be expanded as

[math]\displaystyle{ \phi(x,z)=e^{-\hat{k_0}x}\phi_0+\sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}x}\phi_{m}(z), \;\;x\lt 0 }[/math]

and

[math]\displaystyle{ \phi(x,z)=\sum_{m=-2}^{\infty}b_{m} e^{-\hat{\kappa}_{m}x}\psi_{m}(z), \;\;x\gt 0 }[/math]

where [math]\displaystyle{ \hat{k}_{m} = \sqrt{k_m^2 + k_y^2} }[/math] and [math]\displaystyle{ \hat{\kappa}_{m} = \sqrt{\kappa_m^2 + k_y^2} }[/math] where we always take the positive real root or the root with positive imaginary part.

The equations are derived almost identically to those above and we obtain

[math]\displaystyle{ A_{0}\delta_{0l}+a_{l}A_{l} =\sum_{n=0}^{\infty}b_{m}B_{ml} }[/math]

and

[math]\displaystyle{ -\hat{k_{0}}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l =-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml} }[/math]

and these are solved exactly as before.

Energy Balance

The energy balance equation is (Energy Balance for Two Elastic Plates) given by

[math]\displaystyle{ D |T|^2 + |R|^2 = 1, \, }[/math]

where [math]\displaystyle{ T=b_0 }[/math], [math]\displaystyle{ R=a_0 }[/math], and [math]\displaystyle{ D }[/math] is given by

[math]\displaystyle{ D = \left(\frac{\hat{\kappa}_0 k_0\cosh^2{(k_0 h)}}{\hat{k} \kappa_0\cosh^2{(\kappa_0 h)}}\right) \left(\frac{\frac{\beta}{\alpha}4(\kappa_0)^3((\hat{\kappa}_0)^2 +k_y^2)\sinh^2{(\kappa_0 h)} + \frac{1}{2}{\sinh{(2 \kappa_0 h)}}+\kappa_0 h} { \frac{1}{2}{\sinh{(2 k_0 h)}}+k_0h}\right) }[/math]