Eigenfunction Matching for a Finite Dock

From WikiWaves

Introduction

The problems consists of a region to the left and right with a free surface and a middle region with a rigid surface through which not flow is possible. We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We then consider the case when the waves are incident at an angle. For the later we give the equations in slightly less detail. The theory is based on Eigenfunction Matching for a Semi-Infinite Dock and this should be consulted for many details. The solution here can be straightforwardly extended using Symmetry in Two Dimensions to two docks of the same length and this can be found Two Identical Docks using Symmetry. We also show how the solution can be found using Symmetry in Two Dimensions for the finite dock in Eigenfunction Matching for a Finite Dock using Symmetry.

Governing Equations

Wave scattering by a finite dock

We consider here the Frequency Domain Problem for a finite dock which occupies the region $- L < x < L$ (we assume $e i ω t$ time dependence). The water is assumed to have constant finite depth $h$ and the $z$ -direction points vertically upward with the water surface at $z = 0$ and the sea floor at $z = - h$ . The boundary value problem can therefore be expressed as

$\Delta\phi=0, \,\, -h<z<0,$

$\phi_{z}=0, \,\, z=-h,$

$\partial_z\phi=\alpha\phi, \,\, z=0,\,x <-L, {\rm or} \, x>L$

$\partial_z\phi=0, \,\, z=0,\,-L<x<L,$

We must also apply the Sommerfeld Radiation Condition as $|x|\rightarrow\infty$ . This essentially implies that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave and a wave propagating away.

Solution Method

We use separation of variables in the three regions, exactly as for the Eigenfunction Matching for a Semi-Infinite Dock.

Separation of variables for a free surface

We use separation of variables

We express the potential as

$\phi(x,z) = X(x)Z(z)\,$

and then Laplace's equation becomes

$\frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2$

The separation of variables equation for deriving free surface eigenfunctions is as follows:

$Z^{\prime\prime} + k^2 Z =0.$

subject to the boundary conditions

$Z^{\prime}(-h) = 0$

and

$Z^{\prime}(0) = \alpha Z(0)$

We can then use the boundary condition at $z=-h \,$ to write

$Z = \frac{\cos k(z+h)}{\cos kh}$

where we have chosen the value of the coefficent so we have unit value at $z = 0$ . The boundary condition at the free surface ( $z=0 \,$ ) gives rise to:

$k\tan\left( kh\right) =-\alpha \,$

which is the Dispersion Relation for a Free Surface

We denote the negative imaginary solution of this equation by $k_{0} \,$ and the positive real solutions by $k_{m} \,$ , $m\geq1$ . We define

$\phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0$

as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that

$\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) \mathrm{d} z=A_{n}\delta_{mn}$

where

$A_{n}=\frac{1}{2}\left( \frac{\cos k_{n}h\sin k_{n}h+k_{n}h}{k_{n}\cos ^{2}k_{n}h}\right).$

Separation of Variables for a Dock

The separation of variables equation for a floating dock

$Z^{\prime\prime} + k^2 Z =0,$

subject to the boundary conditions

$Z^{\prime} (-h) = 0,$

and

$Z^{\prime} (0) = 0.$

The solution is $k=\kappa_{m}= \frac{m\pi}{h} \,$ , $m\geq 0$ and

$Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad m\geq 0.$

We note that

$\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn},$

where

$C_{m} = \begin{cases} h,\quad m=0 \\ \frac{1}{2}h,\,\,\,m\neq 0 \end{cases}$

Inner product between free surface and dock modes

$\int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) \mathrm{d} z=B_{mn}$

where

$B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin \kappa_{m}h}{\left( \cos k_{n}h\right) \left( k_{n} ^{2}-\kappa_{m}^{2}\right) }$

Incident potential

The incident potential is a wave of amplitude $A$ in displacement travelling in the positive $x$ -direction. The incident potential can therefore be written as

$\phi^{\mathrm{I}} =e^{-k_{0}x}\phi_{0}\left( z\right)$

Expansion of the Potential

The potential can be expanded as

$\phi(x,z)=e^{-k_{0}(x+L)}\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}(x+L)}\phi_{m}(z), \;\;x<-L$

$\phi(x,z)=b_0 \frac{L-x}{2L}\psi_{0}(z) + \sum_{m=1}^{\infty}b_{m} e^{-\kappa_{m} (x+L)}\psi_{m}(z) +c_0 \frac{L+x}{2L}\psi_{0}(z) + \sum_{m=1}^{\infty}c_{m} e^{\kappa_{m} (x-L)}\psi_{m}(z) , \;\;-L<x<L$

and

$\phi(x,z)=\sum_{m=0}^{\infty}d_{m} e^{-k_{m}(x-L)}\phi_{m}(z), \;\;L<x$

where $a m$ and $d m$ are the coefficients of the potential in the open water regions to the left and right and $c m$ and $d m$ are the coefficients under the dock covered region. We have an incident wave from the left.

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 $x=\pm L$ have to be equal. We obtain

$\phi_{0}\left( z\right) + \sum_{m=0}^{\infty} a_{m} \phi_{m}\left( z\right) =\sum_{m=0}^{\infty}b_{m}\psi_{m}(z) + \sum_{m=1}^{\infty}c_{m}\psi_{m}(z)e^{-2L\kappa_m}$

$-k_{0}\phi_{0}\left( z\right) +\sum _{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right) =-\frac{b_0}{2L}\psi_0(z) -\sum_{m=1}^{\infty}b_{m}\kappa_{m}\psi _{m}(z) + \frac{c_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c_{m}\kappa_{m}\psi _{m}(z)e^{-2L\kappa_m}$

$\sum_{m=1}^{\infty}b_{m}\psi_{m}(z)e^{-2L\kappa_m} + \sum_{m=0}^{\infty}c_{m}\psi_{m}(z) =\sum_{m=0}^{\infty}d_{m} \phi_{m}\left( z\right)$

$-\frac{b_0}{2L}\psi_0(z) -\sum_{m=1}^{\infty}b_{m}\kappa_{m}\psi _{m}(z)e^{-2L\kappa_m} + \frac{c_0}{2L}\psi_0(z)+\sum_{m=1}^{\infty}c_{m}\kappa_{m}\psi _{m}(z) = -\sum_{m=0}^{\infty}d_{m} k_m\phi_{m}\left( z\right)$

We solve these equations by multiplying both equations by $φ l (z)$ and integrating from $- h$ to $0$ to obtain:

$A_{0}\delta_{0l}+a_{l}A_{l} =\sum_{m=0}^{\infty}b_{m}B_{ml} + \sum_{m=1}^{\infty}c_{m}B_{ml}e^{-2L\kappa_m}$

$-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l = - b_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b_{m}\kappa_{m}B_{ml} + c_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m}$

$\sum_{m=1}^{\infty}b_{m}B_{ml}e^{-2L\kappa_m} + \sum_{m=0}^{\infty}c_{m}B_{ml} =d_l A_l$

$- b_0 \frac{B_{0l}}{2L} - \sum_{m=1}^{\infty}b_{m}\kappa_{m}B_{ml} e^{-2L\kappa_m} + c_0 \frac{B_{0l}}{2L} + \sum_{m=1}^{\infty}c_{m}\kappa_{m}B_{ml} = -d_l k_l A_l$

Numerical Solution

To solve the system of equations we set the upper limit of $l$ to be $M$ . We then simply need to solve the linear system of equations.

Solution with Waves Incident at an Angle

We can consider the problem when the waves are incident at an angle $θ$ . When a wave in incident at an angle $θ$ we have the wavenumber in the $y$ direction is $k y = sinθ k 0$ where $k 0$ is as defined previously (note that $k y$ is imaginary).

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

$k^2 = k_x^2 + k_y^2$

where $k$ is the separation constant calculated without an incident angle.

Therefore the potential can be expanded as

$\phi(x,z)=e^{-k_{0}(x+L)}\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m}e^{\hat{k}_{m}(x+L)}\phi_{m}(z), \;\;x<-L$

$\phi(x,z)=\sum_{m=0}^{\infty}b_{m} e^{-\hat{\kappa}_{m} (x+L)}\psi_{m}(z) ++ \sum_{m=1}^{\infty}c_{m} e^{\hat{\kappa}_{m} (x-L)}\psi_{m}(z) , \;\;-L<x<L$

and

$\phi(x,z)=\sum_{m=0}^{\infty}d_{m} e^{-\hat{k}_{m}(x-L)}\phi_{m}(z), \;\;L<x$

where $\hat{k}_{m} = \sqrt{k_m^2 + k_y^2}$ and $\hat{\kappa}_{m} = \sqrt{\kappa_m^2 + k_y^2}$ 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

$A_{0}\delta_{0l}+a_{l}A_{l} =\sum_{m=0}^{\infty}b_{m}B_{ml} + \sum_{m=0}^{\infty}c_{m}B_{ml}e^{-2L\hat{\kappa}_m}$

$-\hat{k}_{0}A_{0}\delta_{0l}+a_{l}\hat{k}_{l}A_l = - \sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml} + \sum_{m=0}^{\infty}c_{m}\hat{\kappa}_{m}B_{ml} e^{-2L\hat{\kappa}_m}$

$\sum_{m=0}^{\infty}b_{m}B_{ml}e^{-2L\hat{\kappa}_m} + \sum_{m=0}^{\infty}c_{m}B_{ml} =d_l A_l$

$-\sum_{m=0}^{\infty}b_{m}\hat{\kappa}_{m}B_{ml} e^{-2L\hat{\kappa}_m} + \sum_{m=0}^{\infty}c_{m}\hat{\kappa}_{m}B_{ml} = -d_l \hat{k}_l A_l$

and these are solved exactly as before.

Matlab Code

A program to calculate the coefficients for the finite dock problems can be found here finite_dock.m

Additional code

This program requires

dispersion_free_surface.m

Eigenfunction Matching for a Finite Dock

From WikiWaves

Contents

Introduction

Governing Equations

Solution Method

Separation of variables for a free surface

Separation of Variables for a Dock

Inner product between free surface and dock modes

Incident potential

Expansion of the Potential

An infinite dimensional system of equations

Numerical Solution

Solution with Waves Incident at an Angle

Matlab Code

Additional code

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