Eigenfunction Matching for a Finite Dock

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.

Governing Equations

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

[math]\displaystyle{ \Delta\phi=0, \,\, -h\lt z\lt 0, }[/math]

[math]\displaystyle{ \phi_{z}=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_z\phi=0, \,\, z=0,\,-L\lt x\lt L, }[/math]

We must also apply the Sommerfeld Radiation Condition as [math]\displaystyle{ |x|\rightarrow\infty }[/math]. 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

[math]\displaystyle{ \phi(x,z) = X(x)Z(z)\, }[/math]

and then Laplace's equation becomes

[math]\displaystyle{ \frac{X^{\prime\prime}}{X} = - \frac{Z^{\prime\prime}}{Z} = k^2 }[/math]

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

[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }[/math]

subject to the boundary conditions

[math]\displaystyle{ Z^{\prime}(-h) = 0 }[/math]

and

[math]\displaystyle{ Z^{\prime}(0) = \alpha Z(0) }[/math]

We can then use the boundary condition at [math]\displaystyle{ z=-h \, }[/math] to write

[math]\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }[/math]

where we have chosen the value of the coefficent so we have unit value at [math]\displaystyle{ z=0 }[/math]. The boundary condition at the free surface ([math]\displaystyle{ z=0 \, }[/math]) gives rise to:

[math]\displaystyle{ k\tan\left( kh\right) =-\alpha \, }[/math]

which is the Dispersion Relation for a Free Surface

The above equation is a transcendental equation. If we solve for all roots in the complex plane we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by [math]\displaystyle{ k_{0}=\pm ik \, }[/math] and the positive real solutions by [math]\displaystyle{ k_{m} \, }[/math], [math]\displaystyle{ m\geq1 }[/math]. The [math]\displaystyle{ k \, }[/math] of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

[math]\displaystyle{ \cos ix = \cosh x, \quad \sin ix = i\sinh x, }[/math]

to arrive at the dispersion relation

[math]\displaystyle{ \alpha = k\tanh kh. }[/math]

We note that for a specified frequency [math]\displaystyle{ \omega \, }[/math] the equation determines the wavenumber [math]\displaystyle{ k \, }[/math].

Finally we define the function [math]\displaystyle{ Z(z) \, }[/math] as

[math]\displaystyle{ \chi_{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. 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

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

where

[math]\displaystyle{ 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). }[/math]

Separation of Variables for a Dock

The separation of variables equation for a floating dock

[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0, }[/math]

subject to the boundary conditions

[math]\displaystyle{ Z^{\prime} (-h) = 0, }[/math]

and

[math]\displaystyle{ Z^{\prime} (0) = 0. }[/math]

The solution is [math]\displaystyle{ k=\kappa_{m}= \frac{m\pi}{h} \, }[/math], [math]\displaystyle{ m\geq 0 }[/math] and

[math]\displaystyle{ Z = \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad m\geq 0. }[/math]

We note that

[math]\displaystyle{ \int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) \mathrm{d} z=C_{m}\delta_{mn}, }[/math]

where

[math]\displaystyle{ C_{m} = \begin{cases} h,\quad m=0 \\ \frac{1}{2}h,\,\,\,m\neq 0 \end{cases} }[/math]

Inner product between free surface and dock modes

[math]\displaystyle{ \int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) \mathrm{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\right) \left( k_{n} ^{2}-\kappa_{m}^{2}\right) } }[/math]

Incident potential

To create meaningful solutions of the velocity potential [math]\displaystyle{ \phi }[/math] in the specified domains we add an incident wave term to the expansion for the domain of [math]\displaystyle{ x \lt 0 }[/math] above. The incident potential is a wave of amplitude [math]\displaystyle{ A }[/math] in displacement travelling in the positive [math]\displaystyle{ x }[/math]-direction. We would only see this in the time domain [math]\displaystyle{ \Phi(x,z,t) }[/math] however, in the frequency domain the incident potential can be written as

[math]\displaystyle{ \phi_{\mathrm{I}}(x,z) =e^{-k_{0}x}\chi_{0}\left( z\right). }[/math]

The total velocity (scattered) potential now becomes [math]\displaystyle{ \phi = \phi_{\mathrm{I}} + \phi_{\mathrm{D}} }[/math] for the domain of [math]\displaystyle{ x \lt 0 }[/math].

The first term in the expansion of the diffracted potential for the domain [math]\displaystyle{ x \lt 0 }[/math] is given by

[math]\displaystyle{ a_{0}e^{k_{0}x}\chi_{0}\left( z\right) }[/math]

which represents the reflected wave.

In any scattering problem [math]\displaystyle{ |R|^2 + |T|^2 = 1 }[/math] where [math]\displaystyle{ R }[/math] and [math]\displaystyle{ T }[/math] are the reflection and transmission coefficients respectively. In our case of the semi-infinite dock [math]\displaystyle{ |a_{0}| = |R| = 1 }[/math] and [math]\displaystyle{ |T| = 0 }[/math] as there are no transmitted waves in the region under the dock.

Expansion of the Potential

The potential can be expanded as

[math]\displaystyle{ \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\lt -L }[/math]

[math]\displaystyle{ \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\lt x\lt L }[/math]

and

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

where [math]\displaystyle{ a_{m} }[/math] and [math]\displaystyle{ d_{m} }[/math] are the coefficients of the potential in the open water regions to the left and right and [math]\displaystyle{ c_m }[/math] and [math]\displaystyle{ d_m }[/math] 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 [math]\displaystyle{ x=\pm L }[/math] have to be equal. We obtain

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ -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} }[/math]

[math]\displaystyle{ \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) }[/math]

[math]\displaystyle{ -\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) }[/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}^{\infty}b_{m}B_{ml} + \sum_{m=1}^{\infty}c_{m}B_{ml}e^{-2L\kappa_m} }[/math]

[math]\displaystyle{ -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} }[/math]

[math]\displaystyle{ \sum_{m=1}^{\infty}b_{m}B_{ml}e^{-2L\kappa_m} + \sum_{m=0}^{\infty}c_{m}B_{ml} =d_l A_l }[/math]

[math]\displaystyle{ - 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 }[/math]

Numerical Solution

To solve the system of equations we set the upper limit of [math]\displaystyle{ l }[/math] to be [math]\displaystyle{ M }[/math]. 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 [math]\displaystyle{ \theta }[/math]. When a wave in incident at an angle [math]\displaystyle{ \theta }[/math] 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

[math]\displaystyle{ k^2 = k_x^2 + k_y^2 }[/math]

where [math]\displaystyle{ k }[/math] is the separation constant calculated without an incident angle.

Therefore the potential can be expanded as

[math]\displaystyle{ \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\lt -L }[/math]

[math]\displaystyle{ \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\lt x\lt L }[/math]

and

[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}d_{m} e^{-\hat{k}_{m}(x-L)}\phi_{m}(z), \;\;L\lt x }[/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_{m=0}^{\infty}b_{m}B_{ml} + \sum_{m=0}^{\infty}c_{m}B_{ml}e^{-2L\hat{\kappa}_m} }[/math]

[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} + \sum_{m=0}^{\infty}c_{m}\hat{\kappa}_{m}B_{ml} e^{-2L\hat{\kappa}_m} }[/math]

[math]\displaystyle{ \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 }[/math]

[math]\displaystyle{ -\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 }[/math]

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

Solution using Symmetry

The finite dock problem is symmetric about the line [math]\displaystyle{ x=0 }[/math] and this allows us to solve the problem using symmetry. This method is numerically more efficient and requires only slight modification of the code for Eigenfunction Matching for a Semi-Infinite Dock, the developed theory here is very close to the semi-infinite solution. We decompose the solution into a symmetric and an anti-symmetric part as is described in Symmetry in Two Dimensions

Symmetric solution

The symmetric potential can be expanded as

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

and

[math]\displaystyle{ \phi(x,z)=\sum_{m=0}^{\infty}b_{m}^{s} \frac{\cosh\kappa_{m} x}{\cosh \kappa_m L}\phi_{m}(z), \;\;-L\lt x\lt 0 }[/math]

where [math]\displaystyle{ a_{m}^{s} }[/math] and [math]\displaystyle{ b_{m}^{s} }[/math] are the coefficients of the potential in the open water regions and the dock covered region respectively.

We now match at [math]\displaystyle{ x=-L }[/math] and multiply 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}^{s}A_{l} =\sum_{n=0}^{\infty}b_{m}^{s}B_{ml} }[/math]

and

[math]\displaystyle{ -k_{0}A_{0}\delta_{0l}+a_{l}^{s}k_{l}A_l =-\sum_{m=0}^{\infty}b_{m}^{s}\kappa_{m}\tanh(\kappa_{m}L) B_{ml} }[/math]

(for full details of this derivation see Eigenfunction Matching for a Semi-Infinite Dock)

Anti-symmetric solution

The anti-symmetric potential can be expanded as

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

and

[math]\displaystyle{ \phi(x,z)=-b_0^{a}\frac{x}{L}+\sum_{m=1}^{\infty}b_{m}^{a} \frac{\sinh\kappa_{m} x}{-\sinh \kappa_m L}\phi_{m}(z), \;\;-L\lt x\lt 0 }[/math]

where [math]\displaystyle{ a_{m}^{a} }[/math] and [math]\displaystyle{ b_{m}^{a} }[/math] are the coefficients of the potential in the open water regions and the dock covered region respectively. Note that the minus sign in the expression for the dock-covered region has been added so that each component is equal to one at [math]\displaystyle{ x=-L }[/math].

We now match at [math]\displaystyle{ x=-L }[/math] and multiply 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}A_{l} =\sum_{n=0}^{\infty}b_{m}^{a}B_{ml} }[/math]

and

[math]\displaystyle{ -k_{0}A_{0}\delta_{0l}+a_{l}^{a}k_{l}A_l = - \frac{b_{0}^{a}}{L}B_{0l} -\sum_{m=1}^{\infty}b_{m}^{a}\kappa_{m}\coth(\kappa_{m}L) B_{ml} }[/math]

(for full details of this derivation see Eigenfunction Matching for a Semi-Infinite Dock)

Solution to the original problem

We can now reconstruct the potential for the finite dock from the two previous symmetric and anti-symmetric solution as explained in Symmetry in Two Dimensions. The amplitude in the left open-water region is simply obtained by the superposition principle

[math]\displaystyle{ a_{m} = \frac{1}{2}\left(a_{m}^{s}+a_{m}^{a}\right) }[/math]

and in the dock-covered region we now consider a potential written as

[math]\displaystyle{ \phi(x,z)=b_0 \psi_{0}(z) + \sum_{m=1}^{\infty}b_{m} \frac{\cos\kappa_{m} x}{\cos \kappa_m L}\psi_{m}(z) -c_0 \frac{x}{L} \psi_{0}(z) + \sum_{m=1}^{\infty}c_{m} \frac{\sin\kappa_{m} x}{-\sin \kappa_m L} \psi_{m}(z) , \;\;-L\lt x\lt L }[/math]

and we obtain

[math]\displaystyle{ b_{m}=\frac{1}{2}\left(b_{m}^{s}+b_{m}^{a}\right) }[/math].

Matlab Code

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

Additional Code

This program requires

• dispersion_free_surface.m