Eigenfunction Matching for a Finite Change in Depth

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Introduction

The problem consists of a region of free water surface with depth [math]h[/math] except between [math]-L[/math] and [math]L[/math] where the depth is [math]d[/math]. The problem with a semi-infinite change in depth is treated in Eigenfunction Matching for a Semi-Infinite Change in Depth

Governing Equations

We begin with the Frequency Domain Problem for a submerged dock which occupies the region [math]x\gt0[/math] (we assume [math]e^{-\mathrm{i}\omega t}[/math] time dependence). The depth of is constant [math]h[/math] for [math]x\lt0[/math] and constant [math]d[/math] for [math]x\gt0[/math]. The [math]z[/math]-direction points vertically upward with the water surface at [math]z=0[/math]. The boundary value problem can therefore be expressed as

[math] \Delta\phi=0, \,\, -h\ltz\lt0, \,\, x\lt-L,\,x\gtL [/math]
[math] \Delta\phi=0, \,\, -d\ltz\lt0, \,\, -L\ltx\ltL [/math]
[math] \partial_z\phi=\alpha\phi, \,\, z=0, [/math]
[math] \partial_x\phi=0, \,\, -d\ltz\lt-h,\,x=\pm L, [/math]
[math] \partial_z\phi=0, \,\, z=-h,\, x\lt-L,\,x\gtL [/math]
[math] \partial_z\phi=0, \,\, z=-d,\, -L\ltx\ltL [/math]

We must also apply the Sommerfeld Radiation Condition as [math]|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 two regions, [math]x\lt0[/math] and [math]x\gt0[/math].

We express the potential as

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

and then Laplace's equation becomes

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

Separation of variables for a free surface

We use separation of variables

We express the potential as

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

and then Laplace's equation becomes

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

subject to the boundary conditions

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

and

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

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

[math] 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]z=0[/math]. The boundary condition at the free surface ([math]z=0 \,[/math]) gives rise to:

[math] 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]k_{0}=\pm ik \,[/math] and the positive real solutions by [math]k_{m} \,[/math], [math]m\geq1[/math]. The [math]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] \cos ix = \cosh x, \quad \sin ix = i\sinh x, [/math]

to arrive at the dispersion relation

[math] \alpha = k\tanh kh. [/math]

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

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

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

where

[math] 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]

Inner product between free surface and dock modes

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

where

[math] B_{mn}= \int\nolimits_{-h}^{0} \frac{\cos(k_{m}^{h}(z+h))\cos(k_{n}^{d}(z+d))}{\cos(k_{m}h)\cos(k_{n}d)} \mathrm{d} z =\frac{k_{m}^{h}\sin(k_{m}^{d} h)\cos(k_{n}^{d}h)-k_{n}^{d}\cos(k_{m}^{h} h)\sin(k_{n}^{d}h)} {\cos(k_{m}h)\cos(k_{n}d)({k_{m}^{d}}^{2}-{k_{n}^{d}}^{2})} [/math]

Governing Equations

We begin with the Frequency Domain Problem for a submerged dock which occupies the region [math]x\gt0[/math] (we assume [math]e^{i\omega t}[/math] time dependence). The depth of is constant [math]h[/math] for [math]x\lt-L[/math] and [math]x\gtL[/math] and constant [math]d[/math] for [math]-L\ltx\gtL[/math]. The [math]z[/math]-direction points vertically upward with the water surface at [math]z=0[/math]. The boundary value problem can therefore be expressed as

[math] \Delta\phi=0, \,\, -h\ltz\lt0, \,\, x\lt-L\,\textrm{or}\, x\gtL, [/math]
[math] \Delta\phi=0, \,\, -d\ltz\lt0, \,\, -L\ltx\ltL, [/math]
[math] \partial_z\phi=\alpha\phi, \,\, z=0, [/math]
[math] \partial_x\phi=0, \,\, -d\ltz\lt-h,\,x=\pm{L}, [/math]
[math] \partial_z\phi=0, \,\, z=-h,\, x\lt-L\,\textrm{or}\, x\gtL, [/math]
[math] \partial_z\phi=0, \,\, z=-d,\, -L\ltx\ltL. [/math]

We must also apply the Sommerfeld Radiation Condition as [math]|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 two regions, [math]x\lt0[/math] and [math]x\gt0[/math].

We express the potential as

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

and then Laplace's equation becomes

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

Separation of variables for a free surface

We use separation of variables

We express the potential as

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

and then Laplace's equation becomes

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

subject to the boundary conditions

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

and

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

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

[math] 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]z=0[/math]. The boundary condition at the free surface ([math]z=0 \,[/math]) gives rise to:

[math] 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]k_{0}=\pm ik \,[/math] and the positive real solutions by [math]k_{m} \,[/math], [math]m\geq1[/math]. The [math]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] \cos ix = \cosh x, \quad \sin ix = i\sinh x, [/math]

to arrive at the dispersion relation

[math] \alpha = k\tanh kh. [/math]

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

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

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

where

[math] 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]

Solution using Symmetry

The finite dock problem is symmetric about the line [math]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] \phi(x,z)=e^{-k_{0}^{h}(x+L)}\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}a_{m}^{s}e^{k_{m}^{h}(x+L)}\phi_{m}^{h}(z) , \;\;x\lt-L [/math]

and

[math] \phi(x,z)=\sum_{m=0}^{\infty}b_{m}^{s} \frac{\cosh k_{m}^{d} x}{\cosh k_{m}^{d} L} \phi_{m}^{d}(z), \;\;-L\ltx\lt0 [/math]

where [math]a_{m}^{s}[/math] and [math]b_{m}^{s}[/math] are the coefficients of the potential in the two regions.

For the first equation we multiply both sides by [math]\phi_{n}^{h}(z) \,[/math] and integrate from [math]-h[/math] and for the second equation we multiply both sides by [math]\phi_{n}^{d}(z) \,[/math] and integrate from [math]-d[/math]. This gives us

[math] A_{0}\delta_{0n}+a_{n}^{s}A_{n}^{h} =\sum_{m=0}^{\infty}b_{m}^{s}B_{mn} [/math]

and

[math] -k_{0}^{h}B_{n0} + \sum_{m=0}^{\infty} a_{m}^{s}k_{m}^{h} B_{nm} = -b_{n}^{s}k_{n}^{d}\tanh(k_{n}^{d}L) A_{n}^{d} [/math]

(for full details of this derivation see Eigenfunction Matching for a Semi-Infinite Change in Depth)

Anti-symmetric solution

The anti-symmetric potential can be expanded as

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

and

[math] \phi(x,z)=\sum_{m=0}^{\infty}b_{m}^{a} \frac{\sinh k_{m}^{h} x}{-\sinh k_{m}^{h} L}\phi_{m}(z), \;\;-L\ltx\lt0 [/math]

where [math]a_{m}^{a}[/math] and [math]b_{m}^{a}[/math] are the coefficients of the potential in the two regions. 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]x=-L[/math].

For the first equation we multiply both sides by [math]\phi_{n}^{h}(z) \,[/math] and integrate from [math]-h[/math] and for the second equation we multiply both sides by [math]\phi_{n}^{d}(z) \,[/math] and integrate from [math]-d[/math]. This gives us

[math] A_{0}^{h}\delta_{0n}+a_{n}^{a}A_{n}^{h} =\sum_{m=0}^{\infty}b_{m}^{a}B_{mn} [/math]

and

[math] -k_{0}^{h}B_{n0} + \sum_{m=0}^{\infty} a_{m}^{a}k_{m}^{h} B_{nm} = -b_{n}^{a}k_{n}^{d}\coth(k_{n}^{d}L) A_{n}^{d} [/math]

(for full details of this derivation see Eigenfunction Matching for a Semi-Infinite Change in Depth)

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

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

Note the formulae for [math]b_m[/math] and [math]c_m[/math] are more complicated but can be derived with some work.


Matlab Code

A program to calculate the coefficients for the semi-infinite dock problems can be found here finite_change_in_depth.m

Additional code

This program requires