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]\displaystyle{ h }[/math] except between [math]\displaystyle{ -L }[/math] and [math]\displaystyle{ L }[/math] where the depth is [math]\displaystyle{ 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]\displaystyle{ x\gt 0 }[/math] (we assume [math]\displaystyle{ e^{-\mathrm{i}\omega t} }[/math] time dependence). The depth of is constant [math]\displaystyle{ h }[/math] for [math]\displaystyle{ x\lt 0 }[/math] and constant [math]\displaystyle{ d }[/math] for [math]\displaystyle{ x\gt 0 }[/math]. The [math]\displaystyle{ z }[/math]-direction points vertically upward with the water surface at [math]\displaystyle{ z=0 }[/math]. The boundary value problem can therefore be expressed as

[math]\displaystyle{ \Delta\phi=0, \,\, -h\lt z\lt 0, \,\, x\lt -L,\,x\gt L }[/math] [math]\displaystyle{ \Delta\phi=0, \,\, -d\lt z\lt 0, \,\, -L\lt x\lt L }[/math] [math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0, }[/math] [math]\displaystyle{ \partial_x\phi=0, \,\, -d\lt z\lt -h,\,x=\pm L, }[/math] [math]\displaystyle{ \partial_z\phi=0, \,\, z=-h,\, x\lt -L,\,x\gt L }[/math] [math]\displaystyle{ \partial_z\phi=0, \,\, z=-d,\, -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 two regions, [math]\displaystyle{ x\lt 0 }[/math] and [math]\displaystyle{ x\gt 0 }[/math].

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]

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]

Inner product between free surface and dock modes

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

where

[math]\displaystyle{ 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]\displaystyle{ x\gt 0 }[/math] (we assume [math]\displaystyle{ e^{i\omega t} }[/math] time dependence). The depth of is constant [math]\displaystyle{ h }[/math] for [math]\displaystyle{ x\lt -L }[/math] and [math]\displaystyle{ x\gt L }[/math] and constant [math]\displaystyle{ d }[/math] for [math]\displaystyle{ -L\lt x\gt L }[/math]. The [math]\displaystyle{ z }[/math]-direction points vertically upward with the water surface at [math]\displaystyle{ z=0 }[/math]. The boundary value problem can therefore be expressed as

[math]\displaystyle{ \Delta\phi=0, \,\, -h\lt z\lt 0, \,\, x\lt -L\,\textrm{or}\, x\gt L, }[/math] [math]\displaystyle{ \Delta\phi=0, \,\, -d\lt z\lt 0, \,\, -L\lt x\lt L, }[/math] [math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0, }[/math] [math]\displaystyle{ \partial_x\phi=0, \,\, -d\lt z\lt -h,\,x=\pm{L}, }[/math] [math]\displaystyle{ \partial_z\phi=0, \,\, z=-h,\, x\lt -L\,\textrm{or}\, x\gt L, }[/math] [math]\displaystyle{ \partial_z\phi=0, \,\, z=-d,\, -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 two regions, [math]\displaystyle{ x\lt 0 }[/math] and [math]\displaystyle{ x\gt 0 }[/math].

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]

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]

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}^{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]\displaystyle{ \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\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 two regions.

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

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

and

[math]\displaystyle{ -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]\displaystyle{ \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]\displaystyle{ \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\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 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]\displaystyle{ x=-L }[/math].

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

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

and

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

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

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

Inner product between free surface and dock modes

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

where

[math]\displaystyle{ 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]

Matlab Code

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

Additional code

This program requires

• dispersion_free_surface.m