Eigenfunction Matching for a Submerged Circular Dock

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Introduction

We present here the theory for a submerged circular dock. The details of the method can be found in Eigenfunction Matching for a Submerged Semi-Infinite Dock and Eigenfunction Matching for a Circular Dock

Governing Equations

We begin with the Frequency Domain Problem. We will use a cylindrical coordinate system, $(r,\theta,z)$, assumed to have its origin at the centre of the circular plate which has radius $a$. 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

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

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

$\phi_{z}=\alpha\phi, \,\, z=0,$

$\phi_{z}=0, \,\, z=-d,\,r<a$

We must also apply the Sommerfeld Radiation Condition as $r\rightarrow\infty$. The subscript $z$ denotes the derivative in $z$-direction.

Solution Method

We use separation of variables in the two regions, $r<a$ and $r>a$.

The solution of the problem for the potential in finite water depth can be found by a separation ansatz,

$\phi (r,\theta,z) =: Y(r,\theta) Z(z).\,$

Substituting this into the equation for $\phi$ yields

$\frac{1}{Y(r,\theta)} \left[ \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial Y}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 Y}{\partial \theta^2} \right] = - \frac{1}{Z(z)} \frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} = k^2.$

The possible separation constants $k$ will be determined by the free surface condition and the bed condition.

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

The above equation is not really the dispersion relation for a free surface, it would be better to refer to it as 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 $k_{0}=\pm ik \,$ and the positive real solutions by $k_{m} \,$, $m\geq1$. The $k \,$ of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations

$\cos ix = \cosh x, \quad \sin ix = i\sinh x,$

to arrive at the dispersion relation

$\alpha = k\tanh kh.$

We note that for a specified frequency $\omega \,$ the equation determines the wavenumber $k \,$.

Finally we define the function $Z(z) \,$ as

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

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

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

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

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

The depth above the plate is $d$ and below the plate is $h-d$. We introduce a new dispersion value $\mu_n$:

$\mu_n = \begin{cases} k_n^{d},\qquad \qquad\mbox{for}\,\, 0 \leq n \leq N-M\\ n\pi/(h-d),\,\,\mbox{otherwise} \end{cases}$

where $k_n^{d}$ are the roots of the Dispersion Relation for a Free Surface with depth $d$. We also order the roots with the first being the positive imaginary solution $k_0^{d}$, the second being zero, then ordering by increasing size. We then define a new function

$\chi_n = \begin{cases} 0,\,\,\, \qquad-d<z< 0 \\ \psi_{n}(z),\,\,\,-h<z<-d \end{cases}$

or

$\chi_{n} = \begin{cases} \phi_{n}^{d}(z),\,\,\,-d<z< 0 \\ 0,\,\,\qquad-h<z<-d \end{cases}$

where

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

depending on whether the root $\mu_n$ is above or below.

Inner product between free surface and submerged plate modes

We define

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

where $B_{mn}$ is either

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

or

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

depending on whether the root $\mu_n$ is above or below.

Separation for Cylindrical Coordinates

We now separate variables, noting that since the problem has circular symmetry we can write the potential as

$\phi(r,\theta,z)=\frac{\cos k(z+h)}{\cos kh}\sum_{n=-\infty}^{\infty}\rho_{n}(r)e^{i n \theta}$

We now solve for the function $\rho_{n}(r)$. Using Laplace's equation in polar coordinates we obtain

$\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}r^{2}}+\frac{1}{r} \frac{\mathrm{d}\rho_{n}}{\mathrm{d}r}-\left( \frac{n^{2}}{r^{2}}+k^{2}\right) \rho_{n}=0.$

We can convert this equation to the standard form by substituting $y=k r$ (provided that $\mu\neq 0$to obtain

$y^{2}\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}y^{2}}+y\frac{\mathrm{d}\rho_{n} }{\rm{d}y}-(n^{2}+y^{2})\rho_{n}=0$

The solution of this equation is a linear combination of the modified Bessel functions of order $n$, $I_{n}(y)$ and $K_{n}(y)$ (Abramowitz and Stegun 1964).

Therefore

$\rho_n(r) = C_1 I_{n}(kr) + C_2 K_{n}(kr)\,$

for some constants $C_1$ and $C_2$

Since the solution must be bounded we know that under the plate the solution will be a linear combination of $I_{n}(y)$ while outside the plate the solution will be a linear combination of $K_{n}(y)$. The case $\mu_1 =0$ is a special case and the solution under the dock is $(r/a)^{|n|}$.

Therefore the potential can be expanded as

$\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}a_{mn}K_{n} (k_{m}r)e^{i n\theta}\phi_{m}(z), \;\;r>a$

and

$\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}b_{1n}(r/a)^{|n|} e^{i n\theta}\chi_{1}(z)+ \sum_{n=-\infty}^{\infty}\sum_{m=0,m\neq 1}^{\infty}b_{mn} I_{n}(\kappa_{m}r)e^{i n\theta}\chi_{m}(z), \;\;r<a$

where $a_{mn}$ and $b_{mn}$ are the coefficients of the potential in the open water and the plate covered region respectively.

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

$\phi^{\mathrm{I}} =e^{k_{0}x}\phi_{0}\left( z\right) =\sum\limits_{n=-\infty}^{\infty} I_{n}(k_{0}r)\phi_{0}\left(z\right) e^{i n \theta}$

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 $r=a$ have to be equal for each angle and we obtain

$I_{n}(k_{0}a)\phi_{0}\left( z\right) + \sum_{m=0}^{\infty} a_{mn} K_{n}(k_{m}a)\phi_{m}\left( z\right) = b_{1n} \chi_{0}(z) +\sum_{m=0,m\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\chi_{m}(z)$

and

$k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum _{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right) =b_{1n} \frac{|n|}{a} \chi_{0}(z) +\sum_{m=0,m\neq 1}^{\infty} b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\chi_{m}(z)$

for each $n$. We solve these equations by multiplying both equations by $\phi_{l}(z)$ and integrating from $-h$ to $0$ to obtain:

$I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l} =b_{1n}B_{0l} + \sum_{m=0,m\neq 1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml}$

and

$k_{0}I_{n}^{\prime}(k_{0}a)A_{0}\delta_{0l}+a_{ln}k_{l}K_{n}^{\prime }(k_{l}a)A_{l} = b_{1n}B_{0l}\frac{|n|}{a} + \sum_{m=0,m\neq 1}^{\infty}b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)B_{ml}$

Numerical Solution

To solve the system of equations we set the upper limit of $l$ to be $M$.

Matlab Code

A program to calculate the coefficients for circular dock problems can be found here circle_submerged_dock_matching_one_n.m Note that this problem solves only for a single n.

Additional code

This program requires dispersion_free_surface.m to run