# Two Identical Submerged Docks using Symmetry

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## Introduction

This is the extension of Eigenfunction Matching for a Submerged Finite Dock using Symmetry in Two Dimensions.. The full theory is not presented here, and details of the matching method can be found in Eigenfunction Matching for a Submerged Semi-Infinite Dock and Two Identical Docks using Symmetry

## Governing Equations

We begin with the Frequency Domain Problem for the submerged dock in the region $x\gt 0$ (we assume $e^{i\omega t}$ 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

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

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

$\partial_z\phi=\alpha\phi, \,\, z=0,$

$\partial_z\phi=0, \,\, z=-d,\,-L_2\lt x\lt -L_1,\,\,{\rm and}\,\,L_1\lt x\lt L_2$

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, $x\lt 0$ $-d\lt z\lt 0,\,\,x\gt 0$, and $-h\lt z\lt -d,\,\,x\gt 0$. The first two regions use the free-surface eigenfunction and the third uses the dock eigenfunctions. Details can be found in Eigenfunction Matching for a Semi-Infinite Dock.

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

$\phi^{\mathrm{I}} =e^{-k^{h}_{0}(x+L_2)}\phi_{0}\left( z\right)$

We use Symmetry in Two Dimensions and express the symmetric solution as

$\phi(x,z)=e^{-k_{0}^h (x+L_2)}\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}a_{m}^{s}e^{k_{m}^h (x+L_2)}\phi_{m}^h(z), \;\;x\lt -L_2$

$\phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{s} e^{-k_{m}^d (x+L_2)}\phi_{m}^d(z) + \sum_{m=0}^{\infty}c_{m}^{s} e^{k_{m}^d (x+L_1)}\phi_{m}^d(z) , \;\;-d\lt z\lt 0,\,\,x\lt -L_2,\,-L_1\lt x\lt L_1, {\rm or} \, x\gt L_2$

and

$\phi(x,z)= d_0^{s}\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{s} e^{\kappa_{m} (x+L_2)}\psi_{m}(z) + e_0^{s}\frac{x+L_2}{L_2-L_1} \sum_{m=0}^{\infty}e_{m}^{s} e^{-\kappa_{m} (x+L_1)}\psi_{m}(z) , \;\;-h\lt z\lt -d,\,\,\,-L_2\lt x\lt -L_1, {\rm or} \, L_1\lt x\lt L_2,$

$\phi(x,z)= \sum_{m=0}^{\infty}f_{m}^{s}\frac{\cosh(k_{m}^h x)}{\cosh(k_m^h L)}\phi_{m}^h(z), \;\;x\gt L$

The definition of all terms can be found in Eigenfunction Matching for Submerged Semi-Infinite Dock, as can the solution method and the method to extend the solution to waves incident at an angle.

The anti-symmetric solution is

$\phi(x,z)=e^{-k_{0}^h (x+L_2)}\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}a_{m}^{a}e^{k_{m}^h (x+L_2)}\phi_{m}^h(z), \;\;x\lt -L_2$

$\phi(x,z)= \sum_{m=0}^{\infty}b_{m}^{a} e^{-k_{m}^d (x+L_2)}\phi_{m}^d(z) + \sum_{m=0}^{\infty}c_{m}^{a} e^{k_{m}^d (x+L_1)}\phi_{m}^d(z) , \;\;-d\lt z\lt 0,\,\,x\lt -L_2,\,-L_1\lt x\lt L_1, {\rm or} \, x\gt L_2$

and

$\phi(x,z)= d_0^{a}\frac{x+L_1}{L_2-L_1} + \sum_{m=1}^{\infty}d_{m}^{a} e^{-\kappa_{m} (x+L_2)}\psi_{m}(z) + e_0^{a}\frac{x+L_2}{L_2-L_1} \sum_{m=1}^{\infty}e_{m}^{a} e^{\kappa_{m} (x+L_1)}\psi_{m}(z) , \;\;-h\lt z\lt -d,\,\,\,-L_2\lt x\lt -L_1, {\rm or} \, L_1\lt x\lt L_2,$

$\phi(x,z)= \sum_{m=0}^{\infty}f_{m}^{a}\frac{\sinh(k_{m}^h x)}{\sinh(k_m^h L)}\phi_{m}^h(z), \;\;x\gt L$

## Matlab Code

A program to calculate the coefficients for the submerged two finite dock problem can be found here two_submerged_finite_docks_symmetry.m