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 [math]x\gt0[/math] (we assume [math]e^{i\omega t}[/math] time dependence). The water is assumed to have constant finite depth [math]h[/math] and the [math]z[/math]-direction points vertically upward with the water surface at [math]z=0[/math] and the sea floor at [math]z=-h[/math]. The boundary value problem can therefore be expressed as

[math] \Delta\phi=0, \,\, -h\ltz\lt0, [/math]

[math] \phi_{z}=0, \,\, z=-h, [/math]

[math] \partial_z\phi=\alpha\phi, \,\, z=0, [/math]

[math] \partial_z\phi=0, \,\, z=-d,\,-L_2\ltx\lt-L_1,\,\,{\rm and}\,\,L_1\ltx\ltL_2 [/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 three regions, [math]x\lt0[/math] [math]-d\ltz\lt0,\,\,x\gt0[/math], and [math]-h\ltz\lt-d,\,\,x\gt0[/math]. 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 [math]A[/math] in displacement travelling in the positive [math]x[/math]-direction. The incident potential can therefore be written as

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

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

[math] \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 [/math]

[math] \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\ltz\lt0,\,\,x\lt-L_2,\,-L_1\ltx\ltL_1, {\rm or} \, x\gtL_2 [/math]

and

[math] \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\ltz\lt-d,\,\,\,-L_2\ltx\lt-L_1, {\rm or} \, L_1\ltx\ltL_2, [/math]

[math] \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\gtL [/math]

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

[math] \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 [/math]

[math] \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\ltz\lt0,\,\,x\lt-L_2,\,-L_1\ltx\ltL_1, {\rm or} \, x\gtL_2 [/math]

and

[math] \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\ltz\lt-d,\,\,\,-L_2\ltx\lt-L_1, {\rm or} \, L_1\ltx\ltL_2, [/math]

[math] \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\gtL [/math]

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

Additional code

This program requires