Two Identical Submerged Docks using Symmetry

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Contents

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 > 0 (we assume eiωt 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<z<0,

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

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

\partial_z\phi=0, \,\, z=-d,\,-L_2<x<-L_1,\,\,{\rm and}\,\,L_1<x<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 < 0 -d<z<0,\,\,x>0, and -h<z<-d,\,\,x>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<-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<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>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<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<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>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<-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<z<0,\,\,x<-L_2,\,-L_1<x<L_1, {\rm or} \, x>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<z<-d,\,\,\,-L_2<x<-L_1, {\rm or} \, L_1<x<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>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

Additional code

This program requires

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