Eigenfunction Matching for a Submerged Finite Dock

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Introduction

This is the finite length version of the Eigenfunction Matching for a Submerged Semi-Infinite Dock. 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 Eigenfunction Matching for a Finite Dock

Governing Equations

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

[math]\displaystyle{ \Delta\phi=0, \,\, -h\lt z\lt 0, }[/math]

[math]\displaystyle{ \partial_{z} \phi=0, \,\, z=-h, }[/math]

[math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0, }[/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 four regions, {[math]\displaystyle{ x\lt -L \, }[/math]}, {[math]\displaystyle{ x\gt L \, }[/math]}, {[math]\displaystyle{ -d\lt z\lt 0,\,\,-L\lt x\lt L }[/math]}, and {[math]\displaystyle{ -h\lt z\lt -d,\,\,-L\lt x\lt L }[/math]}. The first three regions use the free-surface eigenfunction and the last uses dock eigenfunctions. Details can be found in Eigenfunction Matching for a Semi-Infinite Dock.

The incident potential is a wave of amplitude [math]\displaystyle{ A }[/math] in displacement travelling in the positive [math]\displaystyle{ x }[/math]-direction. The incident potential can therefore be written as

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

The potential can be expanded as

[math]\displaystyle{ \phi(x,z)=e^{-k_{0}^h (x+L)}\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty}a_{m}e^{k_{m}^h (x+L)}\phi_{m}^h(z), \;\;x\lt -L }[/math]

[math]\displaystyle{ \phi(x,z)= \sum_{m=0}^{\infty}b_{m} e^{-k_{m}^d (x+L)}\phi_{m}^d(z) + \sum_{m=0}^{\infty}c_{m} e^{k_{m}^d (x-L)}\phi_{m}^d(z) , \;\;-d\lt z\lt 0,\,\,-L\lt x\lt L }[/math]

and

[math]\displaystyle{ \phi(x,z)= d_0 \frac{L-x}{2 L} + \sum_{m=1}^{\infty}d_{m} e^{-\kappa_{m} (x+L)}\psi_{m}(z) + e_0 \frac{x+L}{2 L} + \sum_{m=1}^{\infty}e_{m} e^{\kappa_{m} (x-L)}\psi_{m}(z) , \;\;-h\lt z\lt -d,\,\,-L\lt x\lt L }[/math]

[math]\displaystyle{ \phi(x,z)= \sum_{m=0}^{\infty}f_{m}e^{-k_{m}^h (x-L)}\phi_{m}^h(z), \;\;L\lt x }[/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.

Matlab Code

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

Additional code

This program requires

• dispersion_free_surface.m