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 $x\gt 0$ (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

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

$\partial_{z} \phi=0, \,\, z=-h,$

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

$\partial_z\phi=0, \,\, z=-d,\,-L\lt x\lt L,$

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 four regions, {$x\lt -L \,$}, {$x\gt L \,$}, {$-d\lt z\lt 0,\,\,-L\lt x\lt L$}, and {$-h\lt z\lt -d,\,\,-L\lt x\lt L$}. 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 $A$ in displacement travelling in the positive $x$-direction. The incident potential can therefore be written as

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

The potential can be expanded as

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

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

and

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

$\phi(x,z)= \sum_{m=0}^{\infty}f_{m}e^{-k_{m}^h (x-L)}\phi_{m}^h(z), \;\;L\lt x$

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