Eigenfunction Matching for a Submerged Finite Dock

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

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

[math]\displaystyle{ \partial_z\phi=0, \,\, z=-d,\,x\gt 0, }[/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 three regions, [math]\displaystyle{ x\lt 0 }[/math] [math]\displaystyle{ -d\lt z\lt 0,\,\,x\gt 0 }[/math], and [math]\displaystyle{ -h\lt z\lt -d,\,\,x\gt 0 }[/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]\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}x}\phi_{0}\left( z\right) }[/math]

The potential can be expanded as

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

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

and

[math]\displaystyle{ \phi(x,z)= \sum_{m=0}^{\infty}c_{m} e^{\kappa_{m} x}\psi_{m}(z) , \;\;-h\lt z\lt -d,\,\,x\gt 0 }[/math]

where [math]\displaystyle{ a_{m} }[/math] and [math]\displaystyle{ b_{m} }[/math] are the coefficients of the potential in the open water regions to the left and right and [math]\displaystyle{ c_m }[/math] are the coefficients under the dock covered region. We have an incident wave from the left. [math]\displaystyle{ k_n^l }[/math] are the roots of the Dispersion Relation for a Free Surface

[math]\displaystyle{ k \tan(kl) = -\alpha\, }[/math]

We denote the

positive imaginary solutions by [math]\displaystyle{ k_{0}^l }[/math] and the positive real solutions by [math]\displaystyle{ k_{m}^l }[/math], [math]\displaystyle{ m\geq1 }[/math] (ordered with increasing imaginary part) and [math]\displaystyle{ \kappa_{m}=m\pi/(h-d) }[/math]. We define

[math]\displaystyle{ \phi_{m}^l\left( z\right) = \frac{\cos k_{m}(z+l)}{\cos k_{m}l},\quad m\geq0 }[/math]

as the vertical eigenfunction of the potential in the open water regions and

[math]\displaystyle{ \psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad m\geq 0 }[/math]

as the vertical eigenfunction of the potential in the dock covered region. We define

[math]\displaystyle{ \int\nolimits_{-d}^{0}\phi_{m}^d(z)\phi_{n}^d(z) d z=A_{m}\delta_{mn} }[/math]

where

[math]\displaystyle{ A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}d\sin k_{m}d+k_{m}d}{k_{m}\cos ^{2}k_{m}l}\right) }[/math]

[math]\displaystyle{ \int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) d z=B_{mn} }[/math]

and

[math]\displaystyle{ \int\nolimits_{-h}^{-d}\phi_{n}^h(z)\psi_{m}(z) d z=C_{mn} }[/math]

and

[math]\displaystyle{ \int\nolimits_{-h}^{-d}\psi_{m}(z)\psi_{n}(z) d z=D_{m}\delta_{mn} }[/math]

where

[math]\displaystyle{ D_{m}=\frac{1}{2}(h-d),\quad,m\neq 0 \quad \mathrm{and} \quad D_0 = (h-d) }[/math]

An infinite dimensional system of equations

The potential and its derivative must be continuous across the transition from open water to the dock region. Therefore, the potentials and their derivatives at [math]\displaystyle{ x=0 }[/math] have to be equal. We obtain

[math]\displaystyle{ \phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty} a_{m} \phi_{m}^h\left( z\right) =\sum_{m=0}^{\infty}b_{m}\phi_{m}^d(z),\,\,\,-d\lt z\lt 0 }[/math]

[math]\displaystyle{ \phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty} a_{m} \phi_{m}^h\left( z\right) =\sum_{m=0}^{\infty}c_{m}\psi_{m}(z),\,\,\,-h\lt z\lt -d }[/math]

[math]\displaystyle{ -k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty} k_m^h a_{m} \phi_{m}^h\left( z\right) =\sum_{m=0}^{\infty}k_m^d b_{m}\phi_{m}^d(z),\,\,\,-d\lt z\lt 0 }[/math]

[math]\displaystyle{ -k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty} k_m^h a_{m} \phi_{m}^h\left( z\right) =\sum_{m=0}^{\infty}\kappa_m c_{m}\psi_{m}(z),\,\,\,-h\lt z\lt -d }[/math]

Numerical Solution

The standard method to solve these equations (from Linton and Evans 1991) is to mutiply both equations by [math]\displaystyle{ \phi_{q}^d(z) }[/math] and integrating from [math]\displaystyle{ -d }[/math] to [math]\displaystyle{ 0 }[/math] or by multiplying both equations by [math]\displaystyle{ \psi_{r}(z) }[/math] and integrating from [math]\displaystyle{ -h }[/math] to [math]\displaystyle{ -d }[/math]. However, we use a different method, which is closer to the solution method for Eigenfunction Matching for a Semi-Infinite Dock which allows us to keep the computer code similar. These is no significant difference between the methods numerically and a close connection exists.

We truncate the sum to [math]\displaystyle{ N+1 }[/math] modes and introduce a new function

[math]\displaystyle{ \chi_n = \begin{cases} \psi_{n}(z),\,\,\,-h\lt z\lt -d \\ 0,\,\,\,-d\lt z\lt 0 \end{cases} }[/math]

for [math]\displaystyle{ 0 \leq n \leq M - 1 }[/math]

[math]\displaystyle{ \chi_{n+M} = \begin{cases} 0,\,\,\,-h\lt z\lt -d \\ \phi_{n}^{d}(z),\,\,\,-d\lt z\lt 0 \end{cases} }[/math]

for [math]\displaystyle{ 0 \leq n \leq N-M }[/math] and we choose the values of [math]\displaystyle{ N }[/math] so that we have the [math]\displaystyle{ N+1 }[/math] smallest values of [math]\displaystyle{ k_n }[/math] and [math]\displaystyle{ \kappa_n }[/math] (with the proviso that we have at least one from each).

We truncate the equations and write

[math]\displaystyle{ \phi_{0}^h\left( z\right) + \sum_{m=0}^{N} a_{m} \phi_{m}^h\left( z\right) =\sum_{m=0}^{N}b_{m} \chi_m, }[/math]

[math]\displaystyle{ -k_0^h\phi_{0}^h\left( z\right) + \sum_{m=0}^{\infty} k_m^h a_{m} \phi_{m}^h\left( z\right) =\sum_{m=0}^{N}k^{\prime}_m b_{m}\chi_{m} }[/math]

where [math]\displaystyle{ k^{\prime}_m }[/math] is either [math]\displaystyle{ k^{d}_q }[/math] or [math]\displaystyle{ \kappa_q }[/math] as appropriate.

We multiply each equation by [math]\displaystyle{ \phi_{q}^h(z) }[/math] and integrating from [math]\displaystyle{ -h }[/math] to [math]\displaystyle{ 0 }[/math] to obtain

[math]\displaystyle{ A_{0}\delta_{0q} + a_{q}A_{q} = \sum_{m=0}^{N} b_m B^{\prime}_{mq} }[/math]

[math]\displaystyle{ -k_{0}^h A_{0}\delta_{0q} + k_{q}^h a_{q}A_{q} = \sum_{m=0}^{N} k^{\prime}_m b_m B^{\prime}_{mq} }[/math]

where [math]\displaystyle{ B^{\prime}_{mq} }[/math] is made from [math]\displaystyle{ B_{mq} }[/math] or [math]\displaystyle{ C_{mq} }[/math] as appropriate.

Solution with Waves Incident at an Angle

We can consider the problem when the waves are incident at an angle [math]\displaystyle{ \theta }[/math] but this is not presented here. For details see Eigenfunction Matching for a Semi-Infinite Dock.

Matlab Code

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

Additional code

This program requires dispersion_free_surface.m to run