Eigenfunction Matching for a Submerged Circular Dock

Introduction

We present here very briefly the theory for a submerged circular dock. The details of the method can be found in Eigenfunction Matching for a Submerged Semi-Infinite Dock and Eigenfunction Matching for a Circular Dock

Governing Equations

We begin with the Frequency Domain Problem. We will use a cylindrical coordinate system, [math]\displaystyle{ (r,\theta,z) }[/math], assumed to have its origin at the centre of the circular plate which has radius [math]\displaystyle{ a }[/math]. 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{ \phi_{z}=\alpha\phi, \,\, z=0, }[/math]

[math]\displaystyle{ \phi_{z}=0, \,\, z=-d,\,r\lt a }[/math]

We must also apply the Sommerfeld Radiation Condition as [math]\displaystyle{ r\rightarrow\infty }[/math]. The subscript [math]\displaystyle{ z }[/math] denotes the derivative in [math]\displaystyle{ z }[/math]-direction.

Separation of Variables

We can separate variables and we obtain the following expression for the velocity potential

Therefore the potential can be expanded as

[math]\displaystyle{ \phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}a_{mn}K_{n} (k_{m}^{h}r)e^{i n\theta}\phi^{h}_{m}(z), \;\;r\gt a }[/math]

and

[math]\displaystyle{ \phi(r,\theta,z)= \sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}b_{mn} I_{n}(k_{m}^{\prime}r)e^{i n\theta}\chi_{m}(z), \;\;r\lt a }[/math]

where the definition of [math]\displaystyle{ k_{m}^{h} }[/math], [math]\displaystyle{ k_{m}^{\prime} }[/math], [math]\displaystyle{ \phi^{h}_{m} }[/math] and [math]\displaystyle{ \chi_{m}(z) }[/math] can be found in Eigenfunction Matching for a Submerged Semi-Infinite Dock and the expansion in the cylindrical eigenfunctions can be found in Eigenfunction Matching for a Circular Dock

Equations to solve

[math]\displaystyle{ I_{n}(k_{0}^{h}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}^{h}a)A_{l} =\sum_{m=0}^{\infty}b_{mn}I_{n}(k_{m}^{\prime}a)B_{ml}^{\prime} }[/math]

and

[math]\displaystyle{ k_{0}I_{n}^{\prime}(k_{0}^{h}a)A_{0}\delta_{0l}+a_{ln}k_{l}^{h}K_{n}^{\prime }(k_{l}a)A_{l} = \sum_{m=0}^{\infty}b_{mn}k_{m}^{\prime}I_{n}^{\prime}(k_{m}^{\prime}a)B_{ml}^{\prime} }[/math]

where the definition of [math]\displaystyle{ A_{l} }[/math] and [math]\displaystyle{ B_{ml}^{\prime} }[/math] can be found in Eigenfunction Matching for a Submerged Semi-Infinite Dock

To solve the system of equations we set the upper limit of [math]\displaystyle{ l }[/math] to be [math]\displaystyle{ M }[/math].

Matlab Code

A program to calculate the coefficients for circular dock problems can be found here circle_submerged_dock_matching_one_n.m Note that this problem solves only for a single n.

Additional code

This program requires dispersion_free_surface.m to run