Eigenfunction Matching for a Finite Dock
The problems consists of a region to the left and right with a free surface and a middle region with a rigid surface through which not flow is possible. We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We then consider the case when the waves are incident at an angle. For the later we give the equations in slightly less detail. The theory is based on Eigenfunction Matching for a Semi-Infinite Dock and this should be consulted for many details. The solution here can be straightforwardly extended using Symmetry in Two Dimensions to two docks of the same length and this can be found Two Identical Docks using Symmetry. We also show how the solution can be found using Symmetry in Two Dimensions for the finite dock in Eigenfunction Matching for a Finite Dock using Symmetry.
We consider here the Frequency Domain Problem for a finite dock which occupies the region (we assume time dependence). The water is assumed to have constant finite depth and the -direction points vertically upward with the water surface at and the sea floor at . The boundary value problem can therefore be expressed as
We must also apply the Sommerfeld Radiation Condition as . 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.
Separation of variables for a free surface
We use separation of variables
We express the potential as
and then Laplace's equation becomes
The separation of variables equation for deriving free surface eigenfunctions is as follows:
subject to the boundary conditions
We can then use the boundary condition at to write
where we have chosen the value of the coefficent so we have unit value at . The boundary condition at the free surface () gives rise to:
which is the Dispersion Relation for a Free Surface
The above equation is not really the dispersion relation for a free surface, it would be better to refer to it as a transcendental equation. If we solve for all roots in the complex plane we find that the first root is a pair of imaginary roots. We denote the imaginary solutions of this equation by and the positive real solutions by , . The of the imaginary solution is the wavenumber. We put the imaginary roots back into the equation above and use the hyperbolic relations
to arrive at the dispersion relation
We note that for a specified frequency the equation determines the wavenumber .
Finally we define the function as
as the vertical eigenfunction of the potential in the open water region. From Sturm-Liouville theory the vertical eigenfunctions are orthogonal. They can be normalised to be orthonormal, but this has no advantages for a numerical implementation. It can be shown that
Separation of Variables for a Dock
The separation of variables equation for a floating dock
subject to the boundary conditions
The solution is , and
We note that
Inner product between free surface and dock modes
To create meaningful solutions of the velocity potential in the specified domains we add an incident wave term to the expansion for the domain of above. The incident potential is a wave of amplitude in displacement travelling in the positive -direction. We would only see this in the time domain however, in the frequency domain the incident potential can be written as
The total velocity (scattered) potential now becomes for the domain of .
The first term in the expansion of the diffracted potential for the domain is given by
which represents the reflected wave.
In any scattering problem where and are the reflection and transmission coefficients respectively. In our case of the semi-infinite dock and as there are no transmitted waves in the region under the dock.
Expansion of the Potential
The potential can be expanded as
where and are the coefficients of the potential in the open water regions to the left and right and and are the coefficients under the dock covered region. We have an incident wave from the left.
An infinite dimensional system of equations
The potential and its derivative must be continuous across the transition from open water to the plate covered region. Therefore, the potentials and their derivatives at have to be equal. We obtain
We solve these equations by multiplying both equations by and integrating from to to obtain:
To solve the system of equations we set the upper limit of to be . We then simply need to solve the linear system of equations.
Solution with Waves Incident at an Angle
We can consider the problem when the waves are incident at an angle . When a wave in incident at an angle we have the wavenumber in the direction is where is as defined previously (note that is imaginary).
This means that the potential is now of the form so that when we separate variables we obtain
where is the separation constant calculated without an incident angle.
Therefore the potential can be expanded as
where and where we always take the positive real root or the root with positive imaginary part.
The equations are derived almost identically to those above and we obtain
and these are solved exactly as before.
A program to calculate the coefficients for the finite dock problems can be found here finite_dock.m
This program requires