Free-Surface Green Function
The Free-Surface Green function is one of the most important objects in linear water wave theory. It forms the basis on many of the numerical solutions, especially for bodies of arbitrary geometry. It first appeared in John 1949 and John 1950. It is based on the Frequency Domain Problem. The exact form of the Green function depends on whether we assume the solution is proportional to or . It is the fundamental tool for the Green Function Solution Method There are many different representations for the Green function.
Equations for the Green function
The Free-Surface Green function is a function which satisfies the following equation (in Finite Depth)
where is the wavenumber in Infinite Depth which is given by where is gravity. We also require a condition as which is the Sommerfeld Radiation Condition. This depends on whether we assume that the solution is proportional to or . We assume through out this.
We define and
Two Dimensional Representations
Many expressions for the Green function have been given. We present here a derivation for finite depth based on an Eigenfunction Matching Method. We write the Green function as
are the roots of the Dispersion Relation for a Free Surface
with being purely imaginary with negative imaginary part and are purely real with positive real part ordered with increasing size. is chosen so that the eigenfunctions are orthonormal, i.e.,
and are given by
The Green function as written needs to only satisfy the condition
We can expand the delta function as
Therefore we can derive the equation
so that we must solve
This has solution
The Green function can therefore be written as
It can be written using the expression for as
We can use the Dispersion Relation for a Free Surface which the roots satisfy to show that and so that we can write the Green function in the following forms
Incident at an angle
In some situations the potential may have a simple dependence (so that it is pseudo two-dimensional). This is used to allow waves to be incident at an angle. We require the Green function to satisfy the following equation
The Green function can be derived exactly as before except we have to include
The Green function for infinite depth can be derived from the expression for finite depth by taking the limit as and converting the sum to an integral using the Riemann sum. Alternatively, the expression can be derived using Fourier Transform Mei 1983
Solution for the singularity at the Free-Surface
We can also consider the following problem
It turns out that the solution to this is nothing more than the Green function we found previously restricted to the free surface, i.e.
Code to calculate the Green function in two dimensions without incident angle for point source and field point on the free surface can be found here two_d_finite_depth_Green_surface.m
Three Dimensional Representations
Let be cylindrical coordinates such that
and let and denote the distance from the source point and the distance from the mirror source point respectively, and .
The most important representation of the finite depth free surface Green function is the eigenfunction expansion given by John 1950. He wrote the Green function in the following form
where and denote the Hankel function of the first kind and the modified Bessel function of the second kind, both of order zero as defined in Abramowitz and Stegun 1964, is the positive real solution to the Dispersion Relation for a Free Surface and are the imaginary parts of the solutions with positive imaginary part. This way of writing the equation was primarily to avoid complex values for the Bessel functions, however most computer packages will calculate Bessel functions for complex argument so it makes more sense to write the Green function in the following form
where are as before except .
An expression where both variables are given in cylindrical polar coordinates is the following
In three dimensions and infinite depth the Green function , for , was given by Havelock 1955 as
It should be noted that this Green function can also be written in the following closely related form,
where and are the Bessel functions of order zero of the first and second kind and is the Struve function of order zero.
The expression due to Peter and Meylan 2004 is