Difference between revisions of "Free-Surface Green Function"
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| − | The Free-Surface Green function is a function which satisfies the following equation (in | + | The Free-Surface Green function is a function which satisfies the following equation (in [[Finite Depth]]) |
| − | <math> \ | + | <math> |
| + | \nabla_{\mathbf{y}}^{2}G(\mathbf{x},\mathbf{y})=\delta(\mathbf{x}-\mathbf{y}), \, -\infty<z<0 | ||
| + | </math> | ||
| + | |||
| + | <math> | ||
| + | \frac{\partial G}{\partial z}=0, \, z=h, | ||
| + | </math> | ||
| + | |||
| + | <math> | ||
| + | \frac{\partial G}{\partial z} = k_{\infty}\phi,\,z\in\Gamma_s, | ||
| + | </math> | ||
| + | |||
| + | <math> | ||
| + | \frac{\partial G}{\partial z} = L\phi, \, z\in\Gamma_w. | ||
| + | </math> | ||
| + | |||
| + | where <math>k_{\infty}</math> is the wavenumber in [[Infinite Depth]] which is given by | ||
| + | <math>k_{\infty}=\omega^2/g</math> where <math>g</math> is gravity. | ||
| + | |||
| + | Many expressions for the Green function have been given. In two dimensions it can be written | ||
| + | as | ||
| + | |||
| + | In three dimensions and infinite depth the Green function <math>G</math>, for <math>r>0</math>, was | ||
| + | given by {\em Havelock\/} \cite{havelock55} as | ||
| + | |||
| + | <math> | ||
| + | G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2} e^{\alpha (z+c)} | ||
| + | \, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1} | ||
| + | - \frac{1}{\pi^2} \int\limits_{0}^{\infty} | ||
| + | \frac{\alpha}{\eta^2 + \alpha^2} \big( \alpha \cos \eta (z+c) - \eta \sin | ||
| + | \eta (z+c) \big) K_0(\eta r) d\eta, | ||
| + | </math> | ||
| + | |||
| + | where <math>H^{(1)}_0</math> and <math>K_0</math> denote the Hankel function of the first | ||
| + | kind and the modified Bessel function of the second kind, both of | ||
| + | order zero as defined in {\em Abramowitz \& Stegun} | ||
| + | \cite{abr_ste}. This Green function will be referred to as {\em | ||
| + | Havelock\/}'s Green function. It should be noted that {\em Havelock\/}'s Green | ||
| + | function can also be written in the following closely related form, | ||
| + | |||
| + | <math> | ||
| + | G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2} e^{\alpha (z+c)} | ||
| + | \, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0} | ||
| + | + \frac{1}{2 \pi^2} \int\limits_{0}^{\infty} | ||
| + | \frac{(\eta^2 - \alpha^2) \cos \eta (z+c) + 2 \eta \alpha \sin | ||
| + | \eta (z+c)}{\eta^2 + \alpha^2} K_0(\eta r) d\eta | ||
| + | </math> | ||
| + | |||
| + | \cite{linton01}. An equivalent representation is due to {\em Kim} | ||
| + | \cite{kim65} for <math>r>0</math>, although implicitly given in the work of {\em | ||
| + | Havelock} \cite{havelock55}, and is given by | ||
| + | |||
| + | <math> | ||
| + | G(\mathbf{x};\mathbf{\xi}) = \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1} | ||
| + | - \frac{\alpha}{4} e^{\alpha (z+c)} \Big( &\mathbf{H}_0(\alpha r) + | ||
| + | Y_0(\alpha r) - 2i J_0 (\alpha r) + \frac{2}{\pi} | ||
| + | \int\limits_{z+c}^0 \frac{e^{-\alpha \eta}}{\sqrt{r^2 + \eta^2}} | ||
| + | d\eta \Big), | ||
| + | </math> | ||
| + | |||
| + | where <math>J_0</math> and <math>Y_0</math> are the Bessel functions of order zero | ||
| + | of the first and second | ||
| + | kind and <math>\mathbf{H}_0</math> is the Struve function of order zero. | ||
Revision as of 09:36, 23 May 2006
The Free-Surface Green function is a function which satisfies the following equation (in Finite Depth)
[math]\displaystyle{ \nabla_{\mathbf{y}}^{2}G(\mathbf{x},\mathbf{y})=\delta(\mathbf{x}-\mathbf{y}), \, -\infty\lt z\lt 0 }[/math]
[math]\displaystyle{ \frac{\partial G}{\partial z}=0, \, z=h, }[/math]
[math]\displaystyle{ \frac{\partial G}{\partial z} = k_{\infty}\phi,\,z\in\Gamma_s, }[/math]
[math]\displaystyle{ \frac{\partial G}{\partial z} = L\phi, \, z\in\Gamma_w. }[/math]
where [math]\displaystyle{ k_{\infty} }[/math] is the wavenumber in Infinite Depth which is given by [math]\displaystyle{ k_{\infty}=\omega^2/g }[/math] where [math]\displaystyle{ g }[/math] is gravity.
Many expressions for the Green function have been given. In two dimensions it can be written as
In three dimensions and infinite depth the Green function [math]\displaystyle{ G }[/math], for [math]\displaystyle{ r\gt 0 }[/math], was given by {\em Havelock\/} \cite{havelock55} as
[math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2} e^{\alpha (z+c)} \, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1} - \frac{1}{\pi^2} \int\limits_{0}^{\infty} \frac{\alpha}{\eta^2 + \alpha^2} \big( \alpha \cos \eta (z+c) - \eta \sin \eta (z+c) \big) K_0(\eta r) d\eta, }[/math]
where [math]\displaystyle{ H^{(1)}_0 }[/math] and [math]\displaystyle{ K_0 }[/math] denote the Hankel function of the first kind and the modified Bessel function of the second kind, both of order zero as defined in {\em Abramowitz \& Stegun} \cite{abr_ste}. This Green function will be referred to as {\em Havelock\/}'s Green function. It should be noted that {\em Havelock\/}'s Green function can also be written in the following closely related form,
[math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) = \frac{i \alpha}{2} e^{\alpha (z+c)} \, H_0^{(1)}(\alpha r) + \frac{1}{4 \pi R_0} + \frac{1}{2 \pi^2} \int\limits_{0}^{\infty} \frac{(\eta^2 - \alpha^2) \cos \eta (z+c) + 2 \eta \alpha \sin \eta (z+c)}{\eta^2 + \alpha^2} K_0(\eta r) d\eta }[/math]
\cite{linton01}. An equivalent representation is due to {\em Kim} \cite{kim65} for [math]\displaystyle{ r\gt 0 }[/math], although implicitly given in the work of {\em Havelock} \cite{havelock55}, and is given by
[math]\displaystyle{ G(\mathbf{x};\mathbf{\xi}) = \frac{1}{4 \pi R_0} + \frac{1}{4 \pi R_1} - \frac{\alpha}{4} e^{\alpha (z+c)} \Big( &\mathbf{H}_0(\alpha r) + Y_0(\alpha r) - 2i J_0 (\alpha r) + \frac{2}{\pi} \int\limits_{z+c}^0 \frac{e^{-\alpha \eta}}{\sqrt{r^2 + \eta^2}} d\eta \Big), }[/math]
where [math]\displaystyle{ J_0 }[/math] and [math]\displaystyle{ Y_0 }[/math] are the Bessel functions of order zero of the first and second kind and [math]\displaystyle{ \mathbf{H}_0 }[/math] is the Struve function of order zero.
