Free-Surface Green Function - Revision history

Adi Kurniawan at 08:57, 19 August 2010

2010-08-19T08:57:18Z

← Older revision		Revision as of 08:57, 19 August 2010
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	& - \frac{1}{\pi^2} \int\limits_{0}^{\infty}		& - \frac{1}{\pi^2} \int\limits_{0}^{\infty}
	\frac{\alpha }{\eta^2 + \alpha ^2} \big( \alpha \cos \eta (z+c) - \eta \sin		\frac{\alpha }{\eta^2 + \alpha ^2} \big( \alpha \cos \eta (z+c) - \eta \sin
	\eta (z+c) \big) K_0(\eta r) d\eta.		\eta (z+c) \big) K_0(\eta r) \mathrm{d}\eta.
	\end{matrix}		\end{matrix}
	</math>		</math>
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	& + \frac{1}{2 \pi^2} \int\limits_{0}^{\infty}		& + \frac{1}{2 \pi^2} \int\limits_{0}^{\infty}
	\frac{(\eta^2 - \alpha ^2) \cos \eta (z+c) + 2 \eta \alpha \sin		\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		\eta (z+c)}{\eta^2 + \alpha ^2} K_0(\eta r) \mathrm{d}\eta
	\end{matrix}		\end{matrix}
	</math>		</math>
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	Y_0(\alpha r) - 2i J_0 (\alpha r) + \frac{2}{\pi}		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}}		\int\limits_{z+c}^0 \frac{e^{-\alpha \eta}}{\sqrt{r^2 + \eta^2}}
	d\eta \Big),		\mathrm{d}\eta \Big),
	</math>		</math>
	</center>		</center>
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	\eta z + \frac{\alpha }{\eta} \sin \eta z \Big)		\eta z + \frac{\alpha }{\eta} \sin \eta z \Big)
	\frac{\eta^2}{\eta^2+\alpha ^2} \Big( \cos \eta c +		\frac{\eta^2}{\eta^2+\alpha ^2} \Big( \cos \eta c +
	\frac{\alpha }{\eta} \sin \eta c \Big) K_0(\eta r) d\eta.		\frac{\alpha }{\eta} \sin \eta c \Big) K_0(\eta r) \mathrm{d}\eta.
	</math>		</math>
	</center>		</center>

Adi Kurniawan: /* Two Dimensional Representations */

2010-08-19T08:56:12Z

Two Dimensional Representations

← Older revision		Revision as of 08:56, 19 August 2010
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	<math>N_n</math> is chosen so that the eigenfunctions are orthonormal, i.e.,		<math>N_n</math> is chosen so that the eigenfunctions are orthonormal, i.e.,
	<center><math>		<center><math>
	\int_{-h}^{0} f_m(z) f_n(z)dz = \delta_{mn}.\,		\int_{-h}^{0} f_m(z) f_n(z)\mathrm{d}z = \delta_{mn}.\,
	</math></center>		</math></center>
	and are given by		and are given by

Meylan: /* Two Dimensional Representations */

2010-04-29T01:54:00Z

Two Dimensional Representations

← Older revision		Revision as of 01:54, 29 April 2010
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	\alpha +k_n\tan{(k_n h)}= 0\,		\alpha +k_n\tan{(k_n h)}= 0\,
	</math></center>		</math></center>
	with <math>k_0</math> being purely imaginary with ~~positive~~ imaginary part and		with <math>k_0</math> being purely imaginary with negative imaginary part and
	<math>k_n,</math> <math>n\geq 1</math> are purely real with positive real part ordered with		<math>k_n,</math> <math>n\geq 1</math> are purely real with positive real part ordered with
	increasing size.		increasing size.

Meylan: /* Matlab code */

2009-04-03T01:58:31Z

Matlab code

← Older revision		Revision as of 01:58, 3 April 2009
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	Code to calculate the Green function in two dimensions without incident angle for point source and field		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 {{~~green~~ function surface code}}		point on the free surface can be found here {{Green function source and field free surface code}}

	== Three Dimensional Representations ==		== Three Dimensional Representations ==

Meylan at 01:41, 3 April 2009

2009-04-03T01:41:30Z

← Older revision		Revision as of 01:41, 3 April 2009
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	depends on whether we assume the solution is proportional to <math>\exp(i\omega t)</math>		depends on whether we assume the solution is proportional to <math>\exp(i\omega t)</math>
	or <math>\exp(-i\omega t)</math>.		or <math>\exp(-i\omega t)</math>.

	It is the fundamental tool for the [[Green Function Solution Method]]		It is the fundamental tool for the [[Green Function Solution Method]]

	There are many different representations for the Green function.		There are many different representations for the Green function.

Meylan at 01:41, 3 April 2009

2009-04-03T01:41:08Z

← Older revision		Revision as of 01:41, 3 April 2009
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	~~= Introduction =~~

	The Free-Surface Green function is one of the most important objects in linear		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		water wave theory. It forms the basis on many of the numerical solutions, especially
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	There are many different representations for the Green function.		There are many different representations for the Green function.

	= Equations 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]])		The Free-Surface Green function is a function which satisfies the following equation (in [[Finite Depth]])
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	We define <math>\mathbf{x}=(x,y,z)</math> and <math>\mathbf{\xi}=(a,b,c)</math>		We define <math>\mathbf{x}=(x,y,z)</math> and <math>\mathbf{\xi}=(a,b,c)</math>

	= Two Dimensional Representations =		== Two Dimensional Representations ==

	Many expressions for the Green function have been given. We present here a derivation for finite depth based on an [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]]. We write the Green function as		Many expressions for the Green function have been given. We present here a derivation for finite depth based on an [[:Category:Eigenfunction Matching Method\|Eigenfunction Matching Method]]. We write the Green function as
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	and [[Wehausen and Laitone 1960]] is incorrect (by a factor of -1).		and [[Wehausen and Laitone 1960]] is incorrect (by a factor of -1).

	==Incident at an angle ==		=== Incident at an angle ===

	In some situations the potential may have a simple <math>e^{i k_y y}</math> dependence		In some situations the potential may have a simple <math>e^{i k_y y}</math> dependence
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	</center>		</center>

	== Infinite Depth ==		=== Infinite Depth ===

	The Green function for infinite depth can be derived from the expression for finite depth by taking the limit as <math>h\to\infty</math> and converting the sum to an integral using the [http://en.wikipedia.org/wiki/Riemann_Sum Riemann sum]. Alternatively, the expression can be derived using [http://en.wikipedia.org/wiki/Fourier_transform Fourier Transform] [[Mei 1983]]		The Green function for infinite depth can be derived from the expression for finite depth by taking the limit as <math>h\to\infty</math> and converting the sum to an integral using the [http://en.wikipedia.org/wiki/Riemann_Sum Riemann sum]. Alternatively, the expression can be derived using [http://en.wikipedia.org/wiki/Fourier_transform Fourier Transform] [[Mei 1983]]

	== Solution for the singularity at the Free-Surface ==		=== Solution for the singularity at the Free-Surface ===

	We can also consider the following problem		We can also consider the following problem
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	</center>		</center>

	== Matlab code ==		=== Matlab code ===

	Code to calculate the Green function in two dimensions without incident angle for point source and field		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 {{green function surface code}}		point on the free surface can be found here {{green function surface code}}

	= Three Dimensional Representations =		== Three Dimensional Representations ==

	Let <math>(r,\theta)</math> be cylindrical coordinates such that		Let <math>(r,\theta)</math> be cylindrical coordinates such that
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	(z+c)^2</math>.		(z+c)^2</math>.

	==[[Finite Depth]]==		===[[Finite Depth]]===

	The most important representation of the finite depth free		The most important representation of the finite depth free
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	given by [[Black 1975]] and [[Fenton 1978]] and can be derived by applying [[Graf's Addition Theorem]] to <math>K_0(k_m\|r\mathrm{e}^{\mathrm{i}\theta}-s\mathrm{e}^{\mathrm{i}\varphi}\|)</math> in the definition of <math>G(\mathbf{x};\mathbf{\xi})</math> above.		given by [[Black 1975]] and [[Fenton 1978]] and can be derived by applying [[Graf's Addition Theorem]] to <math>K_0(k_m\|r\mathrm{e}^{\mathrm{i}\theta}-s\mathrm{e}^{\mathrm{i}\varphi}\|)</math> in the definition of <math>G(\mathbf{x};\mathbf{\xi})</math> above.

	== [[Infinite Depth]] ==		=== [[Infinite Depth]] ===

	In three dimensions and infinite depth the Green function <math>G</math>, for <math>r>0</math>, was		In three dimensions and infinite depth the Green function <math>G</math>, for <math>r>0</math>, was
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	</math>		</math>
	</center>		</center>

	[[Category:Linear Water-Wave Theory]]		[[Category:Linear Water-Wave Theory]]
			[[Category:Complete Pages]]

Meylan: /* Matlab code */

2009-04-03T01:39:32Z

Matlab code

← Older revision		Revision as of 01:39, 3 April 2009
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	Code to calculate the Green function in two dimensions without incident angle for point source and field		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 {{~~free~~ surface ~~dispersion equation~~ code}}		point on the free surface can be found here {{green function surface code}}

	= Three Dimensional Representations =		= Three Dimensional Representations =

Meylan: /* Solution for the singularity at the Free-Surface */

2009-04-03T01:38:19Z

Solution for the singularity at the Free-Surface

← Older revision		Revision as of 01:38, 3 April 2009
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	</math>		</math>
	</center>		</center>

			== Matlab code ==

			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 {{free surface dispersion equation code}}

	= Three Dimensional Representations =		= Three Dimensional Representations =

Meylan: /* Introduction */

2009-04-03T01:33:48Z

Introduction

← Older revision		Revision as of 01:33, 3 April 2009
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	depends on whether we assume the solution is proportional to <math>\exp(i\omega t)</math>		depends on whether we assume the solution is proportional to <math>\exp(i\omega t)</math>
	or <math>\exp(-i\omega t)</math>.		or <math>\exp(-i\omega t)</math>.

			It is the fundamental tool for the [[Green Function Solution Method]]

	There are many different representations for the Green function.		There are many different representations for the Green function.

Mike smith: consistent notation edits, minor changes

2009-03-31T05:50:49Z

consistent notation edits, minor changes

← Older revision		Revision as of 05:50, 31 March 2009
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	<center>		<center>
	<math>		<math>
	\nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -~~\infty~~<z<0		\nabla_{\mathbf{x}}^{2}G(\mathbf{x},\mathbf{\xi})=\delta(\mathbf{x}-\mathbf{\xi}), \, -h<z<0
	</math>		</math>
	</center>		</center>
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	The Green function can therefore be written as		The Green function can therefore be written as
	<center>		<center>
	<math>G(x) = \sum_{n=0}^\infty -\frac{e^{-\|x-a\|k_n}}{2 k_n}f_n(c)f_n(z)</math>		<math>G(x,\xi) = \sum_{n=0}^\infty -\frac{e^{-\|x-a\|k_n}}{2 k_n}f_n(c)f_n(z)</math>
	</center>		</center>
	<center>		<center>
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	<center>		<center>
	<math>		<math>
	G(\mathbf{x},\mathbf{\~~zeta~~})		G(\mathbf{x},\mathbf{\xi})
	= \sum_{n=0}^\infty -\frac{e^{-\|x-a\|k_n}}{\cos(k_nh)\sin(k_nh)+k_nh}		= \sum_{n=0}^\infty -\frac{e^{-\|x-a\|k_n}}{\cos(k_nh)\sin(k_nh)+k_nh}
	\cos(k_n(z+h))\cos(k_n(c+h))		\cos(k_n(z+h))\cos(k_n(c+h))
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	<center>		<center>
	<math>		<math>
	G(\mathbf{x},\mathbf{\~~zeta~~})		G(\mathbf{x},\mathbf{\xi})
	= \sum_{n=0}^\infty \frac{e^{-\|x-a\|k_n}}{k_n/\alpha \sin(k_nh) - k_nh}		= \sum_{n=0}^\infty \frac{e^{-\|x-a\|k_n}}{k_n/\alpha \sin(k_nh) - k_nh}
	\cos(k_n(z+h))\cos(k_n(c+h))		\cos(k_n(z+h))\cos(k_n(c+h))
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	<center>		<center>
	<math>		<math>
	G(\mathbf{x},\mathbf{\~~zeta~~})		G(\mathbf{x},\mathbf{\xi})
	= \sum_{n=0}^\infty \frac{(\alpha ^2+k_n^2)e^{-\|x-a\|k_n}}{\alpha - (\alpha ^2+k_n^2)k_nh }		= \sum_{n=0}^\infty \frac{(\alpha ^2+k_n^2)e^{-\|x-a\|k_n}}{\alpha - (\alpha ^2+k_n^2)k_nh }
	\cos(k_n(z+h))\cos(k_n(c+h))		\cos(k_n(z+h))\cos(k_n(c+h))
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	<center>		<center>
	<math>		<math>
	G(\mathbf{x},\mathbf{\~~zeta~~})		G(\mathbf{x},\mathbf{\xi})
	= \sum_{n=0}^\infty -\frac{k_n}{\sqrt{k_n^2+k_y^2}}		= \sum_{n=0}^\infty -\frac{k_n}{\sqrt{k_n^2+k_y^2}}
	\frac{e^{-\|x-a\|\sqrt{k_n^2+k_y^2}}}{\cos(k_nh)\sin(k_nh)+k_nh}		\frac{e^{-\|x-a\|\sqrt{k_n^2+k_y^2}}}{\cos(k_nh)\sin(k_nh)+k_nh}
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	<center>		<center>
	<math>		<math>
	\nabla^{2} G=0, \, -~~\infty~~<z<0		\nabla^{2} G=0, \, -h<z<0
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	G(\mathbf{x},\mathbf{\~~zeta~~})		G(\mathbf{x},\mathbf{\xi})
	= \sum_{n=0}^\infty -\frac{e^{-\|x-a\|k_n}}{2 k_n N_n^2}		= \sum_{n=0}^\infty -\frac{e^{-\|x-a\|k_n}}{2 k_n N_n^2}
	\cos(k_n(z+h))\cos(k_n h)		\cos(k_n(z+h))\cos(k_n h)
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	</math>		</math>
	</center>		</center>
	where <math>H^{(1)}_0</math> and <math>K_0</math> denote the Hankel function of the first		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		kind and the modified Bessel function of the second kind, both of
	order zero as defined in [[Abramowitz_Stegun_1964a\|Abramowitz and Stegun 1964]], <math>k</math> is the positive real solution to the [[Dispersion Relation for a Free Surface]]		order zero as defined in [[Abramowitz_Stegun_1964a\|Abramowitz and Stegun 1964]], <math>k</math> is the positive real solution to the [[Dispersion Relation for a Free Surface]]