Diffraction Transfer Matrix for Infinite Depth - Revision history

Meylan at 08:20, 19 October 2009

2009-10-19T08:20:02Z

← Older revision		Revision as of 08:20, 19 October 2009
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			{{complete pages}}

	==Introduction==		==Introduction==

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	function <math>\varsigma^j</math> can be found by solving an		function <math>\varsigma^j</math> can be found by solving an
	integral equation. The integral equation is described in		integral equation. The integral equation is described in
	[[~~Weh_Lait]]~~ and ~~numerical methods for its solution are outlined in~~		[[Wehausen and Laitone 1960]].
	~~[[Sarp_Isa~~]].
	Substituting the eigenfunction expansion of the Green's function		Substituting the eigenfunction expansion of the Green's function
	(green_inf) into the above integral equation, the scattered potential can		(green_inf) into the above integral equation, the scattered potential can

Adi Kurniawan at 04:08, 23 September 2008

2008-09-23T04:08:13Z

← Older revision		Revision as of 04:08, 23 September 2008
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	=Introduction=		==Introduction==

	This is an extension of the [[Diffraction Transfer Matrix]] (which only		This is an extension of the [[Diffraction Transfer Matrix]] (which only
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	results in [[Peter and Meylan 2004]].		results in [[Peter and Meylan 2004]].

	=Calculation of the diffraction transfer matrix for bodies of arbitrary geometry=		==Calculation of the diffraction transfer matrix for bodies of arbitrary geometry==

	To calculate the diffraction transfer matrix in infinite depth, we		To calculate the diffraction transfer matrix in infinite depth, we
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	for the decaying modes.		for the decaying modes.

	=The diffraction transfer matrix of rotated bodies=		==The diffraction transfer matrix of rotated bodies==

	For a non-axisymmetric body, a rotation about the mean		For a non-axisymmetric body, a rotation about the mean

Mpeter: /* Calculation of the diffraction transfer matrix for bodies of arbitrary geometry */

2006-06-22T11:40:20Z

Calculation of the diffraction transfer matrix for bodies of arbitrary geometry

← Older revision		Revision as of 11:40, 22 June 2006
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	(\theta - \varphi)} \mathrm{d}\eta,		(\theta - \varphi)} \mathrm{d}\eta,
	</math></center>		</math></center>
	<math>r > s</math>, given by [[Peter and Meylan 2004b]].		<math>r > s</math>, given by [[Peter and Meylan 2004b]] where
			<center><math>
			\psi(z,\eta) = \cos \eta z + \frac{\alpha}{\eta} \sin \eta z.
			</math></center>

	We assume that we have represented the scattered potential in terms of		We assume that we have represented the scattered potential in terms of

Meylan: /* The diffraction transfer matrix of rotated bodies */

2006-06-22T10:18:13Z

The diffraction transfer matrix of rotated bodies

Meylan: /* Calculation of the diffraction transfer matrix for bodies of arbitrary geometry */

2006-06-22T10:15:53Z

Calculation of the diffraction transfer matrix for bodies of arbitrary geometry

← Older revision		Revision as of 10:15, 22 June 2006
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	require the representation of the [[Infinite Depth]], [[Free-Surface Green Function]]		require the representation of the [[Infinite Depth]], [[Free-Surface Green Function]]
	in cylindrical eigenfunctions,		in cylindrical eigenfunctions,
	<center><math> ~~(green_inf)\begin{matrix}~~		<center><math>
	G(r,\theta,z;s,\varphi,c) =& \frac{\mathrm{i}\alpha}{2} \, \mathrm{e}^{\alpha (z+c)}		G(r,\theta,z;s,\varphi,c) = \frac{\mathrm{i}\alpha}{2} \, \mathrm{e}^{\alpha (z+c)}
	\sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(\alpha r) J_\nu(\alpha s) \mathrm{e}^{\mathrm{i}\nu		\sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(\alpha r) J_\nu(\alpha s) \mathrm{e}^{\mathrm{i}\nu
	(\theta - \varphi)} \\		(\theta - \varphi)}
	+& \frac{1}{\pi^2} \int\limits_0^{\infty}		</math></center>
			<center><math>
			+ \frac{1}{\pi^2} \int\limits_0^{\infty}
	\psi(z,\eta) \frac{\eta^2}{\eta^2+\alpha^2} \psi(c,\eta)		\psi(z,\eta) \frac{\eta^2}{\eta^2+\alpha^2} \psi(c,\eta)
	\sum_{\nu=-\infty}^{\infty} K_\nu(\eta r) I_\nu(\eta s) \mathrm{e}^{\mathrm{i}\nu		\sum_{\nu=-\infty}^{\infty} K_\nu(\eta r) I_\nu(\eta s) \mathrm{e}^{\mathrm{i}\nu
	(\theta - \varphi)} \mathrm{d}\eta,		(\theta - \varphi)} \mathrm{d}\eta,
	~~\end{matrix}~~
	</math></center>		</math></center>
	<math>r > s</math>, given by [[Peter and Meylan 2004b]].		<math>r > s</math>, given by [[Peter and Meylan 2004b]].
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	the source strength distribution <math>\varsigma^j</math> so that the scattered		the source strength distribution <math>\varsigma^j</math> so that the scattered
	potential can be written as		potential can be written as
	<center><math> ~~(int_eq_1)~~		<center><math>
	\phi_j^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma_j} G		\phi_j^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma_j} G
	(\mathbf{y},\mathbf{\zeta}) \, \varsigma^j (\mathbf{\zeta})		(\mathbf{y},\mathbf{\zeta}) \, \varsigma^j (\mathbf{\zeta})
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	[[Sarp_Isa]].		[[Sarp_Isa]].
	Substituting the eigenfunction expansion of the Green's function		Substituting the eigenfunction expansion of the Green's function
	(green_inf) into ~~(int_eq_1)~~, the scattered potential can		(green_inf) into the above integral equation, the scattered potential can
	be written as		be written as
	<center><math>~~\begin{matrix}~~		<center><math>
	&\phi_j^\mathrm{S}(r_j,\theta_j,z) = \mathrm{e}^{\alpha z} \sum_{\nu = -		\phi_j^\mathrm{S}(r_j,\theta_j,z) = \mathrm{e}^{\alpha z} \sum_{\nu = -
	\infty}^{\infty} \bigg[ \frac{\mathrm{i}\alpha}{2}		\infty}^{\infty} \bigg[ \frac{\mathrm{i}\alpha}{2}
	\int\limits_{\Gamma_j} \mathrm{e}^{\alpha c} J_\nu(\alpha s) \mathrm{e}^{-\mathrm{i}\nu		\int\limits_{\Gamma_j} \mathrm{e}^{\alpha c} J_\nu(\alpha s) \mathrm{e}^{-\mathrm{i}\nu
	\varphi} \varsigma^j(\mathbf{\zeta})		\varphi} \varsigma^j(\mathbf{\zeta})
	\mathrm{d}\sigma_\mathbf{\zeta} \bigg] H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}\\		\mathrm{d}\sigma_\mathbf{\zeta} \bigg] H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}
	~~& \,~~ + \int\limits_{0}^{\infty} \psi(z,\eta) \sum_{\nu = -		</math></center>
			<center><math>
			+ \int\limits_{0}^{\infty} \psi(z,\eta) \sum_{\nu = -
	\infty}^{\infty} \bigg[ \frac{1}{\pi^2} \frac{\eta^2		\infty}^{\infty} \bigg[ \frac{1}{\pi^2} \frac{\eta^2
	}{\eta^2 + \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_\nu(\eta s)		}{\eta^2 + \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_\nu(\eta s)
	\mathrm{e}^{-\mathrm{i}\nu \varphi} \varsigma^j({\mathbf{\zeta}})		\mathrm{e}^{-\mathrm{i}\nu \varphi} \varsigma^j({\mathbf{\zeta}})
	\mathrm{d}\sigma_{\mathbf{\zeta}} \bigg] K_\nu (\eta r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j} \mathrm{d}\eta,		\mathrm{d}\sigma_{\mathbf{\zeta}} \bigg] K_\nu (\eta r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j} \mathrm{d}\eta,
	~~\end{matrix}~~</math></center>		</math></center>
	where		where
	<math>\mathbf{\zeta}=(s,\varphi,c)</math> and <math>r>s</math>.		<math>\mathbf{\zeta}=(s,\varphi,c)</math> and <math>r>s</math>.
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	different incident modes of unit-amplitude. The elements of the		different incident modes of unit-amplitude. The elements of the
	diffraction transfer matrix of a body of arbitrary shape are therefore given by		diffraction transfer matrix of a body of arbitrary shape are therefore given by
	<center><math> ~~(B_elem)~~		<center><math>
	({\mathbf B}_j)_{pq} = \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j}		({\mathbf B}_j)_{pq} = \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j}
	\mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta})		\mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta})
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	<math>\varsigma_q^j(\mathbf{\zeta})</math> is the source strength distribution		<math>\varsigma_q^j(\mathbf{\zeta})</math> is the source strength distribution
	due to an incident potential of mode <math>q</math> of the form		due to an incident potential of mode <math>q</math> of the form
	<center><math> ~~(test_modesinf)~~		<center><math>
	\phi_q^{\mathrm{I}}(s,\varphi,c) = \mathrm{e}^{\alpha c} H_q^{(1)} (\alpha		\phi_q^{\mathrm{I}}(s,\varphi,c) = \mathrm{e}^{\alpha c} H_q^{(1)} (\alpha
	s) \mathrm{e}^{\mathrm{i}q \varphi}		s) \mathrm{e}^{\mathrm{i}q \varphi}

Meylan: /* Introduction */

2006-06-22T10:12:23Z

Introduction

← Older revision		Revision as of 10:12, 22 June 2006
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	=Introduction=		=Introduction=

	This is an extension of the [[~~DIffraction~~ Transfer Matrix]] (which only		This is an extension of the [[Diffraction Transfer Matrix]] (which only
	applied to finite depth) to infinite depth. This is based upon the		applied to finite depth) to infinite depth. This is based upon the
	results in [[Peter and Meylan ~~2004a~~]].		results in [[Peter and Meylan 2004]].

	=Calculation of the diffraction transfer matrix for bodies of arbitrary geometry=		=Calculation of the diffraction transfer matrix for bodies of arbitrary geometry=

Meylan at 10:11, 22 June 2006

2006-06-22T10:11:51Z

New page

=Introduction=

This is an extension of the [[DIffraction Transfer Matrix]] (which only
applied to finite depth) to infinite depth. This is based upon the
results in [[Peter and Meylan 2004a]].

=Calculation of the diffraction transfer matrix for bodies of arbitrary geometry=

To calculate the diffraction transfer matrix in infinite depth, we
require the representation of the [[Infinite Depth]], [[Free-Surface Green Function]]
in cylindrical eigenfunctions,
<center><math> (green_inf)\begin{matrix}
G(r,\theta,z;s,\varphi,c) =& \frac{\mathrm{i}\alpha}{2} \, \mathrm{e}^{\alpha (z+c)}
\sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(\alpha r) J_\nu(\alpha s) \mathrm{e}^{\mathrm{i}\nu
(\theta - \varphi)} \\
+& \frac{1}{\pi^2} \int\limits_0^{\infty}
\psi(z,\eta) \frac{\eta^2}{\eta^2+\alpha^2} \psi(c,\eta)
\sum_{\nu=-\infty}^{\infty} K_\nu(\eta r) I_\nu(\eta s) \mathrm{e}^{\mathrm{i}\nu
(\theta - \varphi)} \mathrm{d}\eta,
\end{matrix}
</math></center>
<math>r > s</math>, given by [[Peter and Meylan 2004b]].

We assume that we have represented the scattered potential in terms of
the source strength distribution <math>\varsigma^j</math> so that the scattered
potential can be written as
<center><math> (int_eq_1)
\phi_j^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma_j} G
(\mathbf{y},\mathbf{\zeta}) \, \varsigma^j (\mathbf{\zeta})
\mathrm{d}\sigma_\mathbf{\zeta}, \quad \mathbf{y} \in D,
</math></center>
where <math>D</math> is the volume occupied by the water and <math>\Gamma_j</math> is the
immersed surface of body <math>\Delta_j</math>. The source strength distribution
function <math>\varsigma^j</math> can be found by solving an
integral equation. The integral equation is described in
[[Weh_Lait]] and numerical methods for its solution are outlined in
[[Sarp_Isa]].
Substituting the eigenfunction expansion of the Green's function
(green_inf) into (int_eq_1), the scattered potential can
be written as
<center><math>\begin{matrix}
&\phi_j^\mathrm{S}(r_j,\theta_j,z) = \mathrm{e}^{\alpha z} \sum_{\nu = -
\infty}^{\infty} \bigg[ \frac{\mathrm{i}\alpha}{2}
\int\limits_{\Gamma_j} \mathrm{e}^{\alpha c} J_\nu(\alpha s) \mathrm{e}^{-\mathrm{i}\nu
\varphi} \varsigma^j(\mathbf{\zeta})
\mathrm{d}\sigma_\mathbf{\zeta} \bigg] H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}\\
& \, + \int\limits_{0}^{\infty} \psi(z,\eta) \sum_{\nu = -
\infty}^{\infty} \bigg[ \frac{1}{\pi^2} \frac{\eta^2
}{\eta^2 + \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_\nu(\eta s)
\mathrm{e}^{-\mathrm{i}\nu \varphi} \varsigma^j({\mathbf{\zeta}})
\mathrm{d}\sigma_{\mathbf{\zeta}} \bigg] K_\nu (\eta r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j} \mathrm{d}\eta,
\end{matrix}</math></center>
where
<math>\mathbf{\zeta}=(s,\varphi,c)</math> and <math>r>s</math>.
This restriction implies that the eigenfunction expansion is only valid
outside the escribed cylinder of the body.

The columns of the diffraction transfer matrix are the coefficients of
the eigenfunction expansion of the scattered wavefield due to the
different incident modes of unit-amplitude. The elements of the
diffraction transfer matrix of a body of arbitrary shape are therefore given by
<center><math> (B_elem)
({\mathbf B}_j)_{pq} = \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j}
\mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta})
\mathrm{d}\sigma_\mathbf{\zeta}
</math></center>
and
<center><math>
({\mathbf B}_j)_{pq} = \frac{1}{\pi^2} \frac{\eta^2}{\eta^2 +
\alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
\varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}
</math></center>
for the propagating and the decaying modes respectively, where
<math>\varsigma_q^j(\mathbf{\zeta})</math> is the source strength distribution
due to an incident potential of mode <math>q</math> of the form
<center><math> (test_modesinf)
\phi_q^{\mathrm{I}}(s,\varphi,c) = \mathrm{e}^{\alpha c} H_q^{(1)} (\alpha
s) \mathrm{e}^{\mathrm{i}q \varphi}
</math></center>
for the propagating modes, and
<center><math>
\phi_q^{\mathrm{I}}(s,\varphi,c) = \psi(c,\eta) K_q (\eta s) \mathrm{e}^{\mathrm{i}q \varphi}
</math></center>
for the decaying modes.

=The diffraction transfer matrix of rotated bodies=

For a non-axisymmetric body, a rotation about the mean
centre position in the <math>(x,y)</math>-plane will result in a
different diffraction transfer matrix. We will show how the
diffraction transfer matrix of a body rotated by an angle <math>\beta</math> can
be easily calculated from the diffraction transfer matrix of the
non-rotated body. The rotation of the body influences the form of the
elements of the diffraction transfer matrices in two ways. Firstly, the
angular dependence in the integral over the immersed surface of the
body is altered and, secondly, the source strength distribution
function is different if the body is rotated. However, the source
strength distribution function of the rotated body can be obtained by
calculating the response of the non-rotated body due to rotated
incident potentials. It will be shown that the additional angular
dependence can be easily factored out of the elements of the
diffraction transfer matrix.

The additional angular dependence caused by the rotation of the
incident potential can be factored out of the normal derivative of the
incident potential such that
<center><math>
\frac{\partial \phi_{q\beta}^{\mathrm{I}}}{\partial n} =
\frac{\partial \phi_{q}^{\mathrm{I}}}{\partial n}
\mathrm{e}^{\mathrm{i}q \beta},
</math></center>
where <math>\phi_{q\beta}^{\mathrm{I}}</math> is the rotated incident potential.
Since the integral equation for the determination of the source
strength distribution function is linear, the source strength
distribution function due to the rotated incident potential is thus just
given by
<center><math>
\varsigma_{q\beta}^j = \varsigma_q^j \, \mathrm{e}^{\mathrm{i}q \beta}.
</math></center>
This is also the source strength distribution function of the rotated
body due to the standard incident modes.

The elements of the diffraction transfer matrix <math>\mathbf{B}_j</math> are
given by equations (B_elem). Keeping in mind that the body is
rotated by the angle <math>\beta</math>, the elements of the diffraction transfer
matrix of the rotated body are given by
<center><math> (B_elemrot)
(\mathbf{B}_j^\beta)_{pq} = \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j}
\mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p (\varphi+\beta)}
\varsigma_{q\beta}^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta},
</math></center>
and
<center><math>
(\mathbf{B}_j^\beta)_{pq} = \frac{1}{\pi^2} \frac{\eta^2}{\eta^2 +
\alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
(\varphi+\beta)} \varsigma_{q\beta}^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta},
</math></center>
for the propagating and decaying modes respectively.

Thus the additional angular dependence caused by the rotation of
the body can be factored out of the elements of the diffraction
transfer matrix. The elements of the diffraction transfer matrix
corresponding to the body rotated by the angle <math>\beta</math>,
<math>\mathbf{B}_j^\beta</math>, are given by
<center><math> (B_rot)
(\mathbf{B}_j^\beta)_{pq} = (\mathbf{B}_j)_{pq} \, \mathrm{e}^{\mathrm{i}(q-p) \beta}.
</math></center>
As before, <math>(\mathbf{B})_{pq}</math> is understood to be the element of
<math>\mathbf{B}</math> which corresponds to the coefficient of the <math>p</math>th scattered
mode due to a unit-amplitude incident wave of mode <math>q</math>. Equation (B_rot) applies to
propagating and decaying modes likewise.

[[Category:Interaction Theory]]

← Older revision		Revision as of 10:18, 22 June 2006
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	The elements of the diffraction transfer matrix <math>\mathbf{B}_j</math> are		The elements of the diffraction transfer matrix <math>\mathbf{B}_j</math> are
	given by equations ~~(B_elem)~~. Keeping in mind that the body is		given by equations in the previous section. Keeping in mind that the body is
	rotated by the angle <math>\beta</math>, the elements of the diffraction transfer		rotated by the angle <math>\beta</math>, the elements of the diffraction transfer
	matrix of the rotated body are given by		matrix of the rotated body are given by
	<center><math> ~~(B_elemrot)~~		<center><math>
	(\mathbf{B}_j^\beta)_{pq} = \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j}		(\mathbf{B}_j^\beta)_{pq} = \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j}
	\mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p (\varphi+\beta)}		\mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p (\varphi+\beta)}
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	</math></center>		</math></center>
	for the propagating and decaying modes respectively.		for the propagating and decaying modes respectively.

	Thus the additional angular dependence caused by the rotation of		Thus the additional angular dependence caused by the rotation of
	the body can be factored out of the elements of the diffraction		the body can be factored out of the elements of the diffraction
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	corresponding to the body rotated by the angle <math>\beta</math>,		corresponding to the body rotated by the angle <math>\beta</math>,
	<math>\mathbf{B}_j^\beta</math>, are given by		<math>\mathbf{B}_j^\beta</math>, are given by
	<center><math> ~~(B_rot)~~		<center><math>
	(\mathbf{B}_j^\beta)_{pq} = (\mathbf{B}_j)_{pq} \, \mathrm{e}^{\mathrm{i}(q-p) \beta}.		(\mathbf{B}_j^\beta)_{pq} = (\mathbf{B}_j)_{pq} \, \mathrm{e}^{\mathrm{i}(q-p) \beta}.
	</math></center>		</math></center>
	As before, <math>(\mathbf{B})_{pq}</math> is understood to be the element of		As before, <math>(\mathbf{B})_{pq}</math> is understood to be the element of
	<math>\mathbf{B}</math> which corresponds to the coefficient of the <math>p</math>th scattered		<math>\mathbf{B}</math> which corresponds to the coefficient of the <math>p</math>th scattered
	mode due to a unit-amplitude incident wave of mode <math>q</math>. ~~Equation (B_rot)~~ applies to		mode due to a unit-amplitude incident wave of mode <math>q</math>. This equation applies to
	propagating and decaying modes likewise.		propagating and decaying modes likewise.


	[[Category:Interaction Theory]]		[[Category:Interaction Theory]]