Interaction Theory for Infinite Arrays - Revision history

Meylan at 08:21, 19 October 2009

2009-10-19T08:21:40Z

← Older revision		Revision as of 08:21, 19 October 2009
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	=Introduction =		{{complete pages}}

			==Introduction ==

	There are two approaches to solution for the [[:Category:Infinite Array\|Infinite Array]],		There are two approaches to solution for the [[:Category:Infinite Array\|Infinite Array]],
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	This is based on [[Peter, Meylan, and Linton 2006]]		This is based on [[Peter, Meylan, and Linton 2006]]

	= System of equations =		== System of equations ==

	We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely		We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely
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	between adjacent bodies greater than <math>Rk</math>.		between adjacent bodies greater than <math>Rk</math>.

	=The efficient computation of the <math>\sigma_{\nu}^0</math> =		==The efficient computation of the <math>\sigma_{\nu}^0</math> ==

	The constants <math>\sigma_{\nu}^0</math> (cf.~\eqref{eq_op_sigma}) appearing in		The constants <math>\sigma_{\nu}^0</math> (cf.~\eqref{eq_op_sigma}) appearing in
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	where <math>\delta_{mn}</math> is the Kronecker delta.		where <math>\delta_{mn}</math> is the Kronecker delta.

	= Acoustic scattering by an infinite array of identical generalized cylinders =		== Acoustic scattering by an infinite array of identical generalized cylinders ==

	The theory above has so far been developed for water-wave scattering		The theory above has so far been developed for water-wave scattering

Meylan at 01:56, 10 September 2006

2006-09-10T01:56:49Z

← Older revision		Revision as of 01:56, 10 September 2006
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	For circular cylinders, i.e. cylinders which have a circular		For circular cylinders, i.e. cylinders which have a circular
	cross-section, this problem has been considered by [[Linton and Evans 1993]].		cross-section, this problem has been considered by [[Linton and Evans 1993]].

			[[Category:Infinite Array]]
			[[Category:Interaction Theory]]

Meylan: /* System of equations */

2006-09-04T20:48:44Z

System of equations

@@ Line 66: / Line 66: @@
 <math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>.
 Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.
 =The far field=

Meylan: /* System of equations */

2006-09-04T20:48:17Z

System of equations

← Older revision		Revision as of 20:48, 4 September 2006
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	[[Far Field Waves]]		[[Far Field Waves]]






	An algebraically exact solution to the problem of linear water-wave		An algebraically exact solution to the problem of linear water-wave
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	present some computations for the case of fixed, rigid and elastic floating		present some computations for the case of fixed, rigid and elastic floating
	bodies of negligible draft concentrating on presenting the		bodies of negligible draft concentrating on presenting the
	amplitudes of the scattered waves as functions of the incident angle.		amplitudes of the scattered waves as functions of the incident angle.







	=The far field=		=The far field=

Meylan: /* Acoustic scattering by an infinite array of identical generalized cylinders */

2006-09-04T20:47:34Z

Acoustic scattering by an infinite array of identical generalized cylinders

@@ Line 329: / Line 329: @@
 theory applies with the following modifications:
+#The [[Dispersion Relation for a Free Surface]] is replaced by <math>k=\omega /
-#The dispersion relation \eqref{eq_k} is replaced by </math>k=\omega /
+c</math> where <math>c</math> is the speed of sound in the medium under consideration
-c<math> where </math>c<math> is the speed of sound in the medium under consideration
+and the [[Dispersion Relation for a Free Surface]] is omitted.
 #All factors <math>\cos k_m(z+d)</math>, <math>\cos k_m(c+d)</math>, <math>\cos k_m d</math>
 and <math>f_0</math> are replaced by 1.
 #The factor <math>N_0</math> in \eqref{green_d} is <math>k/\pi</math>.
-Note that point <math>(a)</math> implies that there are no evanescent modes in this
+Note that there are no evanescent modes in this
-problem, i.e.~the sums over <math>m</math> and <math>n</math> in the eigenfunction expansions
+problem, i.e. the sums over <math>m</math> and <math>n</math> in the eigenfunction expansions
-\eqref{basisrep_out_d} and \eqref{basisrep_in_d}, respectively, only
+only
-contain the terms for <math>m=0</math> and <math>n=0</math>. Moreover, we have </math>k_0 = -
+contain the terms for <math>m=0</math> and <math>n=0</math>. Moreover, we have <math>k_0 = -
-\mathrm{i} \, \omega /c<math>.
+\mathrm{i} \, \omega /c</math>.
-Non-dimensional variables (denoted with an overbar) are introduced,
+For circular cylinders, i.e. cylinders which have a circular
-<center><math>
+cross-section, this problem has been considered by [[Linton and Evans 1993]].

Meylan: /* The efficient computation of the $\sigma_{\nu}^0$ */

2006-09-04T20:42:37Z

The efficient computation of the <math>\sigma_{\nu}^0</math>

← Older revision		Revision as of 20:42, 4 September 2006
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	<center><math>		<center><math>
	\sigma_{\nu}^0 = \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j)		\sigma_{\nu}^0 = \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j)
	K_{\nu} (-\mathrm{i} k j R) = \frac{\pi \i^{\nu+1}}{2} \sum_{j=1}^{\infty}		K_{\nu} (-\mathrm{i} k j R) = \frac{\pi i^{\nu+1}}{2} \sum_{j=1}^{\infty}
	(P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR),		(P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR),
	</math></center>		</math></center>
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	(P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR)		(P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR)
	</math></center>		</math></center>
	as the <math>\sigma^0_\nu</math> are then determined by </math>\sigma^0_\nu =		as the <math>\sigma^0_\nu</math> are then determined by <math>\sigma^0_\nu =
	\pi/2 \,\, \i^{\nu+1} \, \tilde{\sigma}^0_\nu<math>.		\pi/2 \,\, i^{\nu+1} \, \tilde{\sigma}^0_\nu</math>.

	An efficient way of computing the <math>\tilde{\sigma}_{\nu}^0</math>		An efficient way of computing the <math>\tilde{\sigma}_{\nu}^0</math>
	is given in [[~~linton98~~]] and the results are briefly outlined		is given in [[Linton 1998]] and the results are briefly outlined
	in our notation.		in our notation.
	Noting that <math>H^{(1)}_{-\nu} (\,\cdot\,)= (-1)^{\nu}		Noting that <math>H^{(1)}_{-\nu} (\,\cdot\,)= (-1)^{\nu}
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	Referring to [[Linton 1998]], the constants <math>\tilde{\sigma}_{\nu}^0</math> can		Referring to [[Linton 1998]], the constants <math>\tilde{\sigma}_{\nu}^0</math> can
	be written as		be written as

	<center><math>		<center><math>
			\tilde{\sigma}_{0}^0 = -1 -\frac{2 i}{\pi} \left( C + \log \frac{k}{2p}
	\tilde{\sigma}_{0}^0 &= -1 -\frac{2\i}{\pi} \left( C + \log \frac{k}{2p}
	\right) + \frac{2}{R k \sin \chi} - \frac{2 \mathrm{i} (k^2 + 2		\right) + \frac{2}{R k \sin \chi} - \frac{2 \mathrm{i} (k^2 + 2
	\psi^2)}{p^3 R} \zeta(3)~~\\ &\quad~~		\psi^2)}{p^3 R} \zeta(3)
	+ \frac{2}{R} \sum_{m=1}^\infty \left(		+ \frac{2}{R} \sum_{m=1}^\infty \left(
	\frac{1}{k \sin \chi_{-m}} + \frac{1}{k \sin \chi_m} +		\frac{1}{k \sin \chi_{-m}} + \frac{1}{k \sin \chi_m} +
	\frac{2 \i}{p m} + \frac{\mathrm{i} (k^2 + 2 \psi^2)}{p^3 m^3} \right)		\frac{2 i}{p m} + \frac{\mathrm{i} (k^2 + 2 \psi^2)}{p^3 m^3} \right)

	</math></center>		</math></center>
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	as well as		as well as
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	~~&\quad~~		\tilde{\sigma}_{2\nu}^0 &=& 2 (-1)^{\nu} \left( \frac{\mathrm{e}^{2\mathrm{i} \nu
	\tilde{\sigma}_{2\nu}^0 &= 2 (-1)^{\nu} \left( \frac{\mathrm{e}^{2\mathrm{i} \nu		\chi} }{R k \sin \chi} - \frac{ i}{\pi}
	\chi} }{R k \sin \chi} - \frac{\i}{\pi}
	\left( \frac{k}{2 p} \right)^{2\nu} \zeta(2\nu +1) \right) +		\left( \frac{k}{2 p} \right)^{2\nu} \zeta(2\nu +1) \right) +
	\frac{\i}{\nu \pi} \\		\frac{ i}{\nu \pi} \\
	&~~\quad~~ + 2 (-1)^\nu \sum_{m=1}^\infty		& + &2 (-1)^\nu \sum_{m=1}^\infty
	\left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi_m}}{R k \sin \chi_{m}} +		\left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi_m}}{R k \sin \chi_{m}} +
	\frac{\mathrm{e}^{-2 \mathrm{i} \nu \chi_{-m}}}{R k \sin \chi_{-m}} +		\frac{\mathrm{e}^{-2 \mathrm{i} \nu \chi_{-m}}}{R k \sin \chi_{-m}} +
	\frac{\i}{m\pi} \left( \frac{k}{2 m p} \right)^{2\nu}		\frac{ i}{m\pi} \left( \frac{k}{2 m p} \right)^{2\nu}
	\right)\\ &~~\quad~~		\right)\\ &
	+ \frac{\i}{\pi} \sum_{m=1}^\nu		&+& \frac{ i}{\pi} \sum_{m=1}^\nu
	\frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m)! (\nu-m)!} \left( \frac{p}{k} \right)^{2m}		\frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m)! (\nu-m)!} \left( \frac{p}{k} \right)^{2m}
	B_{2m}(\psi/p),		B_{2m}(\psi/p),
	\\		\end{matrix}</math></center>
	&		<center><math>\begin{matrix}
	\tilde{\sigma}_{2\nu-1}^0 &= - 2 (-1)^\nu \left( \frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1)		\tilde{\sigma}_{2\nu-1}^0 &=& - 2 (-1)^\nu \left( \frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1)
	\chi}}{R k \sin \chi} - \frac{ \psi R		\chi}}{R k \sin \chi} - \frac{ \psi R
	\nu}{\pi^2} \left( \frac{k}{2 p} \right)^{2\nu-1}		\nu}{\pi^2} \left( \frac{k}{2 p} \right)^{2\nu-1}
	\zeta(2\nu +1) \right)\\		\zeta(2\nu +1) \right)\\
	&~~\quad~~ - 2 (-1)^\nu \sum_{m=1}^\infty		& -& 2 (-1)^\nu \sum_{m=1}^\infty
	\left(\frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1)\chi_m} }{R k \sin \chi_m} +		\left(\frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1)\chi_m} }{R k \sin \chi_m} +
	\frac{\mathrm{i} \mathrm{e}^{-\mathrm{i} (2\nu-1) \chi_{-m}}}{R k \sin \chi_{-m}} +		\frac{\mathrm{i} \mathrm{e}^{-\mathrm{i} (2\nu-1) \chi_{-m}}}{R k \sin \chi_{-m}} +
	\frac{\psi R \nu}{m^2\pi^2} \left( \frac{k}{2		\frac{\psi R \nu}{m^2\pi^2} \left( \frac{k}{2
	m p} \right)^{2\nu-1} \right)\\ &~~\quad~~ - \frac{2}{\pi} \sum_{m=0}^{\nu-1}		m p} \right)^{2\nu-1} \right)\\
			& -& \frac{2}{\pi} \sum_{m=0}^{\nu-1}
	\frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m+1)! (\nu-m-1)!} \left( \frac{p}{k}		\frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m+1)! (\nu-m-1)!} \left( \frac{p}{k}
	\right)^{2m+1} B_{2m+1}(\psi/p),		\right)^{2m+1} B_{2m+1}(\psi/p),
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	converge like <math>O(m^{-5})</math> as <math>m\rightarrow\infty</math>.		converge like <math>O(m^{-5})</math> as <math>m\rightarrow\infty</math>.

	Note that since <math>\sin \chi_m</math> is purely imaginary for </math>m \notin		Note that since <math>\sin \chi_m</math> is purely imaginary for <math>m \notin
	\mathcal{M}<math>, the computation of the real part of		\mathcal{M}</math>, the computation of the real part of
	<math>\tilde{\sigma}_{2\nu}^0</math> and the imaginary part of <math>\tilde{\sigma}_{2\nu-1}^0</math>		<math>\tilde{\sigma}_{2\nu}^0</math> and the imaginary part of <math>\tilde{\sigma}_{2\nu-1}^0</math>
	is particularly simple. For <math>\nu \geq 0</math>, they are given by		is particularly simple. For <math>\nu \geq 0</math>, they are given by

Meylan: /* The far field */

2006-09-04T20:34:48Z

The far field

← Older revision		Revision as of 20:34, 4 September 2006
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	Letting <math>p=2\pi/R</math>, define the scattering angles <math>\chi_m</math> by		Letting <math>p=2\pi/R</math>, define the scattering angles <math>\chi_m</math> by
	<center><math>		<center><math>
	\chi_m = \cos^{-1} (\psi_m/k) \quad {\mathrm where}		\chi_m = \cos^{-1} (\psi_m/k) \quad \mathrm{where}
	\quad \psi_m = k \cos \chi + m p		\quad \psi_m = k \cos \chi + m p
	</math></center>		</math></center>

Meylan: /* The far field */

2006-09-04T20:34:06Z

The far field

Show changes

Meylan: /* Numerical calculations */

2006-09-04T12:08:31Z

Numerical calculations

Show changes

Meylan: /* The fixed and rigid body cases */

2006-09-04T12:07:57Z

The fixed and rigid body cases

← Older revision		Revision as of 12:07, 4 September 2006
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	~~==The fixed and rigid body cases==~~

	~~The fixed and rigid body cases can easily be solved by the method~~
	~~outlined above since they can be considered special cases.~~
	~~In the problem of a fixed body (dock), the displacement is always~~
	~~zero, <math>w=0</math>, so we simply need to solve equation (46) for~~
	~~<math>\phi=\phi^0</math>. For the case of a rigid body, we need to truncate~~
	~~the sums in \eqref{expanded} to include the first three modes only~~
	~~(which correspond to the three modes of rigid motion of the~~
	~~plate, namely the heave, pitch and roll). Note that for these modes the~~
	~~eigenvalue is <math>\lambda_k=0</math> so that the term involving the stiffness~~
	~~<math>\beta</math> does not appear in equation (50).~~

	=Numerical calculations=		=Numerical calculations=

Interaction Theory for Infinite Arrays - Revision history

Meylan at 08:21, 19 October 2009

Meylan at 01:56, 10 September 2006

Meylan: /* System of equations */

Meylan: /* System of equations */

Meylan: /* Acoustic scattering by an infinite array of identical generalized cylinders */

Meylan: /* The efficient computation of the \sigma_{\nu}^0 */

Meylan: /* The far field */

Meylan: /* The far field */

Meylan: /* Numerical calculations */

Meylan: /* The fixed and rigid body cases */

Meylan: /* The efficient computation of the $\sigma_{\nu}^0$ */