Kagemoto and Yue Interaction Theory - Revision history

Meylan: /* Eigenfunction expansion of the potential */

2010-05-02T10:24:06Z

Eigenfunction expansion of the potential

← Older revision		Revision as of 10:24, 2 May 2010
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	</math></center>		</math></center>
	where <math>k_0</math> is the		where <math>k_0</math> is the
	imaginary root with ~~positive~~ imaginary part		imaginary root with negative imaginary part
	and <math>k_m</math>, <math>m>0</math>, are given the positive real roots ordered		and <math>k_m</math>, <math>m>0</math>, are given the positive real roots ordered
	with increasing size.		with increasing size.

2008-08-18T21:47:23Z

Eigenfunction expansion of the potential

← Older revision		Revision as of 21:47, 18 August 2008
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	are cylindrical polar coordinates centered at each body		are cylindrical polar coordinates centered at each body
	<center><math>		<center><math>
	f_m(z) = \frac{\cos k_m (z+H)}{\cos k_m H}.		f_m(z) = \frac{\cos k_m (z+h)}{\cos k_m h}.
	</math></center>		</math></center>
	where <math>k_m</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]		where <math>k_m</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
	<center><math>		<center><math>
	\alpha + k_m \tan k_m H = 0\,.		\alpha + k_m \tan k_m h = 0\,.
	</math></center>		</math></center>
	where <math>k_0</math> is the		where <math>k_0</math> is the

2008-08-18T21:46:27Z

Equations of Motion

@@ Line 33: / Line 33: @@
 {{standard linear wave scattering equations}}
-Writing <math>\alpha = \omega^2/g</math> where <math>g</math> is the acceleration due to
+The [[Sommerfeld Radiation Condition]] is also imposed.
 =Eigenfunction expansion of the potential=

2008-08-18T21:45:09Z

Equations of Motion

← Older revision		Revision as of 21:45, 18 August 2008
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	other body.		other body.
	To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point		To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point
	in the water, which is assumed to be of [[Finite Depth]] <math>H</math>,		in the water, which is assumed to be of [[Finite Depth]] <math>h</math>,
	while <math>\mathbf{x}</math> always denotes a point of the undisturbed water		while <math>\mathbf{x}</math> always denotes a point of the undisturbed water
	surface assumed at <math>z=0</math>.		surface assumed at <math>z=0</math>.

			{{standard linear wave scattering equations}}

	Writing <math>\alpha = \omega^2/g</math> where <math>g</math> is the acceleration due to		Writing <math>\alpha = \omega^2/g</math> where <math>g</math> is the acceleration due to

2008-08-18T21:41:41Z

Interaction for the case of circular plates

← Older revision		Revision as of 21:41, 18 August 2008
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	</math></center>		</math></center>
	<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.		<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.

	~~= Interaction for the case of circular plates =~~

2008-08-02T03:24:49Z

Final Equations

← Older revision		Revision as of 03:24, 2 August 2008
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	</math></center>		</math></center>
	<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.		<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.

			= Interaction for the case of circular plates =

2008-08-01T03:10:42Z

Derivation of the system of equations

← Older revision		Revision as of 03:10, 1 August 2008
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	<center><math>		<center><math>
	\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +		\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
	\sum_{j=1,j \neq l}^{n} \, \phi_j^{\mathrm{S}}		\sum_{j=1,j \neq l}^{N} \, \phi_j^{\mathrm{S}}
	(r_l,\theta_l,z)		(r_l,\theta_l,z)
	</math></center>		</math></center>
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	= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}		= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
	\Big[ \tilde{D}_{n\nu}^{l} +		\Big[ \tilde{D}_{n\nu}^{l} +
	\sum_{j=1,j \neq l}^{n} \sum_{\tau =		\sum_{j=1,j \neq l}^{N} \sum_{\tau =
	-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n		-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
	R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n		R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n
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	D_{n\nu}^l =		D_{n\nu}^l =
	\tilde{D}_{n\nu}^{l} +		\tilde{D}_{n\nu}^{l} +
	\sum_{j=1,j \neq l}^{n} \sum_{\tau =		\sum_{j=1,j \neq l}^{N} \sum_{\tau =
	-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n		-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
	R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}}		R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}}

2008-07-31T23:14:40Z

Eigenfunction expansion of the potential

← Older revision		Revision as of 23:14, 31 July 2008
Line 71:		Line 71:
	are cylindrical polar coordinates centered at each body		are cylindrical polar coordinates centered at each body
	<center><math>		<center><math>
	f_m(z) = \frac{\cos k_m (z+d)}{\cos k_m d}.		f_m(z) = \frac{\cos k_m (z+H)}{\cos k_m H}.
	</math></center>		</math></center>
	where <math>k_m</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]		where <math>k_m</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
	<center><math>		<center><math>
	\alpha + k_m \tan k_m d = 0\,.		\alpha + k_m \tan k_m H = 0\,.
	</math></center>		</math></center>
	where <math>k_0</math> is the		where <math>k_0</math> is the

2008-07-31T23:14:03Z

Equations of Motion

← Older revision		Revision as of 23:14, 31 July 2008
Line 27:		Line 27:
	other body.		other body.
	To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point		To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point
	in the water, which is assumed to be of [[Finite Depth]] <math>d</math>,		in the water, which is assumed to be of [[Finite Depth]] <math>H</math>,
	while <math>\mathbf{x}</math> always denotes a point of the undisturbed water		while <math>\mathbf{x}</math> always denotes a point of the undisturbed water
	surface assumed at <math>z=0</math>.		surface assumed at <math>z=0</math>.
Line 42:		Line 42:
	</math></center>		</math></center>
	<center><math>		<center><math>
	\frac{\partial \phi}{\partial z} = 0, \; \mathbf{y} \in D, \ z=-d,		\frac{\partial \phi}{\partial z} = 0, \; \mathbf{y} \in D, \ z=-H,
	</math></center>		</math></center>
	where <math>D</math> is the		where <math>D</math> is the

2008-07-30T23:34:48Z

Derivation of the system of equations

← Older revision		Revision as of 23:34, 30 July 2008
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	<center><math>		<center><math>
	\sum_{n=0}^{\infty} f_n(z)		\sum_{n=0}^{\infty} f_n(z)
	\sum_{\nu = - \infty}^{\infty} D_{n\nu}^j I_\nu (k_n ~~r_j~~) \mathrm{e}^{\mathrm{i}\nu \~~theta_j~~}		\sum_{\nu = - \infty}^{\infty} D_{n\nu}^l I_\nu (k_n r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}
	</math></center>		</math></center>
	<center><math>		<center><math>
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	It therefore follows that		It therefore follows that
	<center><math>		<center><math>
	D_{n\nu}^j =		D_{n\nu}^l =
	\tilde{D}_{n\nu}^{l} +		\tilde{D}_{n\nu}^{l} +
	\sum_{j=1,j \neq l}^{n} \sum_{\tau =		\sum_{j=1,j \neq l}^{n} \sum_{\tau =