Diffraction Transfer Matrix - Revision history

Meylan at 08:17, 19 October 2009

2009-10-19T08:17:43Z

← Older revision		Revision as of 08:17, 19 October 2009
Line 1:		Line 1:
	=Introduction=		{{complete pages}}

			== Introduction ==

	The diffraction transfer matrix relates the incident and scattered potential		The diffraction transfer matrix relates the incident and scattered potential
Line 30:		Line 32:
	The theory is extended to [[Infinite Depth]] in [[Diffraction Transfer Matrix for Infinite Depth]]		The theory is extended to [[Infinite Depth]] in [[Diffraction Transfer Matrix for Infinite Depth]]

	=Eigenfunction expansion of the potential=		== Eigenfunction expansion of the potential ==

	The scattered potential of a body		The scattered potential of a body
Line 57:		Line 59:
	terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.		terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.

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

	The scattered and incident potential can therefore be related by a		The scattered and incident potential can therefore be related by a
Line 124:		Line 126:
	[[Wehausen and Laitone 1960]] and in [[Green Function Solution Method]].		[[Wehausen and Laitone 1960]] and in [[Green Function Solution Method]].

	=Calculation of the coefficients <math>A_{m \mu}^j</math>, using a source strength distribution=		== Calculation of the coefficients <math>A_{m \mu}^j</math>, using a source strength distribution ==

	The idea is to represent the scattered potential for the body <math>\Delta_j</math> in terms		The idea is to represent the scattered potential for the body <math>\Delta_j</math> in terms
Line 159:		Line 161:
	</math></center>		</math></center>

	=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
Line 223:		Line 225:

	[[Category:Interaction Theory]]		[[Category:Interaction Theory]]
	~~[[Category:Linear Water-Wave Theory]]~~

Fmontiel: /* Calculation of the coefficients $A_{m \mu}^j$ , using a source strength distribution */

2008-07-31T04:43:05Z

Calculation of the coefficients <math>A_{m \mu}^j</math>, using a source strength distribution

← Older revision		Revision as of 04:43, 31 July 2008
Line 125:		Line 125:

	=Calculation of the coefficients <math>A_{m \mu}^j</math>, using a source strength distribution=		=Calculation of the coefficients <math>A_{m \mu}^j</math>, using a source strength distribution=

			The idea is to represent the scattered potential for the body <math>\Delta_j</math> in terms
			of a source strength distribution <math>\varsigma^j</math>, as we already did in the previous section
			<center><math>
			\phi_j^\mathrm{S}(r_j,\theta_j,z) = \int\limits_{\Gamma_j} G
			(r_j,\theta_j,z;s,\varphi,c) \, \varsigma^j (s,\varphi,c)
			\mathrm{d}\sigma(s,\varphi,c), \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>. As we will use the [[Cylindrical Eigenfunction Expansion]],
			we express all the variables in a cylindrical coordinate system. The expression of the [[Free-Surface Green Function]]
			has already been expanded in cylindrical coordinates in the previous section, so we obtain the expression
			of the scattered potential as follow
			<center><math>
			\phi_j^\mathrm{S}(r_j,\theta_j,z) = \int\limits_{\Gamma_j} \Big[ \frac{1}{\pi} \sum_{m=0}^{\infty}
			\frac{k_m^2+\alpha^2}{H(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+H) \cos
			k_m(c+H) \sum_{\mu=-\infty}^{\infty} K_\mu(k_m r_j) I_\mu(k_m s) \mathrm{e}^{\mathrm{i}\mu
			(\theta_j - \varphi)} \Big] \varsigma^j (s,\varphi,c) \mathrm{d}\sigma(s,\varphi,c)
			</math></center>
			We can rearrange this expression in order to obtain a [[Cylindrical Eigenfunction Expansion]].
			<center><math>
			\phi_j^\mathrm{S}(r_j,\theta_j,z) = \sum_{m=0}^{\infty} \frac{\cos k_m(z+H)}{\cos k_m H}
			\sum_{\mu=-\infty}^{\infty} \Big[ \frac{1}{\pi} \frac{(k_m^2+\alpha^2) \cos k_m H}{H(k_m^2+\alpha^2)-\alpha}
			\int\limits_{\Gamma_j} \cos k_m(c+H) I_\mu(k_m s) \mathrm{e}^{-\mathrm{i}\mu \varphi} \varsigma^j (s,\varphi,c) \mathrm{d}\sigma(s,\varphi,c)
			\Big]
			K_\mu(k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j}
			</math></center>
			An eigenfunction matching method permits us to identify the coefficients of the scattered potential expression
			<center><math>
			A_{m \mu}^j = \frac{1}{\pi} \frac{(k_m^2+\alpha^2) \cos k_m H}{H(k_m^2+\alpha^2)-\alpha}
			\int\limits_{\Gamma_j} \cos k_m(c+H) I_\mu(k_m s) \mathrm{e}^{-\mathrm{i}\mu \varphi} \varsigma^j (s,\varphi,c)
			\mathrm{d}\sigma(s,\varphi,c)
			</math></center>

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

Fmontiel at 03:38, 31 July 2008

2008-07-31T03:38:35Z

← Older revision		Revision as of 03:38, 31 July 2008
Line 123:		Line 123:
	integral equation. The integral equation is described in		integral equation. The integral equation is described in
	[[Wehausen and Laitone 1960]] and in [[Green Function Solution Method]].		[[Wehausen and Laitone 1960]] and in [[Green Function Solution Method]].

			=Calculation of the coefficients <math>A_{m \mu}^j</math>, using a source strength distribution=

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

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

2008-07-31T00:59:30Z