Kagemoto and Yue Interaction Theory for Infinite Depth - Revision history

Meylan at 09:29, 20 October 2009

2009-10-20T09:29:25Z

← Older revision		Revision as of 09:29, 20 October 2009
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	=Introduction=		{{complete pages}}

			==Introduction==

	[[Kagemoto and Yue Interaction Theory]] applies in [[Finite Depth]] water.		[[Kagemoto and Yue Interaction Theory]] applies in [[Finite Depth]] water.
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	and we present this theory here.		and we present this theory here.

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

	The scattered potential of body <math>\Delta_j</math> can be expanded using		The scattered potential of body <math>\Delta_j</math> can be expanded using
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	</math></center>		</math></center>

	=The interaction in water of infinite depth=		==The interaction in water of infinite depth==

	The scattered potential <math>\phi_j^{\mathrm{S}}</math> of body <math>\Delta_j</math> needs to be		The scattered potential <math>\phi_j^{\mathrm{S}}</math> of body <math>\Delta_j</math> needs to be

Meylan: /* The interaction in water of infinite depth */

2006-06-24T00:03:53Z

The interaction in water of infinite depth

← Older revision		Revision as of 00:03, 24 June 2006
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	between propagating and decaying modes, the first superscript denotes the kind		between propagating and decaying modes, the first superscript denotes the kind
	of scattered mode, the second one the kind of incident mode.		of scattered mode, the second one the kind of incident mode.
	If the ~~diffraction transfer operators~~ are ~~known (their calculation~~		If the [Diffraction Transfer Matrix for Infinite Depth]] are substituted we
	~~will be discussed later), the substitution of~~		obtain the
	~~equations (inc_coeff) into equations (diff_op) give~~ the
	required equations to determine the coefficients and coefficient		required equations to determine the coefficients and coefficient
	functions of the scattered wavefields of all bodies,		functions of the scattered wavefields of all bodies,
	<center><math> ~~(eq_op)~~		<center><math>
	~~\begin{matrix}~~		A_{0n}^l = \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{pp}
	A_{0n}^l =& \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{pp}
	\Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{j=1,j		\Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{j=1,j
	\neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j		\neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
	H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathbf{i}		H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathbf{i}
	(\nu - \mu) \vartheta_{jl}} \Big]\\		(\nu - \mu) \vartheta_{jl}} \Big]
	&+ \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty}		</math></center>
			<center><math>
			+ \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty}
	B_{ln\mu}^\mathrm{pd} (\xi) \Big[D_{l\mu}^{\mathrm{In}}(\eta) +		B_{ln\mu}^\mathrm{pd} (\xi) \Big[D_{l\mu}^{\mathrm{In}}(\eta) +
	\sum_{j=1,j \neq l}^{N} \sum_{\nu =		\sum_{j=1,j \neq l}^{N} \sum_{\nu =
	-\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta		-\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta
	R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}} \Big] \mathrm{d}\xi,\\		R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}} \Big] \mathrm{d}\xi,
	A_n^l (\eta) &= \sum_{\mu = -\infty}^{\infty}		</math></center>
			<center><math>
			A_n^l (\eta) = \sum_{\mu = -\infty}^{\infty}
	B_{ln\mu}^\mathrm{dp} (\eta) \Big[		B_{ln\mu}^\mathrm{dp} (\eta) \Big[
	D_{l0\mu}^{\mathrm{In}} + \sum_{j=1,j		D_{l0\mu}^{\mathrm{In}} + \sum_{j=1,j
	\neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j		\neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
	H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathbf{i}		H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathbf{i}
	(\nu - \mu) \vartheta_{jl}}\Big]\\		(\nu - \mu) \vartheta_{jl}}\Big]
	& + \int\limits_{0}^{\infty}		</math></center>
			<center><math>
			+ \int\limits_{0}^{\infty}
	\sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{dd} (\eta;\xi)		\sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{dd} (\eta;\xi)
	\Big[ D_{l\mu}^{\mathrm{In}}(\eta) +		\Big[ D_{l\mu}^{\mathrm{In}}(\eta) +
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	-\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta		-\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta
	R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}}\Big] \mathrm{d}\xi,		R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}}\Big] \mathrm{d}\xi,
	~~\end{matrix}~~
	</math></center>		</math></center>
	<math>n \in \~~mathit~~{Z},\, l = 1, \ldots, N</math>. It has to be noted that all		<math>n \in \mathbb{Z},\, l = 1, \ldots, N</math>. It has to be noted that all
	equations are coupled so that it is necessary to solve for all		equations are coupled so that it is necessary to solve for all
	scattered coefficients and coefficient functions simultaneously.		scattered coefficients and coefficient functions simultaneously.
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	<center><math>		<center><math>
	\mathbf{B}_l = \left[		\mathbf{B}_l = \left[
	\begin{matrix~~}{cc~~} \mathbf{B}_l^{\mathrm{pp}} & \mathbf{B}_l^{\mathrm{pd}}\\		\begin{matrix} \mathbf{B}_l^{\mathrm{pp}} & \mathbf{B}_l^{\mathrm{pd}}\\
	\mathbf{B}_l^{\mathrm{dp}} & \mathbf{B}_l^{\mathrm{dd}}		\mathbf{B}_l^{\mathrm{dp}} & \mathbf{B}_l^{\mathrm{dd}}
	\end{matrix} \right],		\end{matrix} \right],
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	<math>\Delta_l</math>, <math>\mathbf{d}_l^{\mathrm{In}}</math> to be the vector of		<math>\Delta_l</math>, <math>\mathbf{d}_l^{\mathrm{In}}</math> to be the vector of
	coefficients of the ambient wavefield, and making use of a coordinate		coefficients of the ambient wavefield, and making use of a coordinate
	transformation matrix <math>{\mathbf T}_{jl}</math> given by		transformation matrix <math>{\mathbf T}_{lj}</math> given by
	<center><math> ~~(T_elem_deep)~~		<center><math>
	({\mathbf T}_{jl})_{pq} = H_{p-q}^{(1)}(\alpha R_{jl}) \, \mathrm{e}^{\mathrm{i}(p-q)		({\mathbf T}_{lj})_{pq} = H_{p-q}^{(1)}(\alpha R_{jl}) \, \mathrm{e}^{\mathrm{i}(p-q)
	\vartheta_{jl}}		\vartheta_{jl}}
	</math></center>		</math></center>
	for the propagating modes, and		for the propagating modes, and
	<center><math>		<center><math>
	({\mathbf T}_{jl})_{pq} = (-1)^{q} \, K_{p-q} (\eta R_{jl}) \, \mathrm{e}^{\mathbf{i}		({\mathbf T}_{lj})_{pq} = (-1)^{q} \, K_{p-q} (\eta R_{jl}) \, \mathrm{e}^{\mathbf{i}
	(p-q) \vartheta_{jl}}		(p-q) \vartheta_{jl}}
	</math></center>		</math></center>
	for the decaying modes, a linear system of equations		for the decaying modes, a linear system of equations
	for the unknown coefficients follows ~~from equations (eq_op),~~		for the unknown coefficients follows
	<center><math> ~~(eq_Binf)~~		<center><math>
	{\mathbf a}_l =		{\mathbf a}_l =
	{\mathbf {B}}_l \Big(		{\mathbf {B}}_l \Big(
	{\mathbf d}_l^{\mathrm{In}} +		{\mathbf d}_l^{\mathrm{In}} +
	\sum_{j=1,j \neq l}^{N} ~~trans~~ {\mathbf T}_{jl} \,		\sum_{j=1,j \neq l}^{N} {\mathbf T}_{lj} \,
	{\mathbf a}_j \Big), \quad l=1, \ldots, N,		{\mathbf a}_j \Big), \quad l=1, \ldots, N.
	</math></center>		</math></center>
	~~where the left superscript <math>\mathrm{t}</math> indicates transposition.~~
	The matrix <math>{\mathbf \hat{B}}_l</math> denotes the infinite depth diffraction		The matrix <math>{\mathbf \hat{B}}_l</math> denotes the infinite depth diffraction
	transfer matrix <math>{\mathbf B}_l</math> in which the elements associated with		transfer matrix <math>{\mathbf B}_l</math> in which the elements associated with

Meylan: /* The interaction in water of infinite depth */

2006-06-22T21:27:04Z

The interaction in water of infinite depth

Show changes

Meylan: /* Eigenfunction expansion of the potential */

2006-06-22T21:09:24Z

Eigenfunction expansion of the potential

← Older revision		Revision as of 21:09, 22 June 2006
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	=Eigenfunction expansion of the potential=		=Eigenfunction expansion of the potential=

	The scattered potential of body <math>\Delta_j</math> can be expanded in		The scattered potential of body <math>\Delta_j</math> can be expanded using
	~~cylindrical eigenfunctions~~,		the [[Cylindrical Eigenfunction Expansion]] for [[Infinite Depth]],
	<center><math> ~~(basisrep_out)~~		<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} A_{0 \nu}^j H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu		\infty}^{\infty} A_{0 \nu}^j H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu
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	modes are functions. <math>H_\nu^{(1)}</math> and <math>K_\nu</math> are the Hankel function		modes are functions. <math>H_\nu^{(1)}</math> and <math>K_\nu</math> are the Hankel function
	of the first kind and the modified Bessel function of the second kind		of the first kind and the modified Bessel function of the second kind
	respectively, both of order <math>\nu</math> ~~as defined in~~ [~~[Abramowitz and Stegun 1964]~~].		respectively, both of order <math>\nu</math>
			([http://en.wikipedia.org/wiki/Bessel_function : Bessel functions]).
	The incident potential upon body <math>\Delta_j</math> can be expanded in		The incident potential upon body <math>\Delta_j</math> can be expanded in
	cylindrical eigenfunctions,		cylindrical eigenfunctions,
	<center><math> ~~(basisrep_in)~~		<center><math>
	\phi_j^\mathrm{I} (r_j,\theta_j,z) = \mathrm{e}^{\alpha z} \sum_{\mu = -		\phi_j^\mathrm{I} (r_j,\theta_j,z) = \mathrm{e}^{\alpha z} \sum_{\mu = -
	\infty}^{\infty} D_{0 \mu}^j J_\mu (\alpha r_j) \mathrm{e}^{\mathrm{i}\mu		\infty}^{\infty} D_{0 \mu}^j J_\mu (\alpha r_j) \mathrm{e}^{\mathrm{i}\mu

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

2006-06-22T10:10:30Z

The diffraction transfer matrix of rotated bodies

← Older revision		Revision as of 10:10, 22 June 2006
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	~~=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]]		[[Category:Interaction Theory]]

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

2006-06-22T10:10:01Z

Calculation of the diffraction transfer matrix for bodies of arbitrary geometry

← Older revision		Revision as of 10:10, 22 June 2006
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	integration weights depending on the discretisation of the continuous variable.		integration weights depending on the discretisation of the continuous variable.

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

Meylan at 10:31, 18 June 2006

2006-06-18T10:31:07Z

← Older revision		Revision as of 10:31, 18 June 2006
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	[[Category:~~Linear Water-Wave~~ Theory]]		[[Category:Interaction Theory]]

Mpeter at 11:57, 16 June 2006

2006-06-16T11:57:38Z

← Older revision		Revision as of 11:57, 16 June 2006
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	propagating and decaying modes likewise.		propagating and decaying modes likewise.



	~~=Solving the resulting system of equations=~~
	~~After the coefficient vector of the ambient incident wavefield, the~~
	~~diffraction transfer matrices and the coordinate~~
	~~transformation matrices have been calculated, the system of~~
	~~equations (eq_B_inf),~~
	~~has to be solved. This system can be represented by the following~~
	~~matrix equation,~~
	~~<center><math>~~
	~~\left[ \begin{matrix}~~
	~~{\mathbf a}_1\\ {\mathbf a}_2\\ \\ \vdots \\ \\ {\mathbf a}_N~~
	~~\end{matrix} \right]~~
	~~= \left[ \begin{matrix}~~
	~~{{\mathbf B}}_1 {\mathbf d}_1^\mathrm{In}\\ {{\mathbf B}}_2 {\mathbf~~
	~~d}_2^\mathrm{In}\\ \\ \vdots \\ \\ {{\mathbf B}}_N {\mathbf d}_N^\mathrm{In}~~
	~~\end{matrix} \right]~~
	+
	~~\left[ \begin{matrix}~~
	~~\mathbf{0} & {{\mathbf B}}_1 {\mathbf T}_{21} & {{\mathbf B}}_1~~
	~~{\mathbf T}_{31} & \dots & {{\mathbf B}}_1 {\mathbf T}_{N1}\\~~
	~~{{\mathbf B}}_2 {\mathbf T}_{12} & \mathbf{0} & {{\mathbf B}}_2~~
	~~{\mathbf T}_{32} & \dots & {{\mathbf B}}_2 {\mathbf T}_{N2}\\~~
	~~& & \mathbf{0} & &\\~~
	~~\vdots & & & \ddots & \vdots\\~~
	~~& & & & \\~~
	~~{{\mathbf B}}_N {\mathbf T}_{1N} & & \dots &~~
	~~& \mathbf{0}~~
	~~\end{matrix} \right]~~
	~~\left[ \begin{matrix}~~
	~~{\mathbf a}_1\\ {\mathbf a}_2\\ \\ \vdots \\ \\ {\mathbf a}_N~~
	~~\end{matrix} \right],~~
	~~</math></center>~~
	~~where <math>\mathbf{0}</math> denotes the zero-matrix which is of the same~~
	~~dimension as <math>{{\mathbf B}}_j</math>, say <math>n</math>. This matrix equation can be~~
	~~easily transformed into a classical <math>(N \, n)</math>-dimensional linear system of~~
	~~equations.~~

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

Mpeter at 11:38, 12 June 2006

2006-06-12T11:38:07Z

@@ Line 96: / Line 96: @@
 The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
 expanded in the eigenfunctions corresponding to the incident wavefield upon
-<math>\Delta_l</math>. A detailed illustration of how to accomplish this will be given
+<math>\Delta_l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]]). Let <math>D_{l0\mu}^{\mathrm{In}}</math> denote the coefficients of this
 ambient incident wavefield corresponding to the propagating modes and
 <math>D_{l\mu}^{\mathrm{In}} (\cdot)</math>  denote the coefficients functions
@@ Line 386: / Line 385: @@
 propagating and decaying modes likewise.
-=Representation of the ambient wavefield in the eigenfunction representation=
 =Solving the resulting system of equations=

Meylan: /* Representation of the ambient wavefield in the eigenfunction representation */

2006-06-12T10:11:33Z

Representation of the ambient wavefield in the eigenfunction representation

← Older revision		Revision as of 10:11, 12 June 2006
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	</math></center>		</math></center>

	==Solving the resulting system of equations==		=Solving the resulting system of equations=
	After the coefficient vector of the ambient incident wavefield, the		After the coefficient vector of the ambient incident wavefield, the
	diffraction transfer matrices and the coordinate		diffraction transfer matrices and the coordinate