Difference between revisions of "Category:Interaction Theory"

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[[Graf's Addition Theorem]]
 
[[Graf's Addition Theorem]]
 
<center><math> (transf)
 
<center><math> (transf)
−
H_\tau(k r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
+
H^{(1)}_\tau(k r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
−
\sum_{\nu = - \infty}^{\infty} H_{\tau + \nu} (k R_{jl}) \,
+
\sum_{\nu = - \infty}^{\infty} H_^{(1)}{\tau + \nu} (k R_{jl}) \,
 
J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
 
J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
 
</math></center>
 
</math></center>
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= \sum_{\tau = -
 
= \sum_{\tau = -
 
\infty}^{\infty} A_{\tau}^j \sum_{\nu = -\infty}^{\infty}
 
\infty}^{\infty} A_{\tau}^j \sum_{\nu = -\infty}^{\infty}
−
(-1)^\nu H_{\tau-\nu} (k  R_{jl}) J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu
+
(-1)^\nu H^{(1)}_{\tau-\nu} (k  R_{jl}) J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu
 
\theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}}  
 
\theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}}  
 
</math></center>
 
</math></center>
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=  \sum_{\nu =
 
=  \sum_{\nu =
 
-\infty}^{\infty}  \Big[ \sum_{\tau = - \infty}^{\infty} A_{\tau}^j
 
-\infty}^{\infty}  \Big[ \sum_{\tau = - \infty}^{\infty} A_{\tau}^j
−
(-1)^\nu H_{\tau-\nu} (k R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu)
+
(-1)^\nu H^{(1)}_{\tau-\nu} (k R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu)
 
\varphi_{jl}} \Big] J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.   
 
\varphi_{jl}} \Big] J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.   
 
</math></center>
 
</math></center>

Revision as of 15:55, 20 June 2006

Interaction theory is based on calculating a solution for a number of individual scatterers without simply discretising the total problem. THe theory is generally applied in three dimensions. Essentially the Cylindrical Eigenfunction Expansion surrounding each body is used coupled with some way of mapping these. Various approximations were developed until the the Kagemoto and Yue Interaction Theory which contained a solution without any approximation. This solution method is valid, provided only that an escribed circle can be drawn around each body.

Illustrative Example

We present an illustrative example of an interaction theory for the case of [math]\displaystyle{ n }[/math] Bottom Mounted Cylinders. This theory was presented in Linton and Evans 1990 and it can be derived from the Kagemoto and Yue Interaction Theory by simply assuming that each body is a cylinder.

Equations of Motion

After we have Removed the Depth Dependence the problem consists of [math]\displaystyle{ n }[/math] cylinders of radius [math]\displaystyle{ a_j }[/math] subject to Helmholtz's Equation

[math]\displaystyle{ \nabla^2 \phi -k^2\phi= 0, }[/math]

where [math]\displaystyle{ k }[/math] is the positive real root of the Dispersion Relation for a Free Surface

[math]\displaystyle{ k \tanh k d = \alpha\,. }[/math]

Eigenfunction expansion of the potential

Each body is subject to an incident potential and moves in response to this incident potential to produce a scattered potential. Each of these is expanded using the Cylindrical Eigenfunction Expansion The scattered potential of a body [math]\displaystyle{ \Delta_j }[/math] can be expressed as

[math]\displaystyle{ (basisrep_out_d) \phi_j^\mathrm{S} (r_j,\theta_j,z) = \sum_{\mu = - \infty}^{\infty} A_{\mu}^j H^{(1)}_\mu (k r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j}, }[/math]

with discrete coefficients [math]\displaystyle{ A_{\mu}^j }[/math], where [math]\displaystyle{ (r_j,\theta_j) }[/math] are polar coordinates centered at center of the [math]\displaystyle{ j }[/math]th cylinder.

The incident potential upon body [math]\displaystyle{ \Delta_j }[/math] can be also be expanded in regular cylindrical eigenfunctions,

[math]\displaystyle{ (basisrep_in_d) \phi_j^\mathrm{I} (r_j,\theta_j,z) = \sum_{\nu = - \infty}^{\infty} D_{\nu}^j J_\nu (k r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}, }[/math]

with discrete coefficients [math]\displaystyle{ D_{\nu}^j }[/math]. In these expansions, [math]\displaystyle{ J_\nu }[/math] and [math]\displaystyle{ H^{(1)}_\nu }[/math] denote Bessel and Hankel function respectively (: Bessel functions) both of first kind and order [math]\displaystyle{ \nu }[/math]. For comparison with the Kagemoto and Yue Interaction Theory (which is written slightly differently), we remark that, for real [math]\displaystyle{ x }[/math],

[math]\displaystyle{ K_\nu (-\mathrm{i}x) = \frac{\pi \mathrm{i}^{\nu+1}}{2} H_\nu^{(1)}(x) \quad \mathrm{and} \quad I_\nu (-\mathrm{i}x) = \mathrm{i}^{-\nu} J_\nu(x) }[/math]

with [math]\displaystyle{ I_\nu }[/math] and [math]\displaystyle{ K_\nu }[/math] denoting the modified Bessel functions of first and second kind, respectively, both of order [math]\displaystyle{ \nu }[/math].

Derivation of the system of equations

A system of equations for the unknown coefficients (in the expansion (basisrep_out_d)) of the scattered wavefields of all bodies is developed. This system of equations is based on transforming the scattered potential of [math]\displaystyle{ \Delta_j }[/math] into an incident potential upon [math]\displaystyle{ \Delta_l }[/math] ([math]\displaystyle{ j \neq l }[/math]). Doing this for all bodies simultaneously, and relating the incident and scattered potential for each body, a system of equations for the unknown coefficients is developed.

The scattered potential [math]\displaystyle{ \phi_j^{\mathrm{S}} }[/math] of body [math]\displaystyle{ \Delta_j }[/math] needs to be represented in terms of the incident potential [math]\displaystyle{ \phi_l^{\mathrm{I}} }[/math] upon [math]\displaystyle{ \Delta_l }[/math], [math]\displaystyle{ j \neq l }[/math]. This can be accomplished by using Graf's Addition Theorem

[math]\displaystyle{ (transf) H^{(1)}_\tau(k r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} = \sum_{\nu = - \infty}^{\infty} H_^{(1)}{\tau + \nu} (k R_{jl}) \, J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l, }[/math]

Making use of the eigenfunction expansion as well as equation (transf), the scattered potential of [math]\displaystyle{ \Delta_j }[/math] (cf.~ (basisrep_out_d)) can be expressed in terms of the incident potential upon [math]\displaystyle{ \Delta_l }[/math] as

[math]\displaystyle{ \phi_j^{\mathrm{S}} (r_l,\theta_l,z) = \sum_{\tau = - \infty}^{\infty} A_{\tau}^j \sum_{\nu = -\infty}^{\infty} (-1)^\nu H^{(1)}_{\tau-\nu} (k R_{jl}) J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}} }[/math]
[math]\displaystyle{ = \sum_{\nu = -\infty}^{\infty} \Big[ \sum_{\tau = - \infty}^{\infty} A_{\tau}^j (-1)^\nu H^{(1)}_{\tau-\nu} (k R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. }[/math]

The ambient incident wavefield [math]\displaystyle{ \phi^{\mathrm{In}} }[/math] can also be expanded in the eigenfunctions corresponding to the incident wavefield upon [math]\displaystyle{ \Delta_l }[/math]. Let [math]\displaystyle{ \tilde{D}_{n\nu}^{l} }[/math] denote the coefficients of this ambient incident wavefield in the incoming eigenfunction expansion for [math]\displaystyle{ \Delta_l }[/math] (cf. the example in Cylindrical Eigenfunction Expansion).

[math]\displaystyle{ \phi^{\mathrm{In}}(r_l,\theta_l,z) = A \frac{g}{\omega} \, e^{\mathrm{i}\alpha (O_x^l \cos \chi + O_x^l \sin \chi)} \, \mathrm{e}^{\alpha z} \sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 - \chi)} J_\mu(\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}. }[/math]

The total incident wavefield upon body [math]\displaystyle{ \Delta_j }[/math] can now be expressed as

[math]\displaystyle{ \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}} (r_l,\theta_l,z) }[/math]
[math]\displaystyle{ = \sum_{\nu = -\infty}^{\infty} \Big[{D}_\nu^{l} + \sum_{j=1,j \neq l}^{n} \sum_{\tau = -\infty}^{\infty} A_{\tau}^j (-1)^\nu H_{\tau - \nu} (k R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. }[/math]

Final Equations

The scattered and incident potential can be related by the Diffraction Transfer Matrix for a Bottom Mounted Cylinder acting in the following way,

[math]\displaystyle{ (diff_op) A_{\mu}^l = J'_n(k a_j)/H'_n(k a_j)D_{\mu}^l. }[/math]

The substitution of (inc_coeff) into (diff_op) gives the required equations to determine the coefficients of the scattered wavefields of all bodies,

[math]\displaystyle{ (eq_op) A_{\mu}^l = \sum_{\nu = -\infty}^{\infty} B_{\mu\nu} \Big[ \tilde{D}_{\nu}^{l} + \sum_{j=1,j \neq l}^{N} \sum_{\tau = -\infty}^{\infty} A_{\tau}^j (-1)^\nu H_{\tau - \nu} (k R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big], }[/math]

[math]\displaystyle{ m \in \mathbb{N} }[/math], [math]\displaystyle{ \mu \in \mathbb{Z} }[/math], [math]\displaystyle{ l=1,\dots,N }[/math].

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