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| | Interaction theory is based on calculating a solution for a number of individual scatterers | | 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 | + | without simply discretising the total problem. The theory is generally applied in |
| | three dimensions. | | three dimensions. |
| | Essentially the [[Cylindrical Eigenfunction Expansion]] | | Essentially the [[Cylindrical Eigenfunction Expansion]] |
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| | a solution without any approximation. This solution method is valid, provided only that | | a solution without any approximation. This solution method is valid, provided only that |
| | an escribed circle can be drawn around each body. | | an escribed circle can be drawn around each body. |
| − |
| |
| − | = Illustrative Example =
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| − |
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| | We present an illustrative example of an interaction theory for the case of <math>n</math> | | We present an illustrative example of an interaction theory for the case of <math>n</math> |
| − | [[Bottom Mounted Cylinder|Bottom Mounted Cylinders]]. This theory was presented in [[Linton and Evans 1990]] and it
| + | [[Linton and Evans 1990]] presented an [[Interaction Theory for Cylinders]] |
| − | can be derived from the [[Kagemoto and Yue Interaction Theory]] by simply assuming that each
| + | which was [[Kagemoto and Yue Interaction Theory]] simplified by assuming that each |
| − | body is a cylinder.
| + | body is a [[Bottom Mounted Cylinder]]. |
| − | | |
| − | = Equations of Motion =
| |
| − | | |
| − | After we have [[Removing the Depth Dependence|Removed the Depth Dependence]]
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| − | the problem consists of <math>n</math> cylinders of radius <math>a_j</math>
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| − | subject to [[Helmholtz's Equation]]
| |
| − | <center><math>
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| − | \nabla^2 \phi -k^2\phi= 0,
| |
| − | </math></center>
| |
| − | where <math>k</math> is the positive real root of the [[Dispersion Relation for a Free Surface]]
| |
| − | <center><math>
| |
| − | k \tanh k d = \alpha\,.
| |
| − | </math></center>
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| − | | |
| − | =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
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| − | expanded using the [[Cylindrical Eigenfunction Expansion]]
| |
| − | The scattered potential of a body
| |
| − | <math>\Delta_j</math> can be expressed as
| |
| − | <center><math> (basisrep_out_d)
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| − | \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></center>
| |
| − | with discrete coefficients <math>A_{\mu}^j</math>, where <math>(r_j,\theta_j)</math>
| |
| − | are polar coordinates centered at center of the <math>j</math>th cylinder.
| |
| − | | |
| − | The incident potential upon body <math>\Delta_j</math> can be also be expanded in
| |
| − | regular cylindrical eigenfunctions,
| |
| − | <center><math> (basisrep_in_d)
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| − | \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></center>
| |
| − | with discrete coefficients <math>D_{\nu}^j</math>. In these expansions, <math>J_\nu</math>
| |
| − | and <math>H^{(1)}_\nu</math> denote Bessel and Hankel function
| |
| − | respectively ([http://en.wikipedia.org/wiki/Bessel_function : Bessel functions])
| |
| − | both of order <math>\nu</math>. For
| |
| − | comparison with the [[Kagemoto and Yue Interaction Theory]]
| |
| − | (which is written slightly differently), we remark that, for real <math>x</math>,
| |
| − | <center><math>
| |
| − | K_\nu (-\mathrm{i}x) = \frac{\pi i^{\nu+1}}{2} H_\nu^{(1)}(x) \quad
| |
| − | \mathrm{and} \quad
| |
| − | I_\nu (-\mathrm{i}x) = i^{-\nu} J_\nu(x)
| |
| − | </math></center>
| |
| − | with <math>K_\nu^{(1)}</math> and <math>I_\nu</math> denoting the
| |
| − | Bessel functions, respectively, both of first kind and order <math>\nu</math>.
| |
| − | | |
| − | =Derivation of the system of equations=
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| − | | |
| − | 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>\Delta_j</math> into an incident potential upon
| |
| − | <math>\Delta_l</math> (<math>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>\phi_j^{\mathrm{S}}</math> of body <math>\Delta_j</math> needs to be
| |
| − | represented in terms of the incident potential <math>\phi_l^{\mathrm{I}}</math>
| |
| − | upon <math>\Delta_l</math>, <math>j \neq l</math>. This can be accomplished by using
| |
| − | [[Graf's Addition Theorem]]
| |
| − | <center><math> (transf)
| |
| − | H_\tau(k r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
| |
| − | \sum_{\nu = - \infty}^{\infty} H_{\tau + \nu} (k R_{jl}) \,
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| − | J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
| |
| − | </math></center>
| |
| − | | |
| − | Making use of the eigenfunction expansion as well as equation (transf), the scattered potential
| |
| − | of <math>\Delta_j</math> (cf.~ (basisrep_out_d)) can be expressed in terms of the
| |
| − | incident potential upon <math>\Delta_l</math> as
| |
| − | <center><math>
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| − | \phi_j^{\mathrm{S}} (r_l,\theta_l,z)
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| − | = \sum_{\tau = -
| |
| − | \infty}^{\infty} A_{\tau}^j \sum_{\nu = -\infty}^{\infty}
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| − | (-1)^\nu H_{\tau-\nu} (k R_{jl}) J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu
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| − | \theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}}
| |
| − | </math></center>
| |
| − | <center><math>
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| − | = \sum_{\nu =
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| − | -\infty}^{\infty} \Big[ \sum_{\tau = - \infty}^{\infty} A_{\tau}^j
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| − | (-1)^\nu H_{\tau-\nu} (k R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu)
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| − | \varphi_{jl}} \Big] J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
| |
| − | </math></center>
| |
| − | 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>. Let <math>\tilde{D}_{n\nu}^{l}</math> denote the coefficients of this
| |
| − | ambient incident wavefield in the incoming eigenfunction expansion for
| |
| − | <math>\Delta_l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]]).
| |
| − | | |
| − | <center><math>
| |
| − | \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)}
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| − | J_\mu(\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}.
| |
| − | </math></center>
| |
| − | | |
| − | The total
| |
| − | incident wavefield upon body <math>\Delta_j</math> can now be expressed as
| |
| − | <center><math>
| |
| − | \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}}
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| − | (r_l,\theta_l,z)
| |
| − | </math></center>
| |
| − | <center><math>
| |
| − | = \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
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| − | r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
| |
| − | </math></center>
| |
| − | | |
| − | = 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,
| |
| − | <center><math> (diff_op)
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| − | A_{\mu}^l = J'_n(k a_j)/H'_n(k a_j)D_{\mu}^l.
| |
| − | </math></center>
| |
| − | | |
| − | The substitution of (inc_coeff) into (diff_op) gives the
| |
| − | required equations to determine the coefficients of the scattered
| |
| − | wavefields of all bodies,
| |
| − | <center><math> (eq_op)
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| − | A_{\mu}^l =
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| − | \sum_{\nu = -\infty}^{\infty} B_{\mu\nu}
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| − | \Big[ \tilde{D}_{\nu}^{l} +
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| − | \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></center>
| |
| − | <math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.
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| | | | |
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| | [[Category:Linear Water-Wave Theory]] | | [[Category:Linear Water-Wave Theory]] |
