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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. |
− |
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− | = 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 =
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− | | |
− | 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]]
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− | <center><math>
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− | \nabla^2 \phi -k^2\phi= 0,
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− | </math></center>
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− | where <math>k</math> is the positive real root of the [[Dispersion Relation for a Free Surface]]
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− | <center><math>
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− | k_m \tanh k_m d = \alpha\,.
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− | </math></center>
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− | | |
− | =Eigenfunction expansion of the potential=
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− | | |
− | Each body is subject to an incident potential and moves in response to this
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− | incident potential to produce a scattered potential. Each of these is
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− | expanded using the [[Cylindrical Eigenfunction Expansion]]
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− | The scattered potential of a body
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− | <math>\Delta_j</math> can be expressed as
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− | <center><math> (basisrep_out_d)
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− | \phi_j^\mathrm{S} (r_j,\theta_j,z) = \sum_{\mu = -
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− | \infty}^{\infty} A_{\mu}^j H_\mu (k r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j},
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− | </math></center>
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− | with discrete coefficients <math>A_{\mu}^j</math>, where <math>(r_j,\theta_j)</math
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− | are cylindrical polar coordinate centered at center of the jth cylinder.
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− | | |
− | The incident potential upon body <math>\Delta_j</math> can be also be expanded in
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− | regular cylindrical eigenfunctions,
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− | <center><math> (basisrep_in_d)
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− | \phi_j^\mathrm{I} (r_j,\theta_j,z) =
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− | \sum_{\nu = - \infty}^{\infty} D_{\nu}^j I_\nu (k r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j},
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− | </math></center>
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− | with discrete coefficients <math>D_{\nu}^j</math>. In these expansions, <math>J_\nu</math>
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− | and <math>H_\nu</math> denote Bessel and Hankel function
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− | respectively ([http://en.wikipedia.org/wiki/Bessel_function : Bessel functions])
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− | both of order <math>\nu</math>.
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− | | |
− | =Derivation of the system of equations=
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− | | |
− | A system of equations for the unknown
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− | coefficients (in the expansion (basisrep_out_d)) of the
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− | scattered wavefields of all bodies is developed. This system of
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− | equations is based on transforming the
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− | scattered potential of <math>\Delta_j</math> into an incident potential upon
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− | <math>\Delta_l</math> (<math>j \neq l</math>). Doing this for all bodies simultaneously,
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− | and relating the incident and scattered potential for each body, a system
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− | of equations for the unknown coefficients is developed.
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− | Making use of the periodicity of the geometry and of the ambient incident
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− | wave, this system of equations can then be simplified.
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− | | |
− | The scattered potential <math>\phi_j^{\mathrm{S}}</math> of body <math>\Delta_j</math> needs to be
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− | represented in terms of the incident potential <math>\phi_l^{\mathrm{I}}</math>
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− | upon <math>\Delta_l</math>, <math>j \neq l</math>. This can be accomplished by using
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− | [[Graf's Addition Theorem]]
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− | <center><math> (transf)
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− | K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
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− | \sum_{\nu = - \infty}^{\infty} K_{\tau + \nu} (k_m R_{jl}) \,
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− | I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
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− | </math></center>
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− | which is valid provided that <math>r_l < R_{jl}</math>. Here, <math>(R_{jl},\varphi_{jl})</math> are the polar coordinates of the mean centre position of <math>\Delta_{l}</math> in the local coordinates of <math>\Delta_{j}</math>.
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− | | |
− | The limitation <math>r_l < R_{jl}</math> only requires that the escribed cylinder of each body
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− | <math>\Delta_l</math> does not enclose any other origin <math>O_j</math> (<math>j \neq l</math>). However, the
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− | expansion of the scattered and incident potential in cylindrical
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− | eigenfunctions is only valid outside the escribed cylinder of each
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− | body. Therefore the condition that the
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− | escribed cylinder of each body <math>\Delta_l</math> does not enclose any other
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− | origin <math>O_j</math> (<math>j \neq l</math>) is superseded by the more rigorous
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− | restriction that the escribed cylinder of each body may not contain any
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− | other body.
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− | | |
− | Making use of the eigenfunction expansion as well as equation (transf), the scattered potential
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− | of <math>\Delta_j</math> (cf.~ (basisrep_out_d)) can be expressed in terms of the
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− | incident potential upon <math>\Delta_l</math> as
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− | <center><math>
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− | \phi_j^{\mathrm{S}} (r_l,\theta_l,z)
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− | = \sum_{m=0}^\infty f_m(z) \sum_{\tau = -
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− | \infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty}
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− | (-1)^\nu K_{\tau-\nu} (k_m R_{jl}) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu
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− | \theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}}
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− | </math></center>
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− | <center><math>
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− | = \sum_{m=0}^\infty f_m(z) \sum_{\nu =
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− | -\infty}^{\infty} \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j
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− | (-1)^\nu K_{\tau-\nu} (k_m R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu)
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− | \varphi_{jl}} \Big] I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
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− | </math></center>
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− | The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
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− | expanded in the eigenfunctions corresponding to the incident wavefield upon
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− | <math>\Delta_l</math>. Let <math>\tilde{D}_{n\nu}^{l}</math> denote the coefficients of this
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− | ambient incident wavefield in the incoming eigenfunction expansion for
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− | <math>\Delta_l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]]). The total
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− | incident wavefield upon body <math>\Delta_j</math> can now be expressed as
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− | <center><math>
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− | \phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
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− | \sum_{j=-\infty,j \neq l}^{\infty} \, \phi_j^{\mathrm{S}}
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− | (r_l,\theta_l,z)
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− | </math></center>
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− | <center><math>
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− | = \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
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− | \Big[ \tilde{D}_{n\nu}^{l} +
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− | \sum_{j=1,j \neq l}^{N} \sum_{\tau =
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− | -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
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− | R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n
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− | r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
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− | </math></center>
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− | | |
− | = Final Equations =
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− | | |
− | The scattered and incident potential can be related by the
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− | [[Diffraction Transfer Matrix]] acting in the following way,
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− | <center><math> (diff_op)
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− | A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
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− | \mu \nu} D_{n\nu}^l.
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− | </math></center>
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− | | |
− | The substitution of (inc_coeff) into (diff_op) gives the
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− | required equations to determine the coefficients of the scattered
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− | wavefields of all bodies,
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− | <center><math> (eq_op)
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− | A_{m\mu}^l = \sum_{n=0}^{\infty}
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− | \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
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− | \Big[ \tilde{D}_{n\nu}^{l} +
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− | \sum_{j=1,j \neq l}^{N} \sum_{\tau =
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− | -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
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− | R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big],
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− | </math></center>
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− | <math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.
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| [[Category:Linear Water-Wave Theory]] | | [[Category:Linear Water-Wave Theory]] |