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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 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>
| |
| − | subject to [[Helmholtz's equation]]
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| − | <center><math>
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| − | \nabla^2 \phi -k^2\phi= 0,
| |
| − | </math></center>
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| − | where
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| − | <center><math>
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| − | \frac{\partial \phi}{\partial z} = \alpha \phi, \;
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| − | {\mathbf{x}} \in \Gamma^\mathrm{f},
| |
| − | </math></center>
| |
| − | <center><math>
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| − | \frac{\partial \phi}{\partial z} = 0, \; \mathbf{y} \in D, \ z=-d,
| |
| − | </math></center>
| |
| − | where <math>D</math> is the
| |
| − | is the domain occupied by the water and
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| − | <math>\Gamma^\mathrm{f}</math> is the free water surface. At the immersed body
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| − | surface <math>\Gamma_j</math> of body <math>\Delta_j</math>, <math>j=1,\dots,N</math>, the water velocity potential has to
| |
| − | equal the normal velocity of the body <math>\mathbf{v}_j</math>,
| |
| − | <center><math>
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| − | \frac{\partial \phi}{\partial n} = \mathbf{v}_j, \; {\mathbf{y}}
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| − | \in \Gamma_j.
| |
| − | </math></center>
| |
| − | where the normal derivative is given by the particaluar equations of motion of the body.
| |
| − | Moreover, the [[Sommerfeld Radiation Condition]] is imposed.
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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) =
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| − | \sum_{m=0}^{\infty} f_m(z) \sum_{\mu = -
| |
| − | \infty}^{\infty} A_{m \mu}^j K_\mu (k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j},
| |
| − | </math></center>
| |
| − | with discrete coefficients <math>A_{m \mu}^j</math>, where
| |
| − | <center><math>
| |
| − | f_m(z) = \frac{\cos k_m (z+d)}{\cos k_m d}.
| |
| − | </math></center>
| |
| − | and
| |
| − | where <math>k_n</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
| |
| − | <center><math> (eq_km)
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| − | \alpha + k_m \tan k_m d = 0\,.
| |
| − | </math></center>
| |
| − | where <math>k_0</math> is the
| |
| − | imaginary root with positive imaginary part
| |
| − | and <math>k_m</math>, <math>m>0</math>, are given the positive real roots ordered
| |
| − | with increasing size.
| |
| − | | |
| − | The incident potential upon body <math>\Delta_j</math> can be also be expanded in
| |
| − | regular cylindrical eigenfunctions,
| |
| − | <center><math> (basisrep_in_d)
| |
| − | \phi_j^\mathrm{I} (r_j,\theta_j,z) = \sum_{n=0}^{\infty} f_n(z)
| |
| − | \sum_{\nu = - \infty}^{\infty} D_{n\nu}^j I_\nu (k_n r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j},
| |
| − | </math></center>
| |
| − | with discrete coefficients <math>D_{n\nu}^j</math>. In these expansions, <math>I_\nu</math>
| |
| − | and <math>K_\nu</math> denote the modified [http://en.wikipedia.org/wiki/Bessel_function : Bessel functions]
| |
| − | of the first and second kind, respectively, both of order <math>\nu</math>.
| |
| − | Note that in (basisrep_out_d) (and (basisrep_in_d)) the term for <math>m =0</math> or
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| − | <math>n=0</math>) corresponds to the propagating modes while the
| |
| − | terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.
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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,
| |
| − | </math></center>
| |
| − | 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
| |
| − | <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
| |
| − | 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_{m=0}^\infty f_m(z) \sum_{\tau = -
| |
| − | \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}}
| |
| − | </math></center>
| |
| − | <center><math>
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| − | = \sum_{m=0}^\infty f_m(z) \sum_{\nu =
| |
| − | -\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}.
| |
| − | </math></center>
| |
| − | 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
| |
| − | <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
| |
| − | <center><math>
| |
| − | \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)
| |
| − | </math></center>
| |
| − | <center><math>
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| − | = \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
| |
| − | \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}.
| |
| − | </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.
| |
| − | </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],
| |
| − | </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]] |
