Category:Interaction Theory

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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_m \tanh k_m 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_\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)\lt /math are cylindrical polar coordinate centered at center of the jth cylinder. The incident potential upon body \lt math\gt \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 I_\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_\nu }[/math] denote Bessel and Hankel function respectively (: Bessel functions) 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. Making use of the periodicity of the geometry and of the ambient incident wave, this system of equations can then be simplified.

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) K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} = \sum_{\nu = - \infty}^{\infty} K_{\tau + \nu} (k_m R_{jl}) \, I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l, }[/math]

which is valid provided that [math]\displaystyle{ r_l \lt R_{jl} }[/math]. Here, [math]\displaystyle{ (R_{jl},\varphi_{jl}) }[/math] are the polar coordinates of the mean centre position of [math]\displaystyle{ \Delta_{l} }[/math] in the local coordinates of [math]\displaystyle{ \Delta_{j} }[/math].

The limitation [math]\displaystyle{ r_l \lt R_{jl} }[/math] only requires that the escribed cylinder of each body [math]\displaystyle{ \Delta_l }[/math] does not enclose any other origin [math]\displaystyle{ O_j }[/math] ([math]\displaystyle{ j \neq l }[/math]). However, the expansion of the scattered and incident potential in cylindrical eigenfunctions is only valid outside the escribed cylinder of each body. Therefore the condition that the escribed cylinder of each body [math]\displaystyle{ \Delta_l }[/math] does not enclose any other origin [math]\displaystyle{ O_j }[/math] ([math]\displaystyle{ j \neq l }[/math]) is superseded by the more rigorous restriction that the escribed cylinder of each body may not contain any other body.

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_{m=0}^\infty f_m(z) \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty} (-1)^\nu K_{\tau-\nu} (k_m R_{jl}) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}} }[/math]
[math]\displaystyle{ = \sum_{m=0}^\infty f_m(z) \sum_{\nu = -\infty}^{\infty} \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j (-1)^\nu K_{\tau-\nu} (k_m R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_m 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). 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=-\infty,j \neq l}^{\infty} \, \phi_j^{\mathrm{S}} (r_l,\theta_l,z) }[/math]
[math]\displaystyle{ = \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty} \Big[ \tilde{D}_{n\nu}^{l} + \sum_{j=1,j \neq l}^{N} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n 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 acting in the following way,

[math]\displaystyle{ (diff_op) A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n \mu \nu} D_{n\nu}^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_{m\mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu}^{l} + \sum_{j=1,j \neq l}^{N} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n 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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