Interaction Theory for Cylinders

Introduction

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] whose center is at [math]\displaystyle{ (x_j,y_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 = k_\infty,. }[/math]

where [math]\displaystyle{ k_\infty }[/math] is the Infinite Depth wave number and [math]\displaystyle{ d }[/math] is the water depth.

Eigenfunction expansion of the potential

Each cylinder 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 cylinder [math]\displaystyle{ j }[/math] can be expressed as

[math]\displaystyle{ \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 cylinder [math]\displaystyle{ j }[/math] can be also be expanded in regular cylindrical eigenfunctions,

[math]\displaystyle{ \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] which are given explicitly when we define the incident wave below.

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

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

[math]\displaystyle{ 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]

where [math]\displaystyle{ (R_{jl},\varphi_{jl}) }[/math] are the polar coordinates of the mean centre position of cylinder [math]\displaystyle{ l }[/math] in the local coordinates of cylinder [math]\displaystyle{ j }[/math].

Making use of the eigenfunction expansion as well as Graf's Addition Theorem, the scattered potential of cylinder [math]\displaystyle{ j }[/math] can be expressed in terms of the incident potential upon cylinder [math]\displaystyle{ 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} 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 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 cylinder [math]\displaystyle{ 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} k (x_l \cos \chi + y_l \sin \chi)} \sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 - \chi)} J_\mu(k r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}. }[/math]

Let [math]\displaystyle{ \tilde{D}_{\nu}^{l} }[/math] denote the coefficients of this ambient incident wavefield in the incoming eigenfunction expansion for [math]\displaystyle{ \Delta_l }[/math] so that

[math]\displaystyle{ \tilde{D}_{\nu}^{l} = A \frac{g}{\omega} \, e^{\mathrm{i} k (x_l \cos \chi + y_l \sin \chi)} \mathrm{e}^{\mathrm{i}\nu (\pi/2 - \chi)}. }[/math]

The total incident wavefield upon cylinder [math]\displaystyle{ 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]

which can be written as

[math]\displaystyle{ \sum_{\nu = -\infty}^{\infty} {D}_\nu^{l} J_\nu (kr_l) \mathrm{e}^{\mathrm{i}\nu \theta_l} = \sum_{\nu = -\infty}^{\infty} \Big[\tilde{D}_\nu^{l} + \sum_{j=1,j \neq l}^{n} \sum_{\tau = -\infty}^{\infty} A_{\tau}^j 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]

Therefore it follows that

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

Final Equations

The scattered and incident potential can be related by the Diffraction Transfer Matrix for a Bottom Mounted Cylinder so that,

[math]\displaystyle{ A_{\mu}^l= \frac{H^{(1)}_\mu{}'(k a_l)}{J'_\mu(k a_l)} D_{\mu}^l . }[/math]

This gives the required equations to determine the coefficients of the scattered wavefields of all bodies,

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

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