Kagemoto and Yue Interaction Theory

Introduction

This is an interaction theory which provides the exact solution (i.e. it is not based on a Wide Spacing Approximation). The theory uses the Cylindrical Eigenfunction Expansion and Graf's Addition Theorem to represent the potential in local coordinates. The incident and scattered potential of each body are then related by the associated Diffraction Transfer Matrix.

The basic idea is as follows: The scattered potential of each body is represented in the Cylindrical Eigenfunction Expansion associated with the local coordinates centred at the mean centre position of the body. Using Graf's Addition Theorem, the scattered potential of all bodies (given in their local coordinates) can be mapped to an incident potential associated with the coordiates of all other bodies. Doing this, the incident potential of each body (which is given by the ambient incident potential plus the scattered potentials of all other bodies) is given in the Cylindrical Eigenfunction Expansion associated with its local coordinates. Using the Diffraction Transfer Matrix, which relates the incident and scattered potential of each body in isolation, a system of equations for the coefficients of the scattered potentials of all bodies is obtained.

The theory is described in Kagemoto and Yue 1986 and in Peter and Meylan 2004.

The derivation of the theory in Infinite Depth is also presented Kagemoto and Yue Interaction Theory for Infinite Depth

Equations of Motion

We assume the Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math]. To simplify notation, [math]\displaystyle{ \mathbf{y} = (x,y,z) }[/math] always denotes a point in the water, which is assumed to be of Finite Depth [math]\displaystyle{ d }[/math], while [math]\displaystyle{ \mathbf{x} }[/math] always denotes a point of the undisturbed water surface assumed at [math]\displaystyle{ z=0 }[/math].

Writing [math]\displaystyle{ \alpha = \omega^2/g }[/math] where [math]\displaystyle{ g }[/math] is the acceleration due to gravity, the potential [math]\displaystyle{ \phi }[/math] has to satisfy the standard boundary-value problem

[math]\displaystyle{ \nabla^2 \phi = 0, \; \mathbf{y} \in D }[/math] [math]\displaystyle{ \frac{\partial \phi}{\partial z} = \alpha \phi, \; {\mathbf{x}} \in \Gamma^\mathrm{f}, }[/math] [math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \; \mathbf{y} \in D, \ z=-d, }[/math]

where [math]\displaystyle{ D }[/math] is the is the domain occupied by the water and [math]\displaystyle{ \Gamma^\mathrm{f} }[/math] is the free water surface. At the immersed body surface [math]\displaystyle{ \Gamma_j }[/math] of [math]\displaystyle{ \Delta_j }[/math], the water velocity potential has to equal the normal velocity of the body [math]\displaystyle{ \mathbf{v}_j }[/math],

[math]\displaystyle{ \frac{\partial \phi}{\partial n} = \mathbf{v}_j, \; {\mathbf{y}} \in \Gamma_j. }[/math]

Moreover, the Sommerfeld Radiation Condition is imposed

[math]\displaystyle{ \lim_{\tilde{r} \rightarrow \infty} \sqrt{\tilde{r}} \, \Big( \frac{\partial}{\partial \tilde{r}} - \mathrm{i}k \Big) (\phi - \phi^{\mathrm{In}}) = 0, }[/math]

where [math]\displaystyle{ \tilde{r}^2=x^2+y^2 }[/math], [math]\displaystyle{ k }[/math] is the wavenumber and [math]\displaystyle{ \phi^\mathrm{In} }[/math] is the ambient incident potential. The positive wavenumber [math]\displaystyle{ k }[/math] is related to [math]\displaystyle{ \alpha }[/math] by the Dispersion Relation for a Free Surface

[math]\displaystyle{ (eq_k) \alpha = k \tanh k d, }[/math]

and the values of [math]\displaystyle{ k_m }[/math], [math]\displaystyle{ m\gt 0 }[/math], are given as positive real roots of the dispersion relation

[math]\displaystyle{ (eq_km) \alpha + k_m \tan k_m d = 0. }[/math]

For ease of notation, we write [math]\displaystyle{ k_0 = -\mathrm{i}k }[/math]. Note that [math]\displaystyle{ k_0 }[/math] is a (purely imaginary) root of (eq_k_m).

Eigenfunction expansion of the potential

The scattered potential of a body [math]\displaystyle{ \Delta_j }[/math] can be expanded in the Cylindrical Eigenfunction Expansion,

[math]\displaystyle{ (basisrep_out_d) \phi_j^\mathrm{S} (r_j,\theta_j,z) = \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]

with discrete coefficients [math]\displaystyle{ A_{m \mu}^j }[/math], where

[math]\displaystyle{ f_m(z) = \frac{\cos k_m (z+d)}{\cos k_m d}. }[/math]

The incident potential upon body [math]\displaystyle{ \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_{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]

with discrete coefficients [math]\displaystyle{ D_{n\nu}^j }[/math]. In these expansions, [math]\displaystyle{ I_\nu }[/math] and [math]\displaystyle{ K_\nu }[/math] denote the modified Bessel functions of the first and second kind, respectively, both of order [math]\displaystyle{ \nu }[/math]. Note that in (basisrep_out_d) (and (basisrep_in_d)) the term for [math]\displaystyle{ m =0 }[/math] or [math]\displaystyle{ n=0 }[/math]) corresponds to the propagating modes while the terms for [math]\displaystyle{ m\geq 1 }[/math] ([math]\displaystyle{ n\geq 1 }[/math]) correspond to the evanescent modes.

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

which is valid provided that [math]\displaystyle{ r_l \lt R }[/math]. The angles [math]\displaystyle{ \varphi_{n} }[/math] account for the difference in direction depending if the [math]\displaystyle{ j }[/math]th body is located to the left or to the right of the [math]\displaystyle{ l }[/math]th body and are defined by

[math]\displaystyle{ \varphi_n = \begin{cases} \pi, & n \gt 0,\\ 0, & n \lt 0. \end{cases} }[/math]

The limitation [math]\displaystyle{ r_l \lt R }[/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 |j-l| R) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{j-l}} }[/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 |j-l| R) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{j-l}} \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=-\infty,j \neq l}^{\infty} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n |j-l|R) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{j-l}} \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=-\infty,j \neq l}^{\infty} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n |j-l| R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big], }[/math]

[math]\displaystyle{ m \in {N} }[/math], [math]\displaystyle{ l,\mu \in {Z} }[/math].