Kagemoto and Yue Interaction Theory

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.

We extend the finite depth interaction theory of kagemoto86 to water of infinite depth and bodies of arbitrary geometry. The sum over the discrete roots of the dispersion equation in the finite depth theory becomes an integral in the infinite depth theory. This means that the infinite dimensional diffraction transfer matrix in the finite depth theory must be replaced by an integral operator. In the numerical solution of the equations, this integral operator is approximated by a sum and a linear system of equations is obtained. We also show how the calculations of the diffraction transfer matrix for bodies of arbitrary geometry developed by goo90 can be extended to infinite depth, and how the diffraction transfer matrix for rotated bodies can be easily calculated. This interaction theory is applied to the wave forcing of multiple ice floes and a method to solve the full diffraction problem in this case is presented. Convergence studies comparing the interaction method with the full diffraction calculations and the finite and infinite depth interaction methods are carried out.

==The extension of Kagemoto and Yue's interaction theory to bodies of arbitrary shape in water of infinite depth==

kagemoto86 developed an interaction theory for vertically non-overlapping axisymmetric structures in water of finite depth. While their theory was valid for bodies of arbitrary geometry, they did not develop all the necessary details to apply the theory to arbitrary bodies. The only requirements to apply this scattering theory is that the bodies are vertically non-overlapping and that the smallest cylinder which completely contains each body does not intersect with any other body. In this section we will extend their theory to bodies of arbitrary geometry in water of infinite depth. The extension of \citeauthor{kagemoto86}'s finite depth interaction theory to bodies of arbitrary geometry was accomplished by goo90.

The interaction theory begins by representing the scattered potential of each body in the cylindrical eigenfunction expansion. Furthermore, the incoming potential is also represented in the cylindrical eigenfunction expansion. The operator which maps the incoming and outgoing representation is called the diffraction transfer matrix and is different for each body. Since these representations are local to each body, a mapping of the eigenfunction representations between different bodies is required. This operator is called the coordinate transformation matrix.

The cylindrical eigenfunction expansions will be introduced before we derive a system of equations for the coefficients of the scattered wavefields. Analogously to kagemoto86, we represent the scattered wavefield of each body as an incoming wave upon all other bodies. The addition of the ambient incident wave yields the complete incident potential and with the use of diffraction transfer matrices which relate the coefficients of the incident potential to those of the scattered wavefield a system of equations for the unknown coefficients of the scattered wavefields of all bodies is derived.

Eigenfunction expansion of the potential

The equations of motion for the water are derived from the linearised inviscid theory. Under the assumption of irrotational motion the velocity vector field of the water can be written as the gradient field of a scalar velocity potential [math]\displaystyle{ \Phi }[/math]. Assuming that the motion is time-harmonic with the radian frequency [math]\displaystyle{ \omega }[/math] the velocity potential can be expressed as the real part of a complex quantity,

[math]\displaystyle{ (time) \Phi(\mathbf{y},t) = \Re \{\phi (\mathbf{y}) \mathrm{e}^{-\mathrm{i}\omega t} \}. }[/math]

To simplify notation, [math]\displaystyle{ \mathbf{y} = (x,y,z) }[/math] will always denote a point in the water, which is assumed infinitely deep, while [math]\displaystyle{ \mathbf{x} }[/math] will always denote a point of the undisturbed water surface assumed at [math]\displaystyle{ z=0 }[/math].

The problem consists of [math]\displaystyle{ N }[/math] vertically non-overlapping bodies, denoted by [math]\displaystyle{ \Delta_j }[/math], which are sufficiently far apart that there is no intersection of the smallest cylinder which contains each body with any other body. Each body is subject to an incident wavefield which is incoming, responds to this wavefield and produces a scattered wave field which is outgoing. Both the incident and scattered potential corresponding to these wavefields can be represented in the cylindrical eigenfunction expansion valid outside of the escribed cylinder of the body. Let [math]\displaystyle{ (r_j,\theta_j,z) }[/math] be the local cylindrical coordinates of the [math]\displaystyle{ j }[/math]th body, [math]\displaystyle{ \Delta_j }[/math], [math]\displaystyle{ j \in \{1, \ldots , N\} }[/math], and </math>\alpha = \omega^2/g[math]\displaystyle{ where }[/math]g[math]\displaystyle{ is the acceleration due to gravity. Figure (fig:floe_tri) shows these coordinate systems for two bodies. \begin{figure} \begin{center} \includegraphics[height=5.5cm]{floe_tri} \caption{Plan view of the relation between two bodies.} (fig:floe_tri) \end{center} \end{figure} The scattered potential of body \lt math\gt \Delta_j }[/math] can be expanded in cylindrical eigenfunctions,

[math]\displaystyle{ (basisrep_out) \phi_j^\mathrm{S} (r_j,\theta_j,z) &= \mathrm{e}^{\alpha z} \sum_{\nu = - \infty}^{\infty} A_{0 \nu}^j H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}\\ &\quad + \int\limits_0^{\infty} \big( \cos \eta z + \frac{\alpha}{\eta} \sin \eta z \big) \sum_{\nu = - \infty}^{\infty} A_{\nu}^j (\eta) K_\nu (\eta r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j} \mathrm{d}\eta, }[/math]

where the coefficients [math]\displaystyle{ A_{0 \nu}^j }[/math] for the propagating modes are discrete and the coefficients [math]\displaystyle{ A_{\nu}^j (\cdot) }[/math] for the decaying modes are functions. [math]\displaystyle{ H_\nu^{(1)} }[/math] and [math]\displaystyle{ K_\nu }[/math] are the Hankel function of the first kind and the modified Bessel function of the second kind respectively, both of order [math]\displaystyle{ \nu }[/math] as defined in abr_ste. The incident potential upon body [math]\displaystyle{ \Delta_j }[/math] can be expanded in cylindrical eigenfunctions,

[math]\displaystyle{ (basisrep_in) \phi_j^\mathrm{I} (r_j,\theta_j,z) &= \mathrm{e}^{\alpha z} \sum_{\mu = - \infty}^{\infty} D_{0 \mu}^j J_\mu (\alpha r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j}\\ & \quad + \int\limits_0^{\infty} \big( \cos \eta z + \frac{\alpha}{\eta} \sin \eta z \big) \sum_{\mu = - \infty}^{\infty} D_{\mu}^j (\eta) I_\mu (\eta r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j} \mathrm{d}\eta, }[/math]

where the coefficients [math]\displaystyle{ D_{0 \mu}^j }[/math] for the propagating modes are discrete and the coefficients [math]\displaystyle{ D_{\mu}^j (\cdot) }[/math] for the decaying modes are functions. [math]\displaystyle{ J_\mu }[/math] and [math]\displaystyle{ I_\mu }[/math] are the Bessel function and the modified Bessel function respectively, both of the first kind and order [math]\displaystyle{ \mu }[/math]. To simplify the notation, from now on [math]\displaystyle{ \psi(z,\eta) }[/math] will denote the vertical eigenfunctions corresponding to the decaying modes,

[math]\displaystyle{ \psi(z,\eta) = \cos \eta z + \alpha / \eta \, \sin \eta z. }[/math]

The interaction in water of infinite depth

Following the ideas of kagemoto86, a system of equations for the unknown coefficients and coefficient functions of the scattered wavefields will be 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 will be developed.

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]. From figure (fig:floe_tri) we can see that this can be accomplished by using Graf's addition theorem for Bessel functions given in \citet[eq. 9.1.79]{abr_ste},

[math]\displaystyle{ (transf) \lt center\gt \lt math\gt \begin{matrix} (transf_h) H_\nu^{(1)}(\alpha r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} &= \sum_{\mu = - \infty}^{\infty} H^{(1)}_{\nu + \mu} (\alpha R_{jl}) \, J_\mu (\alpha r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l,\\ (transf_k) K_\nu(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} &= \sum_{\mu = - \infty}^{\infty} K_{\nu + \mu} (\eta R_{jl}) \, I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l, \end{matrix} }[/math]

</math>

which is valid provided that [math]\displaystyle{ r_l \lt R_{jl} }[/math]. This limitation 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 equations (transf) the scattered potential of [math]\displaystyle{ \Delta_j }[/math] can be expressed in terms of the incident potential upon [math]\displaystyle{ \Delta_l }[/math],

[math]\displaystyle{ \begin{matrix} \phi_j^{\mathrm{S}} (r_l,\theta_l,z) &= \mathrm{e}^{\alpha z} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j \sum_{\mu = -\infty}^{\infty} H_{\nu - \mu}^{(1)} (\alpha R_{jl}) J_\mu (\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l} \mathrm{e}^{\mathrm{i}(\nu-\mu) \vartheta_{jl}}\\ & \quad + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\nu = - \infty}^{\infty} A_{\nu}^j (\eta) \sum_{\mu = -\infty}^{\infty} (-1)^\mu K_{\nu-\mu} (\eta R_{jl}) I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l} \mathrm{e}^{\mathrm{i}(\nu-\mu) \vartheta_{jl}} \mathrm{d}\eta\\ &= \mathrm{e}^{\alpha z} \sum_{\mu = -\infty}^{\infty} \Big[ \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j H_{\nu - \mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i (\nu-\mu) \vartheta_{jl}} \Big] J_\mu (\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}\\ & + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\mu = -\infty}^{\infty} \Big[ \sum_{\nu = - \infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu-\mu} (\eta R_{jl}) \mathrm{e}^{\mathrm{i}(\nu-\mu) \vartheta_{jl}} \Big] I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l} \mathrm{d}\eta. \end{matrix} }[/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]. A detailed illustration of how to accomplish this will be given later. Let [math]\displaystyle{ D_{l0\mu}^{\mathrm{In}} }[/math] denote the coefficients of this ambient incident wavefield corresponding to the propagating modes and [math]\displaystyle{ D_{l\mu}^{\mathrm{In}} (\cdot) }[/math] denote the coefficients functions corresponding to the decaying modes (which are identically zero) of the incoming eigenfunction expansion for [math]\displaystyle{ \Delta_l }[/math]. The total incident wavefield upon body [math]\displaystyle{ \Delta_j }[/math] can now be expressed as

[math]\displaystyle{ \begin{matrix} &\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \, \phi_j^{\mathrm{S}} (r_l,\theta_l,z)\\ &= \mathrm{e}^{\alpha z} \sum_{\mu = -\infty}^{\infty} \Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i (\nu - \mu) \vartheta_{jl}} \Big] J_\mu (\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}\\ & + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\mu = -\infty}^{\infty} \Big[ D_{l\mu}^{\mathrm{In}}(\eta) + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu = -\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}} \Big] I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l} \mathrm{d}\eta. \end{matrix} }[/math]

The coefficients of the total incident potential upon [math]\displaystyle{ \Delta_l }[/math] are therefore given by

[math]\displaystyle{ (inc_coeff) \lt center\gt \lt math\gt \begin{matrix} D_{0\mu}^l &= D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i (\nu - \mu) \vartheta_{jl}},\\ D_{\mu}^l(\eta) &= D_{l\mu}^{\mathrm{In}}(\eta) + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu = -\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}}. \end{matrix} }[/math]

</math>

In general, it is possible to relate the total incident and scattered partial waves for any body through the diffraction characteristics of that body in isolation. There exist diffraction transfer operators [math]\displaystyle{ B_l }[/math] that relate the coefficients of the incident and scattered partial waves, such that

[math]\displaystyle{ (eq_B) A_l = B_l (D_l), \quad l=1, \ldots, N, }[/math]

where [math]\displaystyle{ A_l }[/math] are the scattered modes due to the incident modes [math]\displaystyle{ D_l }[/math]. In the case of a countable number of modes, (i.e. when the depth is finite), [math]\displaystyle{ B_l }[/math] is an infinite dimensional matrix. When the modes are functions of a continuous variable (i.e. infinite depth), [math]\displaystyle{ B_l }[/math] is the kernel of an integral operator. For the propagating and the decaying modes respectively, the scattered potential can be related by diffraction transfer operators acting in the following ways,

[math]\displaystyle{ (diff_op) \lt center\gt \lt math\gt \begin{matrix} A_{0\nu}^l &= \sum_{\mu = -\infty}^{\infty} B_{l\nu\mu}^\mathrm{pp} D_{0\mu}^l + \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty} B_{l\nu\mu}^\mathrm{pd} (\xi) D_{\mu}^l (\xi) \mathrm{d}\xi,\\ A_\nu^l (\eta) &= \sum_{\mu = -\infty}^{\infty} B_{l\nu\mu}^\mathrm{dp} (\eta) D_{0\mu}^l + \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty} B_{l\nu\mu}^\mathrm{dd} (\eta;\xi) D_{\mu}^l (\xi) \mathrm{d}\xi. \end{matrix} }[/math]

</math>

The superscripts [math]\displaystyle{ \mathrm{p} }[/math] and [math]\displaystyle{ \mathrm{d} }[/math] are used to distinguish between propagating and decaying modes, the first superscript denotes the kind of scattered mode, the second one the kind of incident mode. If the diffraction transfer operators are known (their calculation will be discussed later), the substitution of equations (inc_coeff) into equations (diff_op) give the required equations to determine the coefficients and coefficient functions of the scattered wavefields of all bodies,

[math]\displaystyle{ (eq_op) \lt center\gt \lt math\gt \begin{matrix} &\begin{aligned} &A_{0n}^l = \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{pp} \Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i (\nu - \mu) \vartheta_{jl}} \Big]\\ & \ + \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{pd} (\xi) \Big[D_{l\mu}^{\mathrm{In}}(\eta) + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu = -\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}} \Big] \mathrm{d}\xi, \end{aligned}\\ &\begin{aligned} &A_n^l (\eta) = \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{dp} (\eta) \Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i (\nu - \mu) \vartheta_{jl}}\Big]\\ & \ + \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{dd} (\eta;\xi) \Big[ D_{l\mu}^{\mathrm{In}}(\eta) + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \sum_{\nu = -\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu} (\eta R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}}\Big] \mathrm{d}\xi, \end{aligned} \end{matrix} }[/math]

</math>

[math]\displaystyle{ n \in \mathds{Z},\, l = 1, \ldots, N }[/math]. It has to be noted that all equations are coupled so that it is necessary to solve for all scattered coefficients and coefficient functions simultaneously.

For numerical calculations, the infinite sums have to be truncated and the integrals must be discretised. Implying a suitable truncation, the four different diffraction transfer operators can be represented by matrices which can be assembled in a big matrix [math]\displaystyle{ \mathbf{B}_l }[/math],

[math]\displaystyle{ \mathbf{B}_l = \left[ \begin{matrix}{cc} \mathbf{B}_l^{\mathrm{pp}} & \mathbf{B}_l^{\mathrm{pd}}\\ \mathbf{B}_l^{\mathrm{dp}} & \mathbf{B}_l^{\mathrm{dd}} \end{matrix} \right], }[/math]

the infinite depth diffraction transfer matrix. Truncating the coefficients accordingly, defining [math]\displaystyle{ {\bf a}^l }[/math] to be the vector of the coefficients of the scattered potential of body [math]\displaystyle{ \Delta_l }[/math], [math]\displaystyle{ \mathbf{d}_l^{\mathrm{In}} }[/math] to be the vector of coefficients of the ambient wavefield, and making use of a coordinate transformation matrix [math]\displaystyle{ {\bf T}_{jl} }[/math] given by

[math]\displaystyle{ (T_elem_deep) \lt center\gt \lt math\gt \begin{matrix} ({\bf T}_{jl})_{pq} &= H_{p-q}^{(1)}(\alpha R_{jl}) \, \mathrm{e}^{\mathrm{i}(p-q) \vartheta_{jl}}\\ =for the propagating modes, and= ({\bf T}_{jl})_{pq} &= (-1)^{q} \, K_{p-q} (\eta R_{jl}) \, \mathrm{e}^{\i (p-q) \vartheta_{jl}} \end{matrix} }[/math]

</math>

for the decaying modes, a linear system of equations for the unknown coefficients follows from equations (eq_op),

[math]\displaystyle{ (eq_B_inf) {\bf a}_l = {\bf \hat{B}}_l \Big( {\bf d}_l^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \trans {\bf T}_{jl} \, {\bf a}_j \Big), \quad l=1, \ldots, N, }[/math]

where the left superscript [math]\displaystyle{ \mathrm{t} }[/math] indicates transposition. The matrix [math]\displaystyle{ {\bf \hat{B}}_l }[/math] denotes the infinite depth diffraction transfer matrix [math]\displaystyle{ {\bf B}_l }[/math] in which the elements associated with decaying scattered modes have been multiplied with the appropriate integration weights depending on the discretisation of the continuous variable.

\subsection{Calculation of the diffraction transfer matrix for bodies of arbitrary geometry}

Before we can apply the interaction theory we require the diffraction transfer matrices [math]\displaystyle{ \mathbf{B}_j }[/math] which relate the incident and the scattered potential for a body [math]\displaystyle{ \Delta_j }[/math] in isolation. The elements of the diffraction transfer matrix, [math]\displaystyle{ ({\bf B}_j)_{pq} }[/math], are the coefficients of the [math]\displaystyle{ p }[/math]th partial wave of the scattered potential due to a single unit-amplitude incident wave of mode [math]\displaystyle{ q }[/math] upon [math]\displaystyle{ \Delta_j }[/math].

While \citeauthor{kagemoto86}'s interaction theory was valid for bodies of arbitrary shape, they did not explain how to actually obtain the diffraction transfer matrices for bodies which did not have an axisymmetric geometry. This step was performed by goo90 who came up with an explicit method to calculate the diffraction transfer matrices for bodies of arbitrary geometry in the case of finite depth. Utilising a Green's function they used the standard method of transforming the single diffraction boundary-value problem to an integral equation for the source strength distribution function over the immersed surface of the body. However, the representation of the scattered potential which is obtained using this method is not automatically given in the cylindrical eigenfunction expansion. To obtain such cylindrical eigenfunction expansions of the potential goo90 used the representation of the free surface finite depth Green's function given by black75 and fenton78. \citeauthor{black75} and \citeauthor{fenton78}'s representation of the Green's function was based on applying Graf's addition theorem to the eigenfunction representation of the free surface finite depth Green's function given by john2. Their representation allowed the scattered potential to be represented in the eigenfunction expansion with the cylindrical coordinate system fixed at the point of the water surface above the mean centre position of the body.

It should be noted that, instead of using the source strength distribution function, it is also possible to consider an integral equation for the total potential and calculate the elements of the diffraction transfer matrix from the solution of this integral equation. An outline of this method for water of finite depth is given by kashiwagi00. We will present here a derivation of the diffraction transfer matrices for the case infinite depth based on a solution for the source strength distribution function. However, an equivalent derivation would be possible based on the solution for the total velocity potential.

To calculate the diffraction transfer matrix in infinite depth, we require the representation of the infinite depth free surface Green's function in cylindrical eigenfunctions,

[math]\displaystyle{ (green_inf) G(r,\theta,z;s,\varphi,c) &= \frac{\mathrm{i}\alpha}{2} \, \mathrm{e}^{\alpha (z+c)} \sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(\alpha r) J_\nu(\alpha s) \mathrm{e}^{\mathrm{i}\nu (\theta - \varphi)}\\ &\quad + \frac{1}{\pi^2} \int\limits_0^{\infty} \psi(z,\eta) \frac{\eta^2}{\eta^2+\alpha^2} \psi(c,\eta) \sum_{\nu=-\infty}^{\infty} K_\nu(\eta r) I_\nu(\eta s) \mathrm{e}^{\mathrm{i}\nu (\theta - \varphi)} \mathrm{d}\eta, }[/math]

[math]\displaystyle{ r \gt s }[/math], given by malte03.

We assume that we have represented the scattered potential in terms of the source strength distribution [math]\displaystyle{ \varsigma^j }[/math] so that the scattered potential can be written as

[math]\displaystyle{ (int_eq_1) \phi_j^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma_j} G (\mathbf{y},\mathbf{\zeta}) \, \varsigma^j (\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}, \quad \mathbf{y} \in D, }[/math]

where [math]\displaystyle{ D }[/math] is the volume occupied by the water and [math]\displaystyle{ \Gamma_j }[/math] is the immersed surface of body [math]\displaystyle{ \Delta_j }[/math]. The source strength distribution function [math]\displaystyle{ \varsigma^j }[/math] can be found by solving an integral equation. The integral equation is described in Weh_Lait and numerical methods for its solution are outlined in Sarp_Isa. Substituting the eigenfunction expansion of the Green's function

(green_inf) into  (int_eq_1), the scattered potential can

be written as

[math]\displaystyle{ \begin{matrix} &\phi_j^\mathrm{S}(r_j,\theta_j,z) = \mathrm{e}^{\alpha z} \sum_{\nu = - \infty}^{\infty} \bigg[ \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j} \mathrm{e}^{\alpha c} J_\nu(\alpha s) \mathrm{e}^{-\mathrm{i}\nu \varphi} \varsigma^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta} \bigg] H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}\\ & \, + \int\limits_{0}^{\infty} \psi(z,\eta) \sum_{\nu = - \infty}^{\infty} \bigg[ \frac{1}{\pi^2} \frac{\eta^2 }{\eta^2 + \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_\nu(\eta s) \mathrm{e}^{-\mathrm{i}\nu \varphi} \varsigma^j({\bf{\zeta}}) \mathrm{d}\sigma_{\bf{\zeta}} \bigg] K_\nu (\eta r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j} \mathrm{d}\eta, \end{matrix} }[/math]

where [math]\displaystyle{ \mathbf{\zeta}=(s,\varphi,c) }[/math] and [math]\displaystyle{ r\gt s }[/math]. This restriction implies that the eigenfunction expansion is only valid outside the escribed cylinder of the body.

The columns of the diffraction transfer matrix are the coefficients of the eigenfunction expansion of the scattered wavefield due to the different incident modes of unit-amplitude. The elements of the diffraction transfer matrix of a body of arbitrary shape are therefore given by

[math]\displaystyle{ (B_elem) \lt center\gt \lt math\gt \begin{matrix} ({\bf B}_j)_{pq} &= \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j} \mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}\\ =and= ({\bf B}_j)_{pq} &= \frac{1}{\pi^2} \frac{\eta^2}{\eta^2 + \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta} \end{matrix} }[/math]

</math>

for the propagating and the decaying modes respectively, where [math]\displaystyle{ \varsigma_q^j(\mathbf{\zeta}) }[/math] is the source strength distribution due to an incident potential of mode [math]\displaystyle{ q }[/math] of the form

[math]\displaystyle{ (test_modes_inf) \lt center\gt \lt math\gt \begin{matrix} \phi_q^{\mathrm{I}}(s,\varphi,c) &= \mathrm{e}^{\alpha c} H_q^{(1)} (\alpha s) \mathrm{e}^{\mathrm{i}q \varphi}\\ =for the propagating modes, and= \phi_q^{\mathrm{I}}(s,\varphi,c) &= \psi(c,\eta) K_q (\eta s) \mathrm{e}^{\mathrm{i}q \varphi} \end{matrix} }[/math]

</math>

for the decaying modes.

The diffraction transfer matrix of rotated bodies

For a non-axisymmetric body, a rotation about the mean centre position in the [math]\displaystyle{ (x,y) }[/math]-plane will result in a different diffraction transfer matrix. We will show how the diffraction transfer matrix of a body rotated by an angle [math]\displaystyle{ \beta }[/math] can be easily calculated from the diffraction transfer matrix of the non-rotated body. The rotation of the body influences the form of the elements of the diffraction transfer matrices in two ways. Firstly, the angular dependence in the integral over the immersed surface of the body is altered and, secondly, the source strength distribution function is different if the body is rotated. However, the source strength distribution function of the rotated body can be obtained by calculating the response of the non-rotated body due to rotated incident potentials. It will be shown that the additional angular dependence can be easily factored out of the elements of the diffraction transfer matrix.

The additional angular dependence caused by the rotation of the incident potential can be factored out of the normal derivative of the incident potential such that

[math]\displaystyle{ \frac{\partial \phi_{q\beta}^{\mathrm{I}}}{\partial n} = \frac{\partial \phi_{q}^{\mathrm{I}}}{\partial n} \mathrm{e}^{\mathrm{i}q \beta}, }[/math]

where [math]\displaystyle{ \phi_{q\beta}^{\mathrm{I}} }[/math] is the rotated incident potential. Since the integral equation for the determination of the source strength distribution function is linear, the source strength distribution function due to the rotated incident potential is thus just given by

[math]\displaystyle{ \varsigma_{q\beta}^j = \varsigma_q^j \, \mathrm{e}^{\mathrm{i}q \beta}. }[/math]

This is also the source strength distribution function of the rotated body due to the standard incident modes.

The elements of the diffraction transfer matrix [math]\displaystyle{ \mathbf{B}_j }[/math] are given by equations (B_elem). Keeping in mind that the body is rotated by the angle [math]\displaystyle{ \beta }[/math], the elements of the diffraction transfer matrix of the rotated body are given by

[math]\displaystyle{ (B_elem_rot) \lt center\gt \lt math\gt \begin{matrix} (\mathbf{B}_j^\beta)_{pq} &= \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j} \mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p (\varphi+\beta)} \varsigma_{q\beta}^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta},\\ =and= (\mathbf{B}_j^\beta)_{pq} &= \frac{1}{\pi^2} \frac{\eta^2}{\eta^2 + \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p (\varphi+\beta)} \varsigma_{q\beta}^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}, \end{matrix} }[/math]

</math>

for the propagating and decaying modes respectively.

Thus the additional angular dependence caused by the rotation of the body can be factored out of the elements of the diffraction transfer matrix. The elements of the diffraction transfer matrix corresponding to the body rotated by the angle [math]\displaystyle{ \beta }[/math], [math]\displaystyle{ \mathbf{B}_j^\beta }[/math], are given by

[math]\displaystyle{ (B_rot) (\mathbf{B}_j^\beta)_{pq} = (\mathbf{B}_j)_{pq} \, \mathrm{e}^{\mathrm{i}(q-p) \beta}. }[/math]

As before, [math]\displaystyle{ (\mathbf{B})_{pq} }[/math] is understood to be the element of [math]\displaystyle{ \mathbf{B} }[/math] which corresponds to the coefficient of the [math]\displaystyle{ p }[/math]th scattered mode due to a unit-amplitude incident wave of mode [math]\displaystyle{ q }[/math]. Equation

(B_rot) applies to propagating and decaying modes likewise.

\subsection{Representation of the ambient wavefield in the eigenfunction representation} In Cartesian coordinates centred at the origin, the ambient wavefield is given by

[math]\displaystyle{ \phi^{\mathrm{In}}(x,y,z) = A \frac{g}{\omega} \, \mathrm{e}^{\mathrm{i}\alpha (x \cos \chi + y \sin \chi)+ \alpha z}, }[/math]

where [math]\displaystyle{ A }[/math] is the amplitude (in displacement) and [math]\displaystyle{ \chi }[/math] is the angle between the [math]\displaystyle{ x }[/math]-axis and the direction in which the wavefield travels. The interaction theory requires that the ambient wavefield, which is incident upon all bodies, is represented in the eigenfunction expansion of an incoming wave in the local coordinates of the body. The ambient wave can be represented in an eigenfunction expansion centred at the origin as

[math]\displaystyle{ \phi^{\mathrm{In}}(x,y,z) = A \frac{g}{\omega} \mathrm{e}^{\alpha z} \sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 - \theta + \chi)} J_\mu(\alpha r) }[/math]

\cite[p. 169]{linton01}. Since the local coordinates of the bodies are centred at their mean centre positions [math]\displaystyle{ O_l = (O_x^l,O_y^l) }[/math], a phase factor has to be defined which accounts for the position from the origin. Including this phase factor the ambient wavefield at the [math]\displaystyle{ l }[/math]th body is given by

[math]\displaystyle{ \phi^{\mathrm{In}}(r_l,\theta_l,z) = A \frac{g}{\omega} \, e^{\mathrm{i}\alpha (O_x^l \cos \chi + O_x^l \sin \chi)} \, \mathrm{e}^{\alpha z} \sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 - \chi)} J_\mu(\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}. }[/math]

Solving the resulting system of equations

After the coefficient vector of the ambient incident wavefield, the diffraction transfer matrices and the coordinate transformation matrices have been calculated, the system of equations (eq_B_inf), has to be solved. This system can be represented by the following matrix equation,

[math]\displaystyle{ \left[ \begin{matrix}{c} {\bf a}_1\\ {\bf a}_2\\ \\ \vdots \\ \\ {\bf a}_N \end{matrix} \right] = \left[ \begin{matrix}{c} \hat{{\bf B}}_1 {\bf d}_1^\mathrm{In}\\ \hat{{\bf B}}_2 {\bf d}_2^\mathrm{In}\\ \\ \vdots \\ \\ \hat{{\bf B}}_N {\bf d}_N^\mathrm{In} \end{matrix} \right]+ \left[ \begin{matrix}{ccccc} \mathbf{0} & \hat{{\bf B}}_1 \trans {\bf T}_{21} & \hat{{\bf B}}_1 \trans {\bf T}_{31} & \dots & \hat{{\bf B}}_1 \trans {\bf T}_{N1}\\ \hat{{\bf B}}_2 \trans {\bf T}_{12} & \mathbf{0} & \hat{{\bf B}}_2 \trans {\bf T}_{32} & \dots & \hat{{\bf B}}_2 \trans {\bf T}_{N2}\\ & & \mathbf{0} & &\\ \vdots & & & \ddots & \vdots\\ & & & & \\ \hat{{\bf B}}_N \trans {\bf T}_{1N} & & \dots & & \mathbf{0} \end{matrix} \right] \left[ \begin{matrix}{c} {\bf a}_1\\ {\bf a}_2\\ \\ \vdots \\ \\ {\bf a}_N \end{matrix} \right], }[/math]

where [math]\displaystyle{ \mathbf{0} }[/math] denotes the zero-matrix which is of the same dimension as [math]\displaystyle{ \hat{{\bf B}}_j }[/math], say [math]\displaystyle{ n }[/math]. This matrix equation can be easily transformed into a classical [math]\displaystyle{ (N \, n) }[/math]-dimensional linear system of equations.

Finite Depth Interaction Theory

We will compare the performance of the infinite depth interaction theory with the equivalent theory for finite depth. As we have stated previously, the finite depth theory was developed by kagemoto86 and extended to bodies of arbitrary geometry by goo90. We will briefly present this theory in our notation and the comparisons will be made in a later section.

In water of constant finite depth [math]\displaystyle{ d }[/math], the scattered potential of a body [math]\displaystyle{ \Delta_j }[/math] can be expanded in cylindrical eigenfunctions,

[math]\displaystyle{ (basisrep_out_d) \phi_j^\mathrm{S} (r_j,\theta_j,z) &= \frac{\cosh k(z+d)}{\cosh kd} \sum_{\nu = - \infty}^{\infty} A_{0 \nu}^j H_\nu^{(1)} (k r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}\\ &\quad + \sum_{m=1}^{\infty} \frac{\cos k_m (z+d)}{\cos kd} \sum_{\nu = - \infty}^{\infty} A_{m \nu}^j K_\nu (k_m r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}, }[/math]

with discrete coefficients [math]\displaystyle{ A_{m \nu}^j }[/math]. The positive wavenumber [math]\displaystyle{ k }[/math] is related to [math]\displaystyle{ \alpha }[/math] by the dispersion relation

[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_k_m) \alpha + k_m \tan k_m d = 0. }[/math]

The incident potential upon body [math]\displaystyle{ \Delta_j }[/math] can be also be expanded in cylindrical eigenfunctions,

[math]\displaystyle{ (basisrep_in_d) \phi_j^\mathrm{I} (r_j,\theta_j,z) &= \frac{\cosh k(z+d)}{\cosh kd} \sum_{\mu = - \infty}^{\infty} D_{0 \mu}^j J_\mu (k r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j}\\ & \quad + \sum_{m=1}^{\infty} \frac{\cos k_m (z+d)}{\cos kd} \sum_{\mu = - \infty}^{\infty} D_{m\mu}^j I_\mu (k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j}, }[/math]

with discrete coefficients [math]\displaystyle{ D_{m\mu}^j }[/math]. A system of equations for the coefficients of the scattered wavefields for the bodies are derived in an analogous way to the infinite depth case. The derivation is simpler because all the coefficients are discrete and the diffraction transfer operator can be represented by an infinite dimensional matrix. Truncating the infinite dimensional matrix as well as the coefficient vectors appropriately, the resulting system of equations is given by

[math]\displaystyle{ {\bf a}_l = {\bf B}_l \Big( {\bf d}_l^\mathrm{In} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \trans {\bf T}_{jl} \, {\bf a}_j \Big), \quad l=1, \ldots, N, }[/math]

where [math]\displaystyle{ {\bf a}_l }[/math] is the coefficient vector of the scattered wave, [math]\displaystyle{ {\bf d}_l^\mathrm{In} }[/math] is the coefficient vector of the ambient incident wave, [math]\displaystyle{ {\bf B}_l }[/math] is the diffraction transfer matrix of [math]\displaystyle{ \Delta_l }[/math] and [math]\displaystyle{ {\bf T}_{jl} }[/math] is the coordinate transformation matrix analogous to (T_elem_deep).

The calculation of the diffraction transfer matrices is also similar to the infinite depth case. The finite depth Green's function

[math]\displaystyle{ (green_d) &G(r,\theta,z;s,\varphi,c)\\ &= \frac{\i}{2} \, \frac{\alpha^2-k^2}{d(\alpha^2-k^2)-\alpha}\, \cosh k(z+d) \cosh k(c+d) \sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(k r) J_\nu(k s) \mathrm{e}^{\mathrm{i}\nu (\theta - \varphi)}\\ & \quad + \frac{1}{\pi} \sum_{m=1}^{\infty} \frac{k_m^2+\alpha^2}{d(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+d) \cos k_m(c+d) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r) I_\nu(k_m s) \mathrm{e}^{\mathrm{i}\nu (\theta - \varphi)}, }[/math]

given by black75 and fenton78, needs to be used instead of the infinite depth Green's function (green_inf). The elements of [math]\displaystyle{ {\bf B}_j }[/math] are therefore given by

[math]\displaystyle{ (B_elem_d) \lt center\gt \lt math\gt \begin{matrix} ({\bf B}_j)_{pq} &= \frac{\i}{2} \, \frac{(\alpha^2-k^2)\cosh^2 kd}{d(\alpha^2-k^2)-\alpha} \int\limits_{\Gamma_j} \cosh k(c+d) J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}\\ =and= ({\bf B}_j)_{pq} &= \frac{1}{\pi} \frac{(k_m^2+\alpha^2)\cos^2 k_md}{d(k_m^2+\alpha^2)-\alpha} \int\limits_{\Gamma_j} \cos k_m(c+d) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta} \end{matrix} }[/math]

</math>

for the propagating and the decaying modes respectively, where [math]\displaystyle{ \varsigma_q^j(\mathbf{\zeta}) }[/math] is the source strength distribution due to an incident potential of mode [math]\displaystyle{ q }[/math] of the form

[math]\displaystyle{ (test_modes_d) \lt center\gt \lt math\gt \begin{matrix} \phi_q^{\mathrm{I}}(s,\varphi,c) &= \frac{\cosh k_m(c+d)}{\cosh kd} H_q^{(1)} (k s) \mathrm{e}^{\mathrm{i}q \varphi}\\ =for the propagating modes, and= \phi_q^{\mathrm{I}}(s,\varphi,c) &= \frac{\cos k_m(c+d)}{\cos kd} K_q (k_m s) \mathrm{e}^{\mathrm{i}q \varphi} \end{matrix} }[/math]

</math>

for the decaying modes.

Numerical Results

In this section we will present some calculations using the interaction theory in finite and infinite depth and the full diffraction method in finite and infinite depth. These will be based on calculations for ice floes. We begin with some convergence tests which aim to compare the various methods. It needs to be noted that this comparison is only of numerical nature since the interactions methods as well as the full diffraction calculations are exact in an analytical sense. However, numerical calculations require truncations which affect the different methods in different ways. Especially the dependence on these truncations will be investigated.

Convergence Test

We will present some convergence tests that aim to compare the performance of the interaction theory with the full diffraction calculations and to compare the performance of the finite and infinite depth interaction methods in deep water. The comparisons will be conducted for the case of two square ice floes in three different arrangements. In the full diffraction calculation the ice floes are discretised in [math]\displaystyle{ 24 \times 24 = 576 }[/math] elements. For the full diffraction calculation the resulting linear system of equations to be solved is therefore 1152. As will be seen, once the diffraction transfer matrix has been calculated (and saved), the dimension of the linear system of equations to be solved in the interaction method is considerably smaller. It is given by twice the dimension of the diffraction transfer matrix. The most challenging situation for the interaction theory is when the bodies are close together. For this reason we choose the distance such that the escribed circles of the two ice floes just overlap. It must be recalled that the interaction theory is valid as long as the escribed cylinder of a body does not intersect with any other body.

Both ice floes have non-dimensionalised stiffness [math]\displaystyle{ \beta = 0.02 }[/math], mass [math]\displaystyle{ \gamma = 0.02 }[/math] and Poisson's ratio is chosen as [math]\displaystyle{ \nu=0.3333 }[/math]. The wavelength of the ambient incident wave is [math]\displaystyle{ \lambda = 2 }[/math]. Each ice floe has side length 2. The ambient wavefield is of unit amplitude and propagates in the [math]\displaystyle{ x }[/math]-direction. Three different arrangements are chosen to compare the results of the finite depth interaction method in deep water and the infinite depth interaction method with the corresponding full diffraction calculations. In the first arrangement the second ice floe is located behind the first, in the second arrangement it is located beside, and the third arrangement it is both beside and behind. The exact positions of the ice floes are given in table (tab:pos).

\begin{table} \begin{center} \begin{tabular}{@{}ccc@{}}

arrangement & [math]\displaystyle{ O_1 }[/math] & [math]\displaystyle{ O_2 }[/math]\

[math]\displaystyle{ 3pt] 1 & \lt math\gt (-1.4,0) }[/math] & [math]\displaystyle{ (1.4,0) }[/math]\\

2 & [math]\displaystyle{ (0,-1.4) }[/math] & [math]\displaystyle{ (0,1.4) }[/math]\\ 3 & [math]\displaystyle{ (-1.4,-0.6) }[/math] & [math]\displaystyle{ (1.4,0.6) }[/math] \end{tabular} \caption{Positions of the ice floes in the different arrangements.} (tab:pos) \end{center} \end{table}

Figure (fig:tsf) shows the solutions corresponding to the three arrangements in the case of water of infinite depth. To illustrate the effect on the water in the vicinity of the ice floes, the water displacement is also shown. It is interesting to note that the ice floe in front is barely influenced by the floe behind while the motion of the floe behind is quite different from its motion in the absence of the floe in front.

\begin{figure} \begin{center}

\includegraphics[width=8.2cm]{int_n11_x0_y28_v5_b02_chi2}\[math]\displaystyle{ 0.4cm] \includegraphics[width=8.2cm]{int_n11_x0_y28_v5_b02_chi0}\\lt center\gt \lt math\gt 0.4cm] \includegraphics[width=8.2cm]{int_n11_x12_y28_v5_b02_chi2} \end{center} \caption{Surface displacement of the ice floes and the water in their vicinity,\newline arrangements 1, 2 and 3.} (fig:tsf) \end{figure} To compare the results, a measure of the error from the full diffraction calculation is used. We calculate the full diffraction solution with a sufficient number of points so that we may use it to approximate the exact solution. \lt center\gt \lt math\gt E_2 = \left( \, \int\limits_{\Delta} \big| w_{i}(\mathbf{x})-w_{f}(\mathbf{x}) \big|^2 \mathrm{d}\mathbf{x} \, \right)^{1/2}, }[/math]

where [math]\displaystyle{ w_{i} }[/math] and [math]\displaystyle{ w_{f} }[/math] are the solutions of the interaction method and the corresponding full diffraction calculation respectively. It would also be possible to compare other errors, the maximum difference of the solutions for example, but the results are very similar.

It is worth noting that the finite depth interaction method only converges up to a certain depth if used with the eigenfunction expansion of the finite depth Green's function (green_d). This is because of the factor [math]\displaystyle{ \alpha^2-k^2 }[/math] in the term of propagating modes of the Green's function. The Green's function can be rewritten by making use of the dispersion relation (eq_k) \cite[as suggested by][p. 26, for example]{linton01} and the depth restriction of the finite depth interaction method for bodies of arbitrary geometry can be circumvented.

The truncation parameters for the interaction methods will now be considered for both finite and infinite depth. The number of propagating modes and angular decaying components are free parameters in both methods. In finite depth, the number of decaying roots of the dispersion relation needs to be chosen while in infinite depth the discretisation of a continuous variable must be selected. In the infinite depth case we are free to choose the number of points as well as the points themselves. In water of finite depth, the depth can also be considered a free parameter as long as it is chosen large enough to account for deep water.

Truncating the infinite sums in the eigenfunction expansion of the outgoing water velocity potential for infinite depth with truncation parameters [math]\displaystyle{ T_H }[/math] and [math]\displaystyle{ T_K }[/math] and discretising the integration by defining a set of nodes, </math>0\leq\eta_1 < \ldots < \eta_m < \ldots <

\eta_{_{T_R}}[math]\displaystyle{ , with weights }[/math]h_m[math]\displaystyle{ , the potential for infinite depth can be approximated by \lt center\gt \lt math\gt \phi (r,\theta,z) &= \mathrm{e}^{\alpha z} \sum_{\nu = - T_H}^{T_H} A_{0\nu} H_\nu^{(1)} (\alpha r) \mathrm{e}^{\mathrm{i}\nu \theta}\\ &\quad + \sum_{m=1}^{T_R} h_m \, \psi (z,\eta_m) \sum_{\nu = - T_K}^{T_K} A_{\nu} (\eta_m) K_\nu (\eta_m r) \mathrm{e}^{\mathrm{i}\nu \theta}. }[/math]

In the following, the integration weights are chosen to be </math>h_m = 1/2\,(\eta_{m+1}-\eta_{m-1})[math]\displaystyle{ , }[/math]m=2, \ldots, T_R-1[math]\displaystyle{ and }[/math]h_1 = \eta_2-\eta_1[math]\displaystyle{ as well as }[/math]h_{_{T_R}} =

\eta_{_{T_R}}-\eta_{_{T_R-1}}[math]\displaystyle{ , which corresponds to the mid-point quadrature rule. Different quadrature rules such as Gaussian quadrature could be considered. Although in general this would lead to better results, the mid-point rule allows a clever choice of the discretisation points so that the convergence with Gaussian quadrature is no better. In finite depth, the analogous truncation leads to \lt center\gt \lt math\gt \phi (r,\theta,z) &= \frac{\cosh k (z+d)}{\cosh kd} \sum_{\nu = - T_H}^{T_H} A_{0\nu} H_\nu^{(1)} (k r) \mathrm{e}^{\mathrm{i}\nu \theta}\\ & \quad + \sum_{m = 1}^{T_R} \frac{\cos k_m(z+d)}{\cos k_m d} \sum_{\nu = - T_K}^{T_K} A_{m\nu} K_\nu (k_m r) \mathrm{e}^{\mathrm{i}\nu \theta}. }[/math]

In both cases, the dimension of the diffraction transfer matrix, [math]\displaystyle{ \mathbf{B} }[/math], is given by [math]\displaystyle{ 2 \, T_H+1+T_R \, (2 \, T_K+1) }[/math].

Since the choice of the number of propagating modes and angular decaying components affects the finite and infinite depth methods in similar ways, the dependence on these parameters will not be further presented. Thorough convergence tests have shown that in the settings investigated here, it is sufficient to choose [math]\displaystyle{ T_H }[/math] to be 11 and [math]\displaystyle{ T_K }[/math] to be 5. Further increasing these parameter values does not result in smaller errors (as compared to the full diffraction calculation with 576 elements per floe). We will now compare the convergence of the infinite depth and the finite depth methods if [math]\displaystyle{ T_H }[/math] and [math]\displaystyle{ T_K }[/math] are fixed (with the previously mentioned values) and [math]\displaystyle{ T_R }[/math] is varied. To be able to compare the results, the discretisation of the continuous variable will always be the same for fixed [math]\displaystyle{ T_R }[/math] and these are shown in table (tab:discr). It should be noted that if only one node is used the integration weight is chosen to be 1.

\begin{table} \begin{center} \begin{tabular}{@{}cl@{}}

[math]\displaystyle{ T_R }[/math] & discretisation of [math]\displaystyle{ \eta }[/math]\

[math]\displaystyle{ 3pt] 1 & \{ 2.1 \}\\ 2 & \{ 1.2, 2.7 \}\\ 3 & \{ 0.8, 1.8, 3.0 \}\\ 4 & \{ 0.4, 1.4, 2.2, 3.2 \}\\ 5 & \{ 0.2, 1.0, 1.8, 3.0, 4.6 \} \end{tabular} \caption{The different discretisations used in the convergence tests.} (tab:discr) \end{center} \end{table} Figures (fig:behind), (fig:beside) and (fig:shifted) show the convergence for arrangement 1, arrangement 2 and arrangement 3, respectively, for the infinite depth method and the finite depth method with depth 2 (plot (a)) and depth 4 (plot (b)). Since the ice floes are located beside each other in arrangement 2 the average errors are the same for both floes. As can be seen from figures (fig:behind), (fig:beside) and (fig:shifted) the convergence of the infinite depth method is similar to that of the finite depth method. Used with depth 2, the convergence of the finite depth method is generally better than that of the infinite depth method while used with depth 4, the infinite depth method achieves the better results. Tests with other depths show that the performance of the finite depth method decreases with increasing water depth as expected. In general, since the wavelength is 2, a depth of \lt math\gt d=2 }[/math] should approximate infinite depth and hence there is no

advantage to using the infinite depth theory. However, as mentioned previously, for certain situations such as ice floes it is not necessarily true that [math]\displaystyle{ d=2 }[/math] will approximate infinite depth.

\begin{figure} \begin{center} \begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}} \includegraphics[height=0.38\columnwidth]{behind_d2}&& \includegraphics[height=0.38\columnwidth]{behind_d4} \end{tabular} \caption{Development of the errors as [math]\displaystyle{ T_R }[/math] is increased in arrangement 1.} (fig:behind) \end{center} \end{figure}

\begin{figure} \begin{center} \begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}} \includegraphics[height=0.38\columnwidth]{beside_d2}&& \includegraphics[height=0.38\columnwidth]{beside_d4} \end{tabular} \caption{Development of the errors as [math]\displaystyle{ T_R }[/math] is increased in arrangement 2.} (fig:beside) \end{center} \end{figure}

\begin{figure} \begin{center} \begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}} \includegraphics[height=0.38\columnwidth]{shifted_d2}&& \includegraphics[height=0.38\columnwidth]{shifted_d4} \end{tabular} \caption{Development of the errors as [math]\displaystyle{ T_R }[/math] is increased in arrangement 3.} (fig:shifted) \end{center} \end{figure}

Multiple ice floe results

We will now present results for multiple ice floes of different geometries and in different arrangements on water of infinite depth. We choose the floe arrangements arbitrarily, since there are no known special ice floe arrangements, such as those that give rise to resonances in the infinite limit. In all plots, the wavelength [math]\displaystyle{ \lambda }[/math] has been chosen to be [math]\displaystyle{ 2 }[/math], the stiffness [math]\displaystyle{ \beta }[/math] and the mass [math]\displaystyle{ \gamma }[/math] of the ice floes to be 0.02 and Poisson's ratio [math]\displaystyle{ \nu }[/math] is [math]\displaystyle{ 0.3333 }[/math]. The ambient wavefield of amplitude 1 propagates in the positive direction of the [math]\displaystyle{ x }[/math]-axis, thus it travels from left to right in the plots.

Figure (fig:int_arb) shows the displacements of multiple interacting ice floes of different shapes and in different arrangements. Since square elements have been used to represent the floes, non-rectangular geometries are approximated. All ice floes have an area of 4 and the escribing circles do not intersect with any of the other ice floes. The plots show the displacement of the ice floes at time [math]\displaystyle{ t=0 }[/math].

\begin{figure} \begin{center} \begin{tabular}{p{0.46\columnwidth}p{0.03\columnwidth}p{0.46\columnwidth}} \includegraphics[width=0.45\columnwidth]{mult_n15_tr_tr_tr_tr_in} &&

\includegraphics[width=0.45\columnwidth]{mult_n15_tr_tr_tr_tr_out}\[math]\displaystyle{ 0.2cm] \includegraphics[width=0.45\columnwidth]{mult_n15_rh_rh_rh} && \includegraphics[width=0.45\columnwidth]{mult_n15_sq_sq_sq_sq_sq}\\ \end{tabular} \end{center} \caption{Surface displacement of interacting ice floes of different geometries.} (fig:int_arb) \end{figure} ==Summary== The finite depth interaction theory developed by [[kagemoto86]] has been extended to water of infinite depth. Furthermore, using the eigenfunction expansion of the infinite depth free surface Green's function we have been able to calculate the diffraction transfer matrices for bodies of arbitrary geometry. We also showed how the diffraction transfer matrices can be calculated efficiently for different orientations of the body. The convergence of the infinite depth interaction method is similar to that of the finite depth method. Generally, it can be said that the greater the water depth in the finite depth method the poorer its performance. Since bodies in the water can change the water depth which is required to allow the water to be approximated as infinitely deep (ice floes for example) it is recommendable to use the infinite depth method if the water depth may be considered infinite. Furthermore, the infinite depth method requires the infinite depth single diffraction solutions which are easier to compute than the finite depth solutions. It is also possible that the convergence of the infinite depth method may be further improved by a novel to optimisation of the discretisation of the continuous variable. }[/math]