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.

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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)} \d\eta, \lt math\gt 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}) \d\sigma_\mathbf{\zeta}, \quad \mathbf{y} \in D, where \lt math\gt 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}) \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}}) \d\sigma_{\bf{\zeta}} \bigg] K_\nu (\eta r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j} \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 \begin{subequations} (B_elem)

[math]\displaystyle{ \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}) \d\sigma_\mathbf{\zeta}\\ \intertext{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}) \d\sigma_\mathbf{\zeta} \end{matrix} }[/math]

\end{subequations} 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 \begin{subequations} (test_modes_inf)

[math]\displaystyle{ \begin{matrix} \phi_q^{\mathrm{I}}(s,\varphi,c) &= \mathrm{e}^{\alpha c} H_q^{(1)} (\alpha s) \mathrm{e}^{\mathrm{i}q \varphi}\\ \intertext{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]

\end{subequations} 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}, where \lt math\gt \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}. 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 \lt math\gt \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 \begin{subequations} (B_elem_rot)

[math]\displaystyle{ \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}) \d\sigma_\mathbf{\zeta},\\ \intertext{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}) \d\sigma_\mathbf{\zeta}, \end{matrix} }[/math]

\end{subequations} 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}. As before, \lt math\gt (\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}, where \lt math\gt 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) \cite[p. 169]{linton01}. Since the local coordinates of the bodies are centred at their mean centre positions \lt math\gt 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}. ===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, \lt center\gt \lt math\gt \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], where \lt math\gt \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}, with discrete coefficients \lt math\gt 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, and the values of \lt math\gt 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. The incident potential upon body \lt math\gt \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}, with discrete coefficients \lt math\gt 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, where \lt math\gt {\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)}, given by [[black75]] and [[fenton78]], needs to be used instead of the infinite depth Green's function (green_inf). The elements of \lt math\gt {\bf B}_j }[/math] are therefore given by

\begin{subequations} (B_elem_d)

[math]\displaystyle{ \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}) \d\sigma_\mathbf{\zeta}\\ \intertext{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}) \d\sigma_\mathbf{\zeta} \end{matrix} }[/math]

\end{subequations} 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 \begin{subequations} (test_modes_d)

[math]\displaystyle{ \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}\\ \intertext{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]

\end{subequations} for the decaying modes.

Wave forcing of an ice floe of arbitrary geometry

The interaction theory which has been developed so far has been for arbitrary bodies. No assumption has been made about the body geometry or its equations of motion. However, we will now use this interaction theory to make calculations for the specific case of ice floes. Ice floes form in vast fields consisting of hundreds if not thousands of individual floes and furthermore most ice floe fields occur in the deep ocean. For this reason they are ideally suited to the application of the scattering theory we have just developed. Furthermore, the presence of the ice lengthens the wavelength making it more difficult to determine how deep the water must be to be approximately infinite.

Mathematical model for ice floes

We will briefly describe the mathematical model which is used to describe ice floes. A more detailed account can be found in Squire_review. We assume that the ice floe is sufficiently thin that we may apply the shallow draft approximation, which essentially applies the boundary conditions underneath the floe at the water surface. The ice floe is modelled as a thin plate rather than a rigid body since the floe flexure is significant owing to the ice floe geometry. This model has been applied to a single ice floe by JGR02. Assuming the ice floe is in contact with the water surface at all times, its displacement [math]\displaystyle{ W }[/math] is that of the water surface and [math]\displaystyle{ W }[/math] is required to satisfy the linear plate equation in the area occupied by the ice floe [math]\displaystyle{ \Delta }[/math]. In analogy to

(time), [math]\displaystyle{ w }[/math] denotes the time-independent surface displacement

(with the same radian frequency as the water velocity potential due to linearity) and the plate equation becomes

[math]\displaystyle{ (plate_non) D \, \nabla^4 w - \omega^2 \, \rho_\Delta \, h \, w = \mathrm{i}\, \omega \, \rho \, \phi - \rho \, g \, w, \quad {\bf{x}} \in \Delta, with the density of the water \lt math\gt \rho }[/math], the modulus of rigidity of the

ice floe [math]\displaystyle{ D }[/math], its density [math]\displaystyle{ \rho_\Delta }[/math] and its thickness [math]\displaystyle{ h }[/math]. The right-hand-side of (plate_non) arises from the linearised Bernoulli equation. It needs to be recalled that [math]\displaystyle{ \mathbf{x} }[/math] always denotes a point of the undisturbed water surface. Free edge boundary conditions apply, namely

[math]\displaystyle{ \frac{\partial^2 w}{\partial n^2} + \nu \frac{\partial^2 w}{\partial s^2} = 0 \quad =and= \quad \frac{\partial^3 w}{\partial n^3} + (2 - \nu) \frac{\partial^3 w}{\partial n \partial s^2} = 0, \quad \mathbf{x} \in \partial \Delta, where \lt math\gt n }[/math] and [math]\displaystyle{ s }[/math] denote the normal and tangential directions on

[math]\displaystyle{ \partial \Delta }[/math] (where they exist) respectively and [math]\displaystyle{ \nu }[/math] is Poisson's ratio.

Non-dimensional variables (denoted with an overbar) are introduced,

[math]\displaystyle{ (\bar{x},\bar{y},\bar{z}) = \frac{1}{a} (x,y,z), \quad \bar{w} = \frac{w}{a}, \quad \bar{\alpha} = a\, \alpha, \quad \bar{\omega} = \omega \sqrt{\frac{a}{g}} \quad =and= \quad \bar{\phi} = \frac{\phi}{a \sqrt{a g}}, where \lt math\gt a }[/math] is a length parameter associated with the floe.

In non-dimensional variables, the equation for the ice floe

(plate_non) reduces to

[math]\displaystyle{ (plate_final) \beta \nabla^4 \bar{w} - \bar{\alpha} \gamma \bar{w} = \i \sqrt{\bar{\alpha}} \bar{\phi} - \bar{w}, \quad \bar{\mathbf{x}} \in \bar{\Delta} with \lt center\gt \lt math\gt \beta = \frac{D}{g \rho a^4} \quad =and= \quad \gamma = \frac{\rho_\Delta h}{ \rho a}. The constants \lt math\gt \beta }[/math] and [math]\displaystyle{ \gamma }[/math] represent the stiffness and the

mass of the plate respectively. For convenience, the overbars will be dropped and non-dimensional variables will be assumed in the sequel.

The standard boundary-value problem applies to the water. The water velocity potential must satisfy the boundary value problem \begin{subequations} (water)

[math]\displaystyle{ \begin{matrix} \nabla^2 \phi &= 0, \; & & \mathbf{y} \in D,\\ (water_freesurf) \frac{\partial \phi}{\partial z} &= \alpha \phi, \; & & {\bf{x}} \not\in \Delta,\\ (water_depth) \sup_{\mathbf{y} \in D} \abs{\phi} &\lt \infty. \intertext{The linearised kinematic boundary condition is applied under the ice floe,} (water_body) \frac{\partial \phi}{\partial z} &= - \mathrm{i}\sqrt{\alpha} w, \; && {\bf{x}} \in \Delta, \end{matrix} }[/math]

and the Sommerfeld radiation condition

[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, \end{subequations} where \lt math\gt \tilde{r}^2=x^2+y^2 }[/math] and [math]\displaystyle{ k }[/math] is the wavenumber is imposed.

Since the numerical convergence will be compared to the finite depth theory later, a formulation for the finite depth problem will be required. However, the differences to the infinite depth formulation are few. For water of constant finite depth [math]\displaystyle{ d }[/math], the volume occupied by the water changes, the vertical dimension being reduced to [math]\displaystyle{ (-d,0) }[/math], (still denoted by [math]\displaystyle{ D }[/math]), and the depth condition (water_depth) is replaced by the bed condition,

[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \quad \mathbf{y} \in D,\: z=-d. In water of finite depth, the positive real wavenumber \lt math\gt k }[/math] is related

to the radian frequency by the dispersion relations (eq_k).

===The wavelength under the ice floe=== (sec:kappa) For the case of a floating thin plate of shallow draft, which we have used here to model ice floes, waves can propagate under the plate. These waves can be understood by considering an infinite sheet of ice and they satisfy a complex dispersion relation given by FoxandSquire. In non-dimensional form it states

[math]\displaystyle{ \kappa^* \tan \kappa^* d = - \frac{\alpha}{\beta \kappa^{*4} - \gamma \alpha +1}, where \lt math\gt \kappa^* }[/math] is the wavenumber under the plate. The purely imaginary

roots of this dispersion relation correspond to the propagating modes and their absolute value is given as the positive root of

[math]\displaystyle{ \kappa \tanh \kappa d = \frac{\alpha}{\beta \kappa^4 - \gamma \alpha + 1}. For realistic values of the parameters, the effect of the plate is to make \lt math\gt \kappa }[/math] smaller than [math]\displaystyle{ k }[/math] (the open water wavenumber), which

increases the wavelength. The effect of the increased wavelength is to increase the depth at which the water may be approximated as infinite.

Transformation into an integral equation

The problem for the water is converted to an integral equation in the following way. Let [math]\displaystyle{ G }[/math] be the three-dimensional free surface Green's function for water of infinite depth. The Green's function allows the representation of the scattered water velocity potential in the standard way,

[math]\displaystyle{ (int_eq) \phi^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma} \left( \phi^\mathrm{S} (\mathbf{\zeta}) \, \frac{\partial G}{\partial n_\mathbf{\zeta}} (\mathbf{y};\mathbf{\zeta}) - G (\mathbf{y};\mathbf{\zeta}) \, \frac{\partial \phi^\mathrm{S}}{\partial n_\mathbf{\zeta}} (\mathbf{\zeta}) \right) \d\sigma_\mathbf{\zeta}, \quad \mathbf{y} \in D. In the case of a shallow draft, the fact that the Green's function is symmetric and therefore satisfies the free surface boundary condition with respect to the second variable as well can be used to drastically simplify (int_eq). Due to the linearity of the problem the ambient incident potential can just be added to the equation to obtain the total water velocity potential, \lt math\gt \phi=\phi^{\mathrm{I}}+\phi^{\mathrm{S}} }[/math]. Limiting the result to

the water surface leaves the integral equation for the water velocity potential under the ice floe,

[math]\displaystyle{ (int_eq_hs) \phi(\mathbf{x}) = \phi^{\mathrm{I}}(\mathbf{x}) + \int\limits_{\Delta} G (\mathbf{x};\mathbf{\xi}) \big( \alpha \phi(\mathbf{\xi}) + \mathrm{i}\sqrt{\alpha} w(\mathbf{\xi}) \big) \d\sigma_\mathbf{\xi}, \quad \mathbf{x} \in \Delta. Since the surface displacement of the ice floe appears in this integral equation, it is coupled with the plate equation (plate_final). A method of solution is discussed in detail by [[JGR02]] but a short outline will be given. The surface displacement of the ice floe is expanded into its modes of vibration by calculating the eigenfunctions and eigenvalues of the biharmonic operator. The integral equation for the potential is then solved for every eigenfunction which gives a corresponding potential to each eigenfunction. The expansion in the eigenfunctions simplifies the biharmonic equation and, by using the orthogonality of the eigenfunctions, a system of equations for the unknown coefficients of the eigenfunction expansion is obtained. ===The coupled ice floe - water equations=== Since the operator \lt math\gt \nabla^4 }[/math], subject to the free edge boundary

conditions, is self-adjoint a thin plate must possess a set of modes [math]\displaystyle{ w^k }[/math] which satisfy the free boundary conditions and the eigenvalue equation

[math]\displaystyle{ \nabla^4 w^k = \lambda_k w^k. The modes which correspond to different eigenvalues \lt math\gt \lambda_k }[/math] are

orthogonal and the eigenvalues are positive and real. While the plate will always have repeated eigenvalues, orthogonal modes can still be found and the modes can be normalised. We therefore assume that the modes are orthonormal, i.e.

[math]\displaystyle{ \int\limits_\Delta w^j (\mathbf{\xi}) w^k (\mathbf{\xi}) \d\sigma_{\mathbf{\xi}} = \delta _{jk}, where \lt math\gt \delta _{jk} }[/math] is the Kronecker delta. The eigenvalues [math]\displaystyle{ \lambda_k }[/math]

have the property that [math]\displaystyle{ \lambda_k \rightarrow \infty }[/math] as </math>k \rightarrow \infty[math]\displaystyle{ and we order the modes by increasing eigenvalue. These modes can be used to expand any function over the wetted surface of the ice floe \lt math\gt \Delta }[/math].

We expand the displacement of the floe in a finite number of modes [math]\displaystyle{ M }[/math], i.e.

[math]\displaystyle{ (expansion) w(\mathbf{x}) =\sum_{k=1}^{M} c_k w^k (\mathbf{x}). \gt From the linearity of (int_eq_hs) the potential can be written in the form \lt center\gt \lt math\gt (expansionphi) \phi(\mathbf{x}) =\phi^0(\mathbf{x}) + \sum_{k=1}^{M} c_k \phi^k (\mathbf{x}), where \lt math\gt \phi^0 }[/math] and [math]\displaystyle{ \phi^k }[/math] respectively satisfy the integral equations

\begin{subequations} (phi)

[math]\displaystyle{ (phi0) \phi^0(\mathbf{x}) = \phi^{\mathrm{I}} (\mathbf{x}) + \int\limits_\Delta \alpha G (\mathbf{x};\mathbf{\xi}) \phi^0 (\mathbf{\xi}) d\sigma_\mathbf{\xi} and \lt center\gt \lt math\gt (phii) \phi^k (\mathbf{x}) = \int\limits_{\Delta} G (\mathbf{x};\mathbf{\xi}) \left( \alpha \phi^k (\mathbf{\xi}) + \mathrm{i}\sqrt{\alpha} w^k (\mathbf{\xi})\right) \d\sigma_{\mathbf{\xi}}. \end{subequations} The potential \lt math\gt \phi^0 }[/math] represents the potential due to the incoming wave

assuming that the displacement of the ice floe is zero. The potential [math]\displaystyle{ \phi^k }[/math] represents the potential which is generated by the plate vibrating with the [math]\displaystyle{ k }[/math]th mode in the absence of any input wave forcing.

We substitute equations (expansion) and (expansionphi) into equation (plate_final) to obtain

[math]\displaystyle{ (expanded) \beta \sum_{k=1}^{M} \lambda_k c_k w^k -\alpha \gamma \sum_{k=1}^{M} c_k w^k = \mathrm{i}\sqrt{\alpha} \big( \phi^0 + \sum_{k=1}^{M} c_k \phi^k \big) - \sum_{k=1}^{M} c_k w^k. To solve equation (expanded) we multiply by \lt math\gt w^j }[/math] and integrate over

the plate (i.e. we take the inner product with respect to [math]\displaystyle{ w^j }[/math]) taking into account the orthogonality of the modes [math]\displaystyle{ w^j }[/math] and obtain

[math]\displaystyle{ (final) \beta \lambda_k c_k + \left( 1-\alpha \gamma \right) c_k = \int\limits_{\Delta} \mathrm{i}\sqrt{\alpha} \big( \phi^0 (\mathbf{\xi}) + \sum_{j=1}^{N} c_j \phi^j (\mathbf{\xi}) \big) w^k (\mathbf{\xi}) \d\sigma_{\mathbf{\xi}}, which is a matrix equation in \lt math\gt c_k }[/math].

Equation (final) cannot be solved without determining the modes of vibration of the thin plate [math]\displaystyle{ w^k }[/math] (along with the associated eigenvalues [math]\displaystyle{ \lambda_k }[/math]) and solving the integral equations

(phi). We use the finite element method to

determine the modes of vibration \cite[]{Zienkiewicz} and the integral equations (phi) are solved by a constant panel method \cite[]{Sarp_Isa}. The same set of nodes is used for the finite element method and to define the panels for the integral equation.

Full diffraction calculation for multiple ice floes

The interaction theory is a method to allow more rapid solutions to problems involving multiple bodies. The principle advantage is that the potential is represented in the cylindrical eigenfunctions and therefore fewer unknowns are required. However, every problem which can be solved by the interaction theory can also be solved by applying the full diffraction theory and solving an integral equation over the wetted surface of all the bodies. In this section we will briefly show how this extension can be performed for the ice floe situation. The full diffraction calculation will be used to check the performance and convergence of our interaction theory. Also, because the interaction theory is only valid when the escribed cylinder for each ice floe does not contain any other floe, the full diffraction calculation is required for a very dense arrangement of ice floes.

We can solve the full diffraction problem for multiple ice floes by the following extension. The displacement of the [math]\displaystyle{ j }[/math]th floe is expanded in a finite number of modes [math]\displaystyle{ M_j }[/math] (since the number of modes may not necessarily be the same), i.e.

[math]\displaystyle{ (expansion_f) w_j \left( \mathbf{x}\right) =\sum_{k=1}^{M_j} c_{jk} w_j^k (\mathbf{x}). \gt From the linearity of (int_eq_hs) the potential can be written in the form \lt center\gt \lt math\gt (expansionphi_f) \phi(\mathbf{x}) =\phi_0(\mathbf{x}) + \sum_{n=1}^{N} \sum_{k=1}^{M_n} c_{nk} \phi_n^k(\mathbf{x}), where \lt math\gt \phi _{0} }[/math] and [math]\displaystyle{ \phi_j^k }[/math] respectively satisfy the integral equations

\begin{subequations} (phi_f)

[math]\displaystyle{ (phi0_f) \phi_j^0 (\mathbf{x}) = \phi^{\mathrm{I}} (\mathbf{x}) + \sum_{n=1}^{N} \int\limits_{\Delta_n} \alpha G (\mathbf{x};\mathbf{\xi}) \phi_j^0(\mathbf{\xi}) \d\sigma_{\mathbf{\xi}} and \lt center\gt \lt math\gt (phii_f) \phi_j^k(\mathbf{x}) = \sum_{n=1}^{N} \int\limits_{\Delta_n} G (\mathbf{x};\mathbf{\xi}) \left( \alpha \phi_j^k (\mathbf{\xi}) + \i\sqrt{\alpha} w_j^k (\mathbf{\xi})\right) \d\sigma_{\mathbf{\xi}}. \end{subequations} The potential \lt math\gt \phi_j^{0} }[/math] represents the potential due the incoming wave

assuming that the displacement of the ice floe is zero, [math]\displaystyle{ \phi_j^k }[/math] represents the potential which is generated by the [math]\displaystyle{ j }[/math]th plate vibrating with the [math]\displaystyle{ k }[/math]th mode in the absence of any input wave forcing. It should be noted that [math]\displaystyle{ \phi_j^k(\mathbf{x}) }[/math] is, in general, non-zero for [math]\displaystyle{ \mathbf{x}\in \Delta_{n} }[/math] (since the vibration of the [math]\displaystyle{ j }[/math]th plate will result in potential under the [math]\displaystyle{ n }[/math]th plate).

We substitute equations (expansion_f) and (expansionphi_f) into equation (plate_final) to obtain

[math]\displaystyle{ (expanded_f) \beta_j \sum_{k=1}^{M_j} \lambda_{jk} c_{jk} w_j^k - \alpha \gamma_j \sum_{k=1}^{M_j} c_{jk} w_j^k = \mathrm{i}\sqrt{\alpha} \, \big( \phi_j^0 + \sum_{n=1}^{N} \sum_{k=1}^{M_n} c_{nk} \phi_n^k \big) - \sum_{k=1}^{M_n} c_{jk} w_j^k. To solve equation (expanded_f) we multiply by \lt math\gt w_j^{l} }[/math] and integrate

over the plate (as before) taking into account the orthogonality of the modes [math]\displaystyle{ w_j^l }[/math] and obtain

[math]\displaystyle{ (final_f) \beta_j \lambda_{jk} c_{jk} + \big(1- \alpha \gamma_j \big) c_{jk} = \int\limits_{\Delta_j} \mathrm{i}\sqrt{\alpha} \, \big( \phi_0 (\mathbf{\xi}) + \sum_{n=1}^{N} \sum_{l=1}^{M_n} c_{nl} \phi_n^l (\mathbf{\xi}) \big) \, w_j^l (\mathbf{\xi}) \d\sigma_{\mathbf{\xi}}, which holds for all \lt math\gt j= 1, \ldots, N }[/math] and therefore gives a matrix

equation