Kagemoto and Yue Interaction Theory for Infinite Depth

Introduction

Kagemoto and Yue Interaction Theory applies in Finite Depth water. The theory was extended by Peter and Meylan 2004 to Infinite Depth water and we present this theory here.

Eigenfunction expansion of the potential

The scattered potential of body [math]\displaystyle{ \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} + \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 Abramowitz and Stegun 1964. 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} + \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

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 Abramowitz and Stegun 1964,

[math]\displaystyle{ (transf) \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]

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}^{\mathrm{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] (cf. the example in Cylindrical Eigenfunction Expansion). 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{}_{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_{j=1,j \neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathrm{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_{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) D_{0\mu}^l = D_{l0\mu}^{\mathrm{In}} + \sum_{j=1,j \neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathbf{i} (\nu - \mu) \vartheta_{jl}} }[/math] [math]\displaystyle{ D_{\mu}^l(\eta) = D_{l\mu}^{\mathrm{In}}(\eta) + \sum_{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}}. }[/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) \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]

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) \begin{matrix} A_{0n}^l =& \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{pp} \Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{j=1,j \neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathbf{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_{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,\\ A_n^l (\eta) &= \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{dp} (\eta) \Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{j=1,j \neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathbf{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_{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{matrix} }[/math]

[math]\displaystyle{ n \in \mathit{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{ {\mathbf 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{ {\mathbf T}_{jl} }[/math] given by

[math]\displaystyle{ (T_elem_deep) ({\mathbf T}_{jl})_{pq} = H_{p-q}^{(1)}(\alpha R_{jl}) \, \mathrm{e}^{\mathrm{i}(p-q) \vartheta_{jl}} }[/math]

for the propagating modes, and

[math]\displaystyle{ ({\mathbf T}_{jl})_{pq} = (-1)^{q} \, K_{p-q} (\eta R_{jl}) \, \mathrm{e}^{\mathbf{i} (p-q) \vartheta_{jl}} }[/math]

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

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

where the left superscript [math]\displaystyle{ \mathrm{t} }[/math] indicates transposition. The matrix [math]\displaystyle{ {\mathbf \hat{B}}_l }[/math] denotes the infinite depth diffraction transfer matrix [math]\displaystyle{ {\mathbf 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.

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_elemrot) (\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}, }[/math]

and

[math]\displaystyle{ (\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}, }[/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.