Kagemoto and Yue Interaction Theory

Introduction

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

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

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

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

Equations of Motion

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

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

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

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

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

Moreover, the Sommerfeld Radiation Condition is imposed

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

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

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

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

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

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

Eigenfunction expansion of the potential

The scattered potential of a body [math]\displaystyle{ \Delta_j }[/math] can be expanded in singular cylindrical eigenfunctions,

[math]\displaystyle{ (basisrep_out_d) \phi_j^\mathrm{S} (r_j,\theta_j,z) = \sum_{m=0}^{\infty} f_m(z) \sum_{\mu = - \infty}^{\infty} A_{m \mu}^j K_\mu (k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j}, }[/math]

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

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

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

[math]\displaystyle{ (basisrep_in_d) \phi_j^\mathrm{I} (r_j,\theta_j,z) = \sum_{n=0}^{\infty} f_n(z) \sum_{\nu = - \infty}^{\infty} D_{n\nu}^j I_\nu (k_n r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}, }[/math]

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

[math]\displaystyle{ (H_K) K_\nu (-\mathrm{i}x) = \frac{\pi i^{\nu+1}}{2} H_\nu^{(1)}(x) \quad =and= \quad I_\nu (-\mathrm{i}x) = i^{-\nu} J_\nu(x) }[/math]

with [math]\displaystyle{ H_\nu^{(1)} }[/math] and [math]\displaystyle{ J_\nu }[/math] denoting the Hankel function and the Bessel function, respectively, both of first kind and order [math]\displaystyle{ \nu }[/math].

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) = \frac{A g}{\omega} f_0(z) \mathrm{e}^{\mathrm{i}k (x \cos \chi + y \sin \chi)}, }[/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 (also cf.~figure (fig:floes)). This expression can be written in the eigenfunction expansion centred at the origin as

[math]\displaystyle{ \phi^{\mathrm{In}}(r,\theta,z) = \frac{A g}{\omega} f_0(z) \sum_{\nu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\nu (\pi/2 - \theta + \chi)} J_\nu(k r) }[/math]

\cite[p.~169]{linton01}. The local coordinates of each body are centred at their mean-centre positions [math]\displaystyle{ O_l = (l R,0) }[/math]. In order to represent the ambient wavefield, which is incident upon all bodies, in the eigenfunction expansion of an incoming wave in the local coordinates of the body, a phase factor has to be defined,

[math]\displaystyle{ (phase_factor) P_l = \mathrm{e}^{\mathrm{i}l R k \cos \chi}, }[/math]

which accounts for the position from the origin. Including this phase factor and making use of (H_K), the ambient wavefield at the [math]\displaystyle{ l }[/math]th body is given by

[math]\displaystyle{ \phi^{\mathrm{In}}(r_l,\theta_l,z) = \frac{A g}{\omega} \, P_l \, f_0(z) \sum_{\nu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\nu (\pi - \chi)} I_\nu(k_0 r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. }[/math]

We can therefore define the coefficients of the ambient wavefield in the eigenfunction expansion of an incident wave,

[math]\displaystyle{ \tilde{D}^l_{n\nu} = \begin{cases} \frac{A g}{\omega} P_l \mathrm{e}^{\mathrm{i}\nu (\pi - \chi)}, & n=0,\\ 0, & n \gt 0. \end{cases} }[/math]

Note that the evanescent coefficients are all zero due to the propagating nature of the ambient wave.

Derivation of the system of equations

A system of equations for the unknown coefficients (in the expansion (basisrep_out_d)) of the scattered wavefields of all bodies is developed. This system of equations is based on transforming the scattered potential of [math]\displaystyle{ \Delta_j }[/math] into an incident potential upon [math]\displaystyle{ \Delta_l }[/math] ([math]\displaystyle{ j \neq l }[/math]). Doing this for all bodies simultaneously, and relating the incident and scattered potential for each body, a system of equations for the unknown coefficients is developed. Making use of the periodicity of the geometry and of the ambient incident wave, this system of equations can then be simplified.

The scattered potential [math]\displaystyle{ \phi_j^{\mathrm{S}} }[/math] of body [math]\displaystyle{ \Delta_j }[/math] needs to be represented in terms of the incident potential [math]\displaystyle{ \phi_l^{\mathrm{I}} }[/math] upon [math]\displaystyle{ \Delta_l }[/math], [math]\displaystyle{ j \neq l }[/math]. From figure (fig:floes) we can see that this can be accomplished by using Graf's Addition Theorem

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

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

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

The limitation [math]\displaystyle{ r_l \lt R }[/math] only requires that the escribed cylinder of each body [math]\displaystyle{ \Delta_l }[/math] does not enclose any other origin [math]\displaystyle{ O_j }[/math] ([math]\displaystyle{ j \neq l }[/math]). However, the expansion of the scattered and incident potential in cylindrical eigenfunctions is only valid outside the escribed cylinder of each body. Therefore the condition that the escribed cylinder of each body [math]\displaystyle{ \Delta_l }[/math] does not enclose any other origin [math]\displaystyle{ O_j }[/math] ([math]\displaystyle{ j \neq l }[/math]) is superseded by the more rigorous restriction that the escribed cylinder of each body may not contain any other body.

Making use of the eigenfunction expansion as well as equation (transf), the scattered potential of [math]\displaystyle{ \Delta_j }[/math] (cf.~ (basisrep_out_d)) can be expressed in terms of the incident potential upon [math]\displaystyle{ \Delta_l }[/math] as

[math]\displaystyle{ \begin{matrix} \phi_j^{\mathrm{S}} (r_l,\theta_l,z) &= \sum_{m=0}^\infty f_m(z) \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty} (-1)^\nu K_{\tau-\nu} (k_m |j-l| R) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{j-l}} \\ &= \sum_{m=0}^\infty f_m(z) \sum_{\nu = -\infty}^{\infty} \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j (-1)^\nu K_{\tau-\nu} (k_m |j-l| R) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{j-l}} \Big] I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. \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]. Let [math]\displaystyle{ \tilde{D}_{n\nu}^{l} }[/math] denote the coefficients of this ambient incident wavefield in the incoming eigenfunction expansion for [math]\displaystyle{ \Delta_l }[/math] (cf.~\S (sec:ambient)). 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=-\infty,j \neq l}^{\infty} \, \phi_j^{\mathrm{S}} (r_l,\theta_l,z)\\ &= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty} \Big[ \tilde{D}_{n\nu}^{l} + \sum_{j=-\infty,j \neq l}^{\infty} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n |j-l|R) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{j-l}} \Big] \times I_\nu (k_n r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. \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_{n\nu}^l = \tilde{D}_{n\nu}^{l} + \sum_{j=-\infty,j \neq l}^{\infty} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n |j-l| R) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{j-l}}. }[/math]

Calculation of the diffraction transfer matrix for bodies of arbitrary geometry

The scattered and incident potential can therefore be related by a diffraction transfer operator acting in the following way,

[math]\displaystyle{ (diff_op) A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n \mu \nu} D_{n\nu}^l. }[/math]

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{ ({\mathbf 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.

The Free-Surface Green Function for Finite Depth in cylindrical polar coordinates

[math]\displaystyle{ G(r,\theta,z;s,\varphi,c)= \frac{1}{\pi} \sum_{m=0}^{\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 Black 1975 and Fenton 1978 is used. The elements of [math]\displaystyle{ {\mathbf B}_j }[/math] are therefore given by

[math]\displaystyle{ ({\mathbf 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} }[/math]

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{ \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} }[/math]

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.

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] [math]\displaystyle{ ({\mathbf 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} }[/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{ ({\mathbf B}_j^\beta)_{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+\beta)} \varsigma_{q\beta}^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta} }[/math]

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]

Final Equations

If the diffraction transfer operator is known (its calculation is discussed later), the substitution of (inc_coeff) into (diff_op) gives the required equations to determine the coefficients of the scattered wavefields of all bodies,

[math]\displaystyle{ (eq_op) A_{m\mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu}^{l} + \sum_{j=-\infty,j \neq l}^{\infty} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n |j-l| R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big], }[/math]

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