Difference between revisions of "Kagemoto and Yue Interaction Theory"

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= Final Equations =
 
= Final Equations =
  
The scattered and incident potential can therefore be related by a
+
The scattered and incident potential can be related by the
diffraction transfer operator acting in the following way,
+
[[Diffraction Transfer Matrix]] acting in the following way,
 
<center><math> (diff_op)
 
<center><math> (diff_op)
 
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
 
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
Line 180: Line 180:
 
</math></center>  
 
</math></center>  
  
If the diffraction transfer operator is known (its calculation
+
The substitution of (inc_coeff) into  (diff_op) gives the
is discussed later), the substitution of (inc_coeff) into  (diff_op) gives the
 
 
required equations to determine the coefficients of the scattered
 
required equations to determine the coefficients of the scattered
 
wavefields of all bodies,  
 
wavefields of all bodies,  
Line 193: Line 192:
 
</math></center>
 
</math></center>
 
<math>m \in {N}</math>, <math>l,\mu \in {Z}</math>.
 
<math>m \in {N}</math>, <math>l,\mu \in {Z}</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,
 
<center><math> (diff_op)
 
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
 
\mu \nu} D_{n\nu}^l.
 
</math></center>
 
 
Before we can apply the interaction theory we require the diffraction
 
transfer matrices <math>\mathbf{B}_j</math> which relate the incident and the
 
scattered potential for a body <math>\Delta_j</math> in isolation.
 
The elements of the diffraction transfer matrix, <math>({\mathbf B}_j)_{pq}</math>,
 
are the coefficients of the
 
<math>p</math>th partial wave of the scattered potential due to a single
 
unit-amplitude incident wave of mode <math>q</math> upon <math>\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
 
<center><math>
 
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></center>
 
given by [[Black 1975]] and [[Fenton 1978]] is used.
 
The elements of <math>{\mathbf B}_j</math> are therefore given by
 
<center><math>
 
({\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></center>
 
where
 
<math>\varsigma_q^j(\mathbf{\zeta})</math> is the source strength distribution
 
due to an incident potential of mode <math>q</math> of the form
 
<center><math>
 
\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></center>
 
 
We assume that we have represented the scattered potential in terms of
 
the source strength distribution <math>\varsigma^j</math> so that the scattered
 
potential can be written as
 
<center><math> (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></center>
 
where <math>D</math> is the volume occupied by the water and <math>\Gamma_j</math> is the
 
immersed surface of body <math>\Delta_j</math>. The source strength distribution
 
function <math>\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>(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>\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
 
<center><math>
 
\frac{\partial \phi_{q\beta}^{\mathrm{I}}}{\partial n} =
 
\frac{\partial \phi_{q}^{\mathrm{I}}}{\partial n}
 
\mathrm{e}^{\mathrm{i}q \beta},
 
</math></center>
 
where <math>\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
 
<center><math>
 
\varsigma_{q\beta}^j = \varsigma_q^j \, \mathrm{e}^{\mathrm{i}q \beta}.
 
</math></center>
 
<center><math>
 
({\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></center>
 
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>\mathbf{B}_j</math> are
 
given by equations  (B_elem). Keeping in mind that the body is
 
rotated by the angle <math>\beta</math>, the elements of the diffraction transfer
 
matrix of the rotated body are given by
 
<center><math>
 
({\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></center>
 
 
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>\beta</math>,
 
<math>\mathbf{B}_j^\beta</math>, are given by
 
<center><math> (B_rot)
 
(\mathbf{B}_j^\beta)_{pq} = (\mathbf{B}_j)_{pq} \, \mathrm{e}^{\mathrm{i}(q-p) \beta}.
 
</math></center>
 

Revision as of 00:27, 16 June 2006

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.

Derivation of the system of equations

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

The scattered potential [math]\displaystyle{ \phi_j^{\mathrm{S}} }[/math] of body [math]\displaystyle{ \Delta_j }[/math] needs to be represented in terms of the incident potential [math]\displaystyle{ \phi_l^{\mathrm{I}} }[/math] upon [math]\displaystyle{ \Delta_l }[/math], [math]\displaystyle{ j \neq l }[/math]. This can be accomplished by using Graf's Addition Theorem

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

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

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

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

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

[math]\displaystyle{ \phi_j^{\mathrm{S}} (r_l,\theta_l,z) = \sum_{m=0}^\infty f_m(z) \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty} (-1)^\nu K_{\tau-\nu} (k_m |j-l| R) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{j-l}} }[/math]
[math]\displaystyle{ = \sum_{m=0}^\infty f_m(z) \sum_{\nu = -\infty}^{\infty} \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j (-1)^\nu K_{\tau-\nu} (k_m |j-l| R) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{j-l}} \Big] I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. }[/math]

The ambient incident wavefield [math]\displaystyle{ \phi^{\mathrm{In}} }[/math] can also be expanded in the eigenfunctions corresponding to the incident wavefield upon [math]\displaystyle{ \Delta_l }[/math]. Let [math]\displaystyle{ \tilde{D}_{n\nu}^{l} }[/math] denote the coefficients of this ambient incident wavefield in the incoming eigenfunction expansion for [math]\displaystyle{ \Delta_l }[/math] (cf. the example in Cylindrical Eigenfunction Expansion). The total incident wavefield upon body [math]\displaystyle{ \Delta_j }[/math] can now be expressed as

[math]\displaystyle{ \phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) + \sum_{j=-\infty,j \neq l}^{\infty} \, \phi_j^{\mathrm{S}} (r_l,\theta_l,z) }[/math]
[math]\displaystyle{ = \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty} \Big[ \tilde{D}_{n\nu}^{l} + \sum_{j=-\infty,j \neq l}^{\infty} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n |j-l|R) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{j-l}} \Big] I_\nu (k_n r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. }[/math]

Final Equations

The scattered and incident potential can be related by the Diffraction Transfer Matrix acting in the following way,

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

The substitution of (inc_coeff) into (diff_op) gives the required equations to determine the coefficients of the scattered wavefields of all bodies,

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

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