Kagemoto and Yue Interaction Theory

2007-04-12T14:19:45Z

Tf3W8q:

= 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]]. [[Interaction Theory for Cylinders]] presents a simplified version for cylinders.

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, see
[[Kagemoto and Yue Interaction Theory for Infinite Depth]].

[[Category:Interaction Theory]]

= Equations of Motion =

The problem consists of <math>n</math> bodies
<math>\Delta_j</math> with immersed body
surface <math>\Gamma_j</math>. Each body is subject to
the [[Standard Linear Wave Scattering Problem]] and the particluar
equations of motion for each body (e.g. rigid, or freely floating)
can be different for each body.
It is a [[Frequency Domain Problem]] with frequency <math>\omega</math>.
The solution is exact, up to the
restriction that the escribed cylinder of each body may not contain any
other body.
To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point
in the water, which is assumed to be of [[Finite Depth]] <math>d</math>,
while <math>\mathbf{x}</math> always denotes a point of the undisturbed water
surface assumed at <math>z=0</math>.

Writing <math>\alpha = \omega^2/g</math> where <math>g</math> is the acceleration due to
gravity, the potential <math>\phi</math> has to
satisfy the standard boundary-value problem
<center><math>
\nabla^2 \phi = 0, \; \mathbf{y} \in D
</math></center>
<center><math>
\frac{\partial \phi}{\partial z} = \alpha \phi, \;
{\mathbf{x}} \in \Gamma^\mathrm{f},
</math></center>
<center><math>
\frac{\partial \phi}{\partial z} = 0, \; \mathbf{y} \in D, \ z=-d,
</math></center>
where <math>D</math> is the
is the domain occupied by the water and
<math>\Gamma^\mathrm{f}</math> is the free water surface. At the immersed body
surface <math>\Gamma_j</math> of body <math>\Delta_j</math>, <math>j=1,\dots,N</math>, the water velocity potential has to
equal the normal velocity of the body <math>\mathbf{v}_j</math>,
<center><math>
\frac{\partial \phi}{\partial n} = \mathbf{v}_j, \; {\mathbf{y}}
\in \Gamma_j.
</math></center>
where the normal derivative is given by the particaluar equations of motion of the body.
Moreover, the [[Sommerfeld Radiation Condition]] is imposed.

=Eigenfunction expansion of the potential=

Each body is subject to an incident potential and moves in response to this
incident potential to produce a scattered potential. Each of these is
expanded using the [[Cylindrical Eigenfunction Expansion]]
The scattered potential of a body
<math>\Delta_j</math> can be expressed as
<center><math>
\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></center>
with discrete coefficients <math>A_{m \mu}^j</math>, where <math>(r_j,\theta_j,z)</math>
are cylinderical polar coordinates centered at each body
<center><math>
f_m(z) = \frac{\cos k_m (z d)}{\cos k_m d}.
</math></center>
and
where <math>k_n</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
<center><math>
\alpha k_m \tan k_m d = 0\,.
</math></center>
where <math>k_0</math> is the
imaginary root with positive imaginary part
and <math>k_m</math>, <math>m>0</math>, are given the positive real roots ordered
with increasing size.

The incident potential upon body <math>\Delta_j</math> can be also be expanded in
regular cylindrical eigenfunctions,
<center><math>
\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></center>
with discrete coefficients <math>D_{n\nu}^j</math>. In these expansions, <math>I_\nu</math>
and <math>K_\nu</math> denote the modified [http://en.wikipedia.org/wiki/Bessel_function : Bessel functions]
of the first and second kind, respectively, both of order <math>\nu</math>.
Note that the term for <math>m =0</math> or
<math>n=0</math>) corresponds to the propagating modes while the
terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.

=Derivation of the system of equations=

A system of equations for the unknown
coefficients of the
scattered wavefields of all bodies is developed. This system of
equations is based on transforming the
scattered potential of <math>\Delta_j</math> into an incident potential upon
<math>\Delta_l</math> (<math>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>\phi_j^{\mathrm{S}}</math> of body <math>\Delta_j</math> needs to be
represented in terms of the incident potential <math>\phi_l^{\mathrm{I}}</math>
upon <math>\Delta_l</math>, <math>j \neq l</math>. This can be accomplished by using
[[Graf's Addition Theorem]]
<center><math>
K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
\sum_{\nu = - \infty}^{\infty} K_{\tau \nu} (k_m R_{jl}) \,
I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l \varphi_{jl})}, \quad j \neq l,
</math></center>
which is valid provided that <math>r_l < R_{jl}</math>. Here, <math>(R_{jl},\varphi_{jl})</math> are the polar coordinates of the mean centre position of <math>\Delta_{l}</math> in the local coordinates of <math>\Delta_{j}</math>.

The limitation <math>r_l < R_{jl}</math> only requires that the escribed cylinder of each body
<math>\Delta_l</math> does not enclose any other origin <math>O_j</math> (<math>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>\Delta_l</math> does not enclose any other
origin <math>O_j</math> (<math>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 [[Graf's Addition Theorem]], the scattered potential
of <math>\Delta_j</math> can be expressed in terms of the
incident potential upon <math>\Delta_l</math> as
<center><math>
\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 R_{jl}) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu
\theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}}
</math></center>
<center><math>
= \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 R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu)
\varphi_{jl}} \Big] I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
</math></center>
The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
expanded in the eigenfunctions corresponding to the incident wavefield upon
<math>\Delta_l</math>. Let <math>\tilde{D}_{n\nu}^{l}</math> denote the coefficients of this
ambient incident wavefield in the incoming eigenfunction expansion for
<math>\Delta_l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]]).
<center><math>
\phi^{\mathrm{In}}(r_l,\theta_l,z)= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
\tilde{D}_{n\nu}^{l} I_\nu (k_n
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
</math></center>
The total
incident wavefield upon body <math>\Delta_j</math> can now be expressed as
<center><math>
\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)
</math></center>
This allows us to write
<center><math>
\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></center>
<center><math>
= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
\Big[ \tilde{D}_{n\nu}^{l}
\sum_{j=1,j \neq l}^{n} \sum_{\tau =
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
</math></center>
It therefore follows tha
<center><math>
D_{n\nu}^j =
\tilde{D}_{n\nu}^{l}
\sum_{j=1,j \neq l}^{n} \sum_{\tau =
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}}
</math></center>

= Final Equations =

The scattered and incident potential of each body <math>\Delta_l</math> can be related by the
[[Diffraction Transfer Matrix]] acting in the following way,
<center><math>
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
\mu \nu}^l D_{n\nu}^l.
</math></center>

The substitution of this into the equation for relating
the coefficients <math>D_{n\nu}^l</math> and
<math>A_{m \mu}^l</math>gives the
required equations to determine the coefficients of the scattered
wavefields of all bodies,
<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\Big[ \tilde{D}_{n\nu}^{l}
\sum_{j=1,j \neq l}^{N} \sum_{\tau =
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big],
</math></center>
<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.

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Kagemoto and Yue Interaction Theory