Difference between revisions of "Category:Interaction Theory"

From WikiWaves
Jump to navigationJump to search
 
(41 intermediate revisions by 3 users not shown)
Line 1: Line 1:
 +
{{complete pages}}
 +
 
Interaction theory is based on calculating a solution for a number of individual scatterers
 
Interaction theory is based on calculating a solution for a number of individual scatterers
without simply discretising the total problem. THe theory is generally applied in
+
without simply discretising the total problem. The theory is generally applied in
 
three dimensions.
 
three dimensions.
 
Essentially the [[Cylindrical Eigenfunction Expansion]]
 
Essentially the [[Cylindrical Eigenfunction Expansion]]
Line 7: Line 9:
 
a solution without any approximation. This solution method is valid, provided only that
 
a solution without any approximation. This solution method is valid, provided only that
 
an escribed circle can be drawn around each body.  
 
an escribed circle can be drawn around each body.  
 
= Illustrative Example =
 
 
 
We present an illustrative example of an interaction theory for the case of <math>n</math>
 
We present an illustrative example of an interaction theory for the case of <math>n</math>
[[Bottom Mounted Cylinders]]. This theory was presented in [[Linton and Evans 1990]] and it
+
[[Linton and Evans 1990]] presented an [[Interaction Theory for Cylinders]]
can be derived from the [[Kagemoto and Yue Interaction Theory]] by simply assuming that each
+
which was [[Kagemoto and Yue Interaction Theory]] simplified by assuming that each
body is a cylinder.
+
body is a [[Bottom Mounted Cylinder]].  
 
 
= Equations of Motion =
 
 
 
After we have [[Removing the Depth Dependence|Removed the Depth Dependence]]
 
the problem consists of <math>n</math> cylinders of radius <math>a_j</math>
 
subject to [[Helmholtz's equation]]
 
<center><math>
 
\nabla^2 \phi -k^2\phi= 0,
 
</math></center>
 
where
 
<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> (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></center>
 
with discrete coefficients <math>A_{m \mu}^j</math>, where
 
<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> (eq_km)
 
\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> (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></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 in (basisrep_out_d) (and  (basisrep_in_d)) 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 (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>\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> (transf)
 
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 equation (transf), the scattered potential
 
of <math>\Delta_j</math> (cf.~ (basisrep_out_d)) 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]]). 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=-\infty,j \neq l}^{\infty} \, \phi_j^{\mathrm{S}}
 
(r_l,\theta_l,z)
 
</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>
 
 
 
= Final Equations =
 
 
 
The scattered and incident potential can be related by the
 
[[Diffraction Transfer Matrix]] 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>
 
 
 
The substitution of (inc_coeff) into  (diff_op) gives the
 
required equations to determine the coefficients of the scattered
 
wavefields of all bodies,
 
<center><math> (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=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>.
 
  
  
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]

Latest revision as of 08:16, 19 October 2009


Interaction theory is based on calculating a solution for a number of individual scatterers without simply discretising the total problem. The theory is generally applied in three dimensions. Essentially the Cylindrical Eigenfunction Expansion surrounding each body is used coupled with some way of mapping these. Various approximations were developed until the the Kagemoto and Yue Interaction Theory which contained a solution without any approximation. This solution method is valid, provided only that an escribed circle can be drawn around each body. We present an illustrative example of an interaction theory for the case of [math]\displaystyle{ n }[/math] Linton and Evans 1990 presented an Interaction Theory for Cylinders which was Kagemoto and Yue Interaction Theory simplified by assuming that each body is a Bottom Mounted Cylinder.