Difference between revisions of "Interaction Theory for Infinite Arrays"

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=Introduction =
+
{{complete pages}}
 +
 
 +
==Introduction ==
  
 
There are two approaches to solution for the [[:Category:Infinite Array|Infinite Array]],
 
There are two approaches to solution for the [[:Category:Infinite Array|Infinite Array]],
Line 7: Line 9:
 
This is based on [[Peter, Meylan, and Linton 2006]]
 
This is based on [[Peter, Meylan, and Linton 2006]]
  
= System of equations =
+
== System of equations ==
  
 
We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely
 
We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely
Line 66: Line 68:
 
<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>.
 
<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>.
 
Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.
 
Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.
 
[[Far Field Waves]]
 
 
 
 
 
 
 
An algebraically exact solution to the problem of linear water-wave
 
scattering by a periodic array of scatterers is presented in which the
 
scatterers may be of arbitrary shape. The method of solution is based
 
on an interaction theory
 
 
in which the incident wave on each body from all the other bodies in
 
the array is expressed in the respective local cylindrical
 
eigenfunction expansion. We show how to calculate the
 
slowly convergent terms efficiently which arise in the formulation and
 
how to calculate the
 
scattered field far from the array. The application to the problem of
 
linear acoustic scattering by cylinders with arbitrary cross-section
 
is also discussed. Numerical calculations are presented to
 
show that our results agree with previous calculations. We
 
present some computations for the case of fixed, rigid and elastic floating
 
bodies of negligible draft concentrating on presenting the 
 
amplitudes of the scattered waves as functions of the incident angle.
 
 
 
 
 
 
 
  
 
=The far field=
 
=The far field=
Line 106: Line 77:
 
Letting <math>p=2\pi/R</math>, define the scattering angles <math>\chi_m</math> by
 
Letting <math>p=2\pi/R</math>, define the scattering angles <math>\chi_m</math> by
 
<center><math>
 
<center><math>
\chi_m = \cos^{-1} (\psi_m/k) \quad =where
+
\chi_m = \cos^{-1} (\psi_m/k) \quad \mathrm{where}
 
\quad \psi_m = k \cos \chi + m p
 
\quad \psi_m = k \cos \chi + m p
 
</math></center>
 
</math></center>
 
 
and write <math>\psi</math> for <math>\psi_0</math>. Also note that <math>\chi_0 = \chi</math> by definition.
 
and write <math>\psi</math> for <math>\psi_0</math>. Also note that <math>\chi_0 = \chi</math> by definition.
If <math>\abs{\psi_m}<k</math>, i.e.~if
+
If <math>|\psi_m|<k</math>, i.e. if
 
<center><math>
 
<center><math>
 
-1 < \cos \chi +\frac{mp}{k}<1,
 
-1 < \cos \chi +\frac{mp}{k}<1,
Line 118: Line 88:
 
(see below) that these angles (<math>\pm \chi_m</math> for <math>m \in \mathcal{M}</math>)
 
(see below) that these angles (<math>\pm \chi_m</math> for <math>m \in \mathcal{M}</math>)
 
are the directions in which plane waves propagate away from the array.
 
are the directions in which plane waves propagate away from the array.
If <math>\abs{\psi_m}>k</math> then <math>\chi_m</math> is no longer real and the
+
If <math>|\psi_m|>k</math> then <math>\chi_m</math> is no longer real and the
 
appropriate branch of the <math>\arccos</math> function is given by
 
appropriate branch of the <math>\arccos</math> function is given by
 
<center><math>
 
<center><math>
\arccos t =
+
\cos^{-1} t =
 
\begin{cases}  
 
\begin{cases}  
\mathrm{i} \arccosh t, & t> 1,\\
+
\mathrm{i} \cosh^{-1} t, & t> 1,\\
\pi-\mathrm{i} \arccosh (-t) & t<-1,
+
\pi-\mathrm{i} \cosh^{-1} (-t) & t<-1,
 
\end{cases}
 
\end{cases}
 
</math></center>
 
</math></center>
with <math>\arccosh t = \log \left(t+\sqrt{t^2-1}\right)</math> for <math>t>1</math>.
+
with <math>\cosh^{-1} t = \log \left(t+\sqrt{t^2-1}\right)</math> for <math>t>1</math>.
  
 
For the total potential we have
 
For the total potential we have
<center><math>\begin{matrix} \notag
+
<center><math>\begin{matrix}  
 
\phi &=\phi^\mathrm{In}+ \sum_{m=0}^{\infty}  
 
\phi &=\phi^\mathrm{In}+ \sum_{m=0}^{\infty}  
 
 
f_m(z) \sum_{j=-\infty}^{\infty} P_j
 
f_m(z) \sum_{j=-\infty}^{\infty} P_j
 
\sum_{\mu = -\infty}^{\infty} A_{m\mu} K_\mu(k_m r_j)\mathrm{e}^{\mathrm{i} \mu\theta_j} \\
 
\sum_{\mu = -\infty}^{\infty} A_{m\mu} K_\mu(k_m r_j)\mathrm{e}^{\mathrm{i} \mu\theta_j} \\
Line 138: Line 107:
  
 
f_0(z) \sum_{j=-\infty}^{\infty} P_j
 
f_0(z) \sum_{j=-\infty}^{\infty} P_j
\sum_{\mu = -\infty}^{\infty} A_{0\mu} \i^{\mu+1} H^{(1)}_\mu (kr_j)
+
\sum_{\mu = -\infty}^{\infty} A_{0\mu} i^{\mu+1} H^{(1)}_\mu (kr_j)
 
\mathrm{e}^{\mathrm{i} \mu\theta_j},  
 
\mathrm{e}^{\mathrm{i} \mu\theta_j},  
(26)
 
 
\end{matrix}</math></center>
 
\end{matrix}</math></center>
 
as <math>kr\to\infty</math>, away from the array axis <math>y=0</math>, where we have used
 
as <math>kr\to\infty</math>, away from the array axis <math>y=0</math>, where we have used
Line 149: Line 117:
 
<center><math>
 
<center><math>
 
H^{(1)}_\mu (kr) \mathrm{e}^{\mathrm{i} \mu \theta}=
 
H^{(1)}_\mu (kr) \mathrm{e}^{\mathrm{i} \mu \theta}=
\frac{(-\i)^{\mu+1}}{\pi} \int\limits_{-\infty}^{\infty}
+
\frac{(-i)^{\mu+1}}{\pi} \int\limits_{-\infty}^{\infty}
\frac{\mathrm{e}^{-k\gamma(t)\abs{y}}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt}\,\mathrm{e}^{\i
+
\frac{\mathrm{e}^{-k\gamma(t)|y|}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt}\,\mathrm{e}^{i
\mu \sgn(y)\arccos t} \,\mathrm{d} t,
+
\mu \sgn(y)\cos^{-1} t} \,\mathrm{d} t,
  
 
</math></center>
 
</math></center>
Line 159: Line 127:
 
\gamma(t) =  
 
\gamma(t) =  
 
\begin{cases}  
 
\begin{cases}  
-\mathrm{i} \sqrt{1-t^2} & \abs{t} \leq 1 \\
+
-\mathrm{i} \sqrt{1-t^2} & |t| \leq 1 \\
\sqrt{t^2-1} & \abs{t}>1,  
+
\sqrt{t^2-1} & |t|>1,  
 
\end{cases}
 
\end{cases}
 
</math></center>
 
</math></center>
into (26) we get
+
we get
 
<center><math>\begin{matrix}
 
<center><math>\begin{matrix}
 
\phi & \sim\phi^\mathrm{In}+ \frac{1}{2}
 
\phi & \sim\phi^\mathrm{In}+ \frac{1}{2}
Line 169: Line 137:
 
f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty
 
f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty
 
\int\limits_{-\infty}^{\infty}  
 
\int\limits_{-\infty}^{\infty}  
\frac{\mathrm{e}^{-k \gamma(t)\abs{y}}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt}
+
\frac{\mathrm{e}^{-k \gamma(t)|y|}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt}
\,\mathrm{e}^{\i(\psi-kt) jR}\,\mathrm{e}^{\i
+
\,\mathrm{e}^{i(\psi-kt) jR}\,\mathrm{e}^{i
\mu \sgn(y)\arccos t} \,\mathrm{d} t \\
+
\mu \sgn(y) \cos^{-1} t} \,\mathrm{d} t \\
 
& =\phi^\mathrm{In}+ \frac{\pi}{kR}
 
& =\phi^\mathrm{In}+ \frac{\pi}{kR}
  
 
f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu}  \sum_{j=-\infty}^\infty
 
f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu}  \sum_{j=-\infty}^\infty
\frac{\mathrm{e}^{-k\gamma(\psi_j/k)\abs{y}}}{\gamma(\psi_j/k)}
+
\frac{\mathrm{e}^{-k\gamma(\psi_j/k)|y|}}{\gamma(\psi_j/k)}
\,\mathrm{e}^{\mathrm{i} x\psi_j}\,\mathrm{e}^{\i
+
\,\mathrm{e}^{\mathrm{i} x\psi_j}\,\mathrm{e}^{i
\mu\sgn(y)\arccos \psi_j/k} \\
+
\mu\sgn(y)\cos^{-1} \psi_j/k} \\
& =\phi^\mathrm{In}+ \frac{\pi\i}{kR}
+
& =\phi^\mathrm{In}+ \frac{\pi i}{kR}
  
 
f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu}  \sum_{j=-\infty}^\infty
 
f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu}  \sum_{j=-\infty}^\infty
\frac{1}{\sin\chi_j} \,\mathrm{e}^{\mathrm{i} kr\cos(\abs{\theta}-\chi_j)}\,\mathrm{e}^{\i
+
\frac{1}{\sin\chi_j} \,\mathrm{e}^{\mathrm{i} kr\cos(|\theta|-\chi_j)}\,\mathrm{e}^{i
 
\mu \sgn(\theta)\chi_j},  
 
\mu \sgn(\theta)\chi_j},  
 
\end{matrix}</math></center>
 
\end{matrix}</math></center>
Line 191: Line 159:
 
</math></center>
 
</math></center>
 
The only terms which contribute to the far field are those for which
 
The only terms which contribute to the far field are those for which
<math>\abs{\psi_m}<k</math>. Thus, as <math>y\to\pm\infty</math>, the far field consists of
+
<math>|\psi_m|<k</math>. Thus, as <math>y\to\pm\infty</math>, the far field consists of
 
a set of plane waves propagating in the directions <math>\theta=\pm\chi_m</math>:  
 
a set of plane waves propagating in the directions <math>\theta=\pm\chi_m</math>:  
 
<center><math>
 
<center><math>
\phi\sim \phi^\mathrm{In}+ \frac{\pi \i}{kR}  
+
\phi\sim \phi^\mathrm{In}+ \frac{\pi i}{kR}  
  
 
f_0(z) \sum_{m\in\mathcal{M}} \frac{1}{\sin\chi_m}
 
f_0(z) \sum_{m\in\mathcal{M}} \frac{1}{\sin\chi_m}
 
\,\mathrm{e}^{\mathrm{i} kr\cos(\theta\mp\chi_m)}
 
\,\mathrm{e}^{\mathrm{i} kr\cos(\theta\mp\chi_m)}
 
\sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}.
 
\sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}.
(30)
+
 
 
</math></center>
 
</math></center>
>From \eqref{eqn:inffar} the amplitudes of the
+
From \eqref{eqn:inffar} the amplitudes of the
 
scattered waves for each scattering angle <math>\pm \chi_m</math> are given in terms
 
scattered waves for each scattering angle <math>\pm \chi_m</math> are given in terms
 
of the coefficients <math>A_{0\mu}</math> by
 
of the coefficients <math>A_{0\mu}</math> by
 
<center><math>(31)
 
<center><math>(31)
A^\pm_m =  \frac{\pi \i}{kR} \frac{1}{\sin\chi_m}
+
A^\pm_m =  \frac{\pi i}{kR} \frac{1}{\sin\chi_m}
 
\sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}.
 
\sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}.
 
</math></center>
 
</math></center>
Line 215: Line 183:
 
scattered plane waves propagates along the array. We will not consider
 
scattered plane waves propagates along the array. We will not consider
 
this resonant case here except for stating that then, the scattered field is
 
this resonant case here except for stating that then, the scattered field is
dominated by waves travelling along the array, either towards </math>x =
+
dominated by waves travelling along the array, either towards <math>x =
\infty<math> (if </math>\chi_m = 0<math>) or towards </math>x=-\infty<math> (if </math>\chi_m = \pi<math>).  
+
\infty</math> (if <math>\chi_m = 0</math>) or towards <math>x=-\infty</math> (if <math>\chi_m = \pi</math>).  
Also, we will not consider the excitation of Rayleigh-Bloch waves, which
+
Also, we will not consider the excitation of [[Rayleigh-Bloch Waves]], which
 
are waves which travel along the array with a phase difference
 
are waves which travel along the array with a phase difference
between adjacent bodies greater than <math>Rk</math> (include refs). Both the resonant
+
between adjacent bodies greater than <math>Rk</math>.
and Rayleigh-Bloch case are important but beyond the scope of the
 
present work.
 
 
  
 
+
==The efficient computation of the <math>\sigma_{\nu}^0</math> ==
 
 
 
 
=The efficient computation of the <math>\sigma_{\nu}^0</math> =
 
  
 
The constants <math>\sigma_{\nu}^0</math> (cf.~\eqref{eq_op_sigma}) appearing in
 
The constants <math>\sigma_{\nu}^0</math> (cf.~\eqref{eq_op_sigma}) appearing in
Line 237: Line 199:
 
<center><math>  
 
<center><math>  
 
\sigma_{\nu}^0  =  \sum_{j=1}^{\infty}  (P_{-j}+ (-1)^\nu P_j)
 
\sigma_{\nu}^0  =  \sum_{j=1}^{\infty}  (P_{-j}+ (-1)^\nu P_j)
K_{\nu} (-\mathrm{i} k j R) =  \frac{\pi \i^{\nu+1}}{2} \sum_{j=1}^{\infty}
+
K_{\nu} (-\mathrm{i} k j R) =  \frac{\pi i^{\nu+1}}{2} \sum_{j=1}^{\infty}
 
(P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR),  
 
(P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR),  
 
</math></center>
 
</math></center>
Line 247: Line 209:
 
(P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR)
 
(P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR)
 
</math></center>
 
</math></center>
as the <math>\sigma^0_\nu</math> are then determined by </math>\sigma^0_\nu =
+
as the <math>\sigma^0_\nu</math> are then determined by <math>\sigma^0_\nu =
\pi/2 \,\, \i^{\nu+1} \, \tilde{\sigma}^0_\nu<math>.
+
\pi/2 \,\, i^{\nu+1} \, \tilde{\sigma}^0_\nu</math>.
  
 
An efficient way of computing the <math>\tilde{\sigma}_{\nu}^0</math>
 
An efficient way of computing the <math>\tilde{\sigma}_{\nu}^0</math>
is given in [[linton98]] and the results are briefly outlined
+
is given in [[Linton 1998]] and the results are briefly outlined
 
in our notation.
 
in our notation.
Noting that </math>H^{(1)}_{-\nu} (\,\cdot\,)= (-1)^{\nu}
+
Noting that <math>H^{(1)}_{-\nu} (\,\cdot\,)= (-1)^{\nu}
H^{(1)}_{\nu} (\,\cdot\,)<math>, it suffices to discuss the computation of the
+
H^{(1)}_{\nu} (\,\cdot\,)</math>, it suffices to discuss the computation of the
 
<math>\sigma_{\nu}^0</math> for non-negative <math>\nu</math>.
 
<math>\sigma_{\nu}^0</math> for non-negative <math>\nu</math>.
  
Referring to [[linton98]], the constants <math>\tilde{\sigma}_{\nu}^0</math> can
+
Referring to [[Linton 1998]], the constants <math>\tilde{\sigma}_{\nu}^0</math> can
 
be written as  
 
be written as  
 
 
<center><math>
 
<center><math>
 
+
\tilde{\sigma}_{0}^0 = -1 -\frac{2 i}{\pi} \left( C + \log \frac{k}{2p}
\tilde{\sigma}_{0}^0 &= -1 -\frac{2\i}{\pi} \left( C + \log \frac{k}{2p}
 
 
\right) + \frac{2}{R k \sin \chi} - \frac{2 \mathrm{i} (k^2 + 2
 
\right) + \frac{2}{R k \sin \chi} - \frac{2 \mathrm{i} (k^2 + 2
\psi^2)}{p^3 R} \zeta(3)\\ &\quad
+
\psi^2)}{p^3 R} \zeta(3)
 
+ \frac{2}{R} \sum_{m=1}^\infty \left(
 
+ \frac{2}{R} \sum_{m=1}^\infty \left(
 
\frac{1}{k \sin \chi_{-m}} + \frac{1}{k \sin \chi_m} +
 
\frac{1}{k \sin \chi_{-m}} + \frac{1}{k \sin \chi_m} +
\frac{2 \i}{p m} + \frac{\mathrm{i} (k^2 + 2 \psi^2)}{p^3 m^3} \right)
+
\frac{2 i}{p m} + \frac{\mathrm{i} (k^2 + 2 \psi^2)}{p^3 m^3} \right)
  
 
</math></center>
 
</math></center>
Line 277: Line 237:
 
as well as
 
as well as
 
<center><math>\begin{matrix}
 
<center><math>\begin{matrix}
&\quad
+
\tilde{\sigma}_{2\nu}^0 &=& 2 (-1)^{\nu} \left( \frac{\mathrm{e}^{2\mathrm{i} \nu
\tilde{\sigma}_{2\nu}^0 &= 2 (-1)^{\nu} \left( \frac{\mathrm{e}^{2\mathrm{i} \nu
+
\chi} }{R k \sin \chi} - \frac{ i}{\pi}
\chi} }{R k \sin \chi} - \frac{\i}{\pi}
 
 
\left( \frac{k}{2 p} \right)^{2\nu} \zeta(2\nu +1) \right) +
 
\left( \frac{k}{2 p} \right)^{2\nu} \zeta(2\nu +1) \right) +
\frac{\i}{\nu \pi} \\
+
\frac{ i}{\nu \pi} \\
&\quad + 2 (-1)^\nu \sum_{m=1}^\infty
+
& + &2 (-1)^\nu \sum_{m=1}^\infty
 
\left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi_m}}{R k \sin \chi_{m}} +
 
\left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi_m}}{R k \sin \chi_{m}} +
 
\frac{\mathrm{e}^{-2 \mathrm{i} \nu \chi_{-m}}}{R k \sin \chi_{-m}} +
 
\frac{\mathrm{e}^{-2 \mathrm{i} \nu \chi_{-m}}}{R k \sin \chi_{-m}} +
\frac{\i}{m\pi} \left( \frac{k}{2 m p} \right)^{2\nu}
+
\frac{ i}{m\pi} \left( \frac{k}{2 m p} \right)^{2\nu}
\right)\\ &\quad
+
\right)\\ &
+ \frac{\i}{\pi} \sum_{m=1}^\nu  
+
&+& \frac{ i}{\pi} \sum_{m=1}^\nu  
 
\frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m)! (\nu-m)!} \left( \frac{p}{k} \right)^{2m}
 
\frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m)! (\nu-m)!} \left( \frac{p}{k} \right)^{2m}
 
B_{2m}(\psi/p),
 
B_{2m}(\psi/p),
\\
+
\end{matrix}</math></center>
&
+
<center><math>\begin{matrix}
\tilde{\sigma}_{2\nu-1}^0 &= - 2 (-1)^\nu \left( \frac{\mathrm{i}  \mathrm{e}^{\mathrm{i} (2\nu-1)
+
\tilde{\sigma}_{2\nu-1}^0 &=& - 2 (-1)^\nu \left( \frac{\mathrm{i}  \mathrm{e}^{\mathrm{i} (2\nu-1)
 
\chi}}{R k \sin \chi} - \frac{ \psi R
 
\chi}}{R k \sin \chi} - \frac{ \psi R
 
\nu}{\pi^2} \left( \frac{k}{2 p} \right)^{2\nu-1}
 
\nu}{\pi^2} \left( \frac{k}{2 p} \right)^{2\nu-1}
 
\zeta(2\nu +1) \right)\\
 
\zeta(2\nu +1) \right)\\
&\quad - 2 (-1)^\nu \sum_{m=1}^\infty
+
& -& 2 (-1)^\nu \sum_{m=1}^\infty
 
\left(\frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1)\chi_m} }{R k \sin \chi_m} +
 
\left(\frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1)\chi_m} }{R k \sin \chi_m} +
 
\frac{\mathrm{i} \mathrm{e}^{-\mathrm{i} (2\nu-1) \chi_{-m}}}{R k \sin \chi_{-m}} +
 
\frac{\mathrm{i} \mathrm{e}^{-\mathrm{i} (2\nu-1) \chi_{-m}}}{R k \sin \chi_{-m}} +
 
\frac{\psi R \nu}{m^2\pi^2} \left( \frac{k}{2
 
\frac{\psi R \nu}{m^2\pi^2} \left( \frac{k}{2
m p} \right)^{2\nu-1} \right)\\ &\quad - \frac{2}{\pi} \sum_{m=0}^{\nu-1}  
+
m p} \right)^{2\nu-1} \right)\\  
 +
& -& \frac{2}{\pi} \sum_{m=0}^{\nu-1}  
 
\frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m+1)! (\nu-m-1)!} \left( \frac{p}{k}
 
\frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m+1)! (\nu-m-1)!} \left( \frac{p}{k}
 
\right)^{2m+1} B_{2m+1}(\psi/p),
 
\right)^{2m+1} B_{2m+1}(\psi/p),
Line 312: Line 272:
 
converge like <math>O(m^{-5})</math> as <math>m\rightarrow\infty</math>.
 
converge like <math>O(m^{-5})</math> as <math>m\rightarrow\infty</math>.
  
Note that since <math>\sin \chi_m</math> is purely imaginary for </math>m \notin
+
Note that since <math>\sin \chi_m</math> is purely imaginary for <math>m \notin
\mathcal{M}<math>, the computation of the real part of
+
\mathcal{M}</math>, the computation of the real part of
 
<math>\tilde{\sigma}_{2\nu}^0</math> and the imaginary part of <math>\tilde{\sigma}_{2\nu-1}^0</math>
 
<math>\tilde{\sigma}_{2\nu}^0</math> and the imaginary part of <math>\tilde{\sigma}_{2\nu-1}^0</math>
 
is particularly simple. For <math>\nu \geq 0</math>, they are given by
 
is particularly simple. For <math>\nu \geq 0</math>, they are given by
Line 326: Line 286:
 
where <math>\delta_{mn}</math> is the Kronecker delta.
 
where <math>\delta_{mn}</math> is the Kronecker delta.
  
 +
== Acoustic scattering by an infinite array of identical generalized cylinders ==
  
 
 
 
\section{Acoustic scattering by an infinite array of identical generalized
 
cylinders}
 
 
The theory above has so far been developed for water-wave scattering
 
The theory above has so far been developed for water-wave scattering
 
of a plane wave by an infinite array of identical arbitrary bodies. It
 
of a plane wave by an infinite array of identical arbitrary bodies. It
Line 344: Line 300:
 
theory applies with the following modifications:
 
theory applies with the following modifications:
  
 
+
#The [[Dispersion Relation for a Free Surface]] is replaced by <math>k=\omega /
#The dispersion relation \eqref{eq_k} is replaced by </math>k=\omega /
+
c</math> where <math>c</math> is the speed of sound in the medium under consideration
c<math> where </math>c<math> is the speed of sound in the medium under consideration
+
and the [[Dispersion Relation for a Free Surface]] is omitted.
and the dispersion relation is \eqref{eq_k_m} omitted.
 
 
#All factors <math>\cos k_m(z+d)</math>, <math>\cos k_m(c+d)</math>, <math>\cos k_m d</math>
 
#All factors <math>\cos k_m(z+d)</math>, <math>\cos k_m(c+d)</math>, <math>\cos k_m d</math>
 
and <math>f_0</math> are replaced by 1.
 
and <math>f_0</math> are replaced by 1.
 
#The factor <math>N_0</math> in \eqref{green_d} is <math>k/\pi</math>.
 
#The factor <math>N_0</math> in \eqref{green_d} is <math>k/\pi</math>.
  
Note that point <math>(a)</math> implies that there are no evanescent modes in this
+
Note that there are no evanescent modes in this
problem, i.e.~the sums over <math>m</math> and <math>n</math> in the eigenfunction expansions
+
problem, i.e. the sums over <math>m</math> and <math>n</math> in the eigenfunction expansions
\eqref{basisrep_out_d} and \eqref{basisrep_in_d}, respectively, only
+
only
contain the terms for <math>m=0</math> and <math>n=0</math>. Moreover, we have </math>k_0 = -
+
contain the terms for <math>m=0</math> and <math>n=0</math>. Moreover, we have <math>k_0 = -
\mathrm{i} \, \omega /c<math>.
+
\mathrm{i} \, \omega /c</math>.
 
 
For circular cylinders, i.e.~cylinders which have a circular
 
cross-section, this problem has been considered by [[linton93]]. In
 
\S 52 we numerically compare our results for this
 
problem with theirs.
 
 
 
 
 
 
 
 
 
 
 
 
 
\section{Wave forcing of a fixed, rigid and flexible body
 
of shallow draft}(37)
 
 
 
The theory which has been developed so far has been
 
for arbitrary bodies. No assumption has been made about the body
 
geometry or its equations of motion. However, we want use this
 
theory to make calculations for the specific case of bodies of
 
shallow draft which may be fixed (which we shall refer to as
 
a dock), rigid, or elastic (modelled as a thin plate).
 
In the formulation, we concentrate on the elastic case of which
 
the other two situations are subcases. This allows us to present
 
a range of results while focusing on the geophysical problem which
 
motivates our work, namely the wave scattering by a field of ice floes.
 
 
 
==Mathematical model for an elastic plate.==
 
We briefly describe the mathematical model of a floating
 
elastic plate. A more detailed account can be found in
 
[[JGR02,JFM04]]. We assume that the elastic plate is sufficiently thin
 
that we may apply the shallow-draft approximation, which essentially
 
applies the boundary conditions underneath the plate at the water
 
surface. Assuming the
 
elastic plate to be in contact with the water
 
surface at all times, its displacement
 
<math>W</math> is that of the water surface and <math>W</math> is required to satisfy the linear
 
plate equation in the area occupied by the elastic plate <math>\Delta</math>. In
 
analogy to \eqref{time}, denoting the time-independent surface displacement
 
(with the same radian frequency as the water velocity potential due to
 
linearity) by <math>w</math> (<math>W=\Re\{w \exp(-\mathrm{i}\omega t)\}</math>), the plate
 
equation becomes 
 
<center><math>(38)
 
D \, \nabla^4 w - \omega^2 \, \rho_\Delta \, h \, w = \mathrm{i} \, \omega \, \rho
 
\, \phi - \rho \, g \, w, \quad {\mathbf{x}} \in \Delta,
 
</math></center>
 
with the density of the water <math>\rho</math>, the modulus of rigidity of the
 
elastic plate <math>D</math>, its density <math>\rho_\Delta</math> and its
 
thickness <math>h</math>. The right-hand side of \eqref{plate_non} arises from the
 
linearized Bernoulli equation. It needs to be recalled that
 
<math>\mathbf{x}</math> always denotes a point of the undisturbed water surface.
 
Free-edge boundary conditions apply, namely
 
<center><math>(39)
 
\left[ \nabla^2 - (1-\nu)
 
\left(\frac{\partial^2}{\partial s^2} + \kappa(s)
 
\frac{\partial}{\partial n} \right) \right] w = 0,
 
</math></center>
 
<center><math>(40)
 
\left[ \frac{\partial}{\partial n} \nabla^2 +(1-\nu)
 
\frac{\partial}{\partial s}
 
\left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}
 
-\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0,
 
</math></center>
 
where <math>\nu</math> is Poisson's ratio and
 
\begin{gather}
 
\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}
 
= \frac{\partial^2}{\partial n^2} + \frac{\partial^2}{\partial s^2}
 
+ \kappa(s) \frac{\partial}{\partial n}.
 
\end{gather}
 
Here, <math>\kappa(s)</math> is the curvature of the boundary, <math>\partial \Delta</math>,
 
as a function of arclength <math>s</math> along <math>\partial \Delta</math>;
 
<math>\partial/\partial s</math> and <math>\partial/\partial n</math> represent derivatives
 
tangential and normal to the boundary <math>\partial \Delta</math>, respectively.
 
 
 
Non-dimensional variables (denoted with an overbar) are introduced,
 
<center><math>
 
(\bar{x},\bar{y},\bar{z}) = \frac{1}{L} (x,y,z), \quad \bar{w} =
 
\frac{w}{L}, \quad \bar{\alpha} = L\, \alpha, \quad \bar{\omega} = \omega
 
\sqrt{\frac{L}{g}} \quad =and=  \quad \bar{\phi} = \frac{\phi}{L
 
\sqrt{L g}},
 
</math></center>
 
where <math>L</math> is a length parameter associated with the plate.
 
In non-dimensional variables, the equation for the elastic plate
 
\eqref{plate_non} reduces to
 
<center><math>(41)
 
\beta \nabla^4 \bar{w} - \bar{\alpha} \gamma \bar{w} = \i
 
\sqrt{\bar{\alpha}}  \bar{\phi} - \bar{w}, \quad
 
\bar{\mathbf{x}} \in \bar{\Delta},
 
</math></center>
 
with
 
<center><math>
 
\beta = \frac{D}{g \rho L^4} \quad =and=  \quad \gamma =
 
\frac{\rho_\Delta h}{ \rho L}.
 
</math></center>
 
The constants <math>\beta</math> and <math>\gamma</math> represent the stiffness and the
 
mass of the plate, respectively. For convenience, the overbars are
 
dropped and non-dimensional variables are assumed in what follows.
 
 
 
 
 
==Method of solution==
 
We briefly outline our method of solution for the coupled water--elastic plate
 
problem \cite[details can be found in][]{JGR02}.
 
The problem for the water velocity potential is converted to an
 
integral equation in the following way. Let <math>G</math> be the
 
three-dimensional free-surface Green's function for water of finite depth.
 
The Green's function allows the representation of the scattered water
 
velocity potential in the standard way,
 
<center><math>(42)
 
\phi^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma}
 
\left( \phi^\mathrm{S} (\mathbf{\zeta}) \, \frac{\partial G}{\partial
 
n_\mathbf{\zeta}} (\mathbf{y};\mathbf{\zeta}) - G
 
(\mathbf{y};\mathbf{\zeta}) \, \frac{\partial
 
\phi^\mathrm{S}}{\partial n_\mathbf{\zeta}} (\mathbf{\zeta}) \right)
 
\d\sigma_\mathbf{\zeta}, \quad \mathbf{y} \in D.
 
</math></center>
 
In the case of a shallow draft, the fact that the Green's function is
 
symmetric and therefore satisfies the free-surface boundary condition
 
with respect to the second variable as well can be used to
 
simplify \eqref{int_eq} drastically. Due to the linearity of the problem,
 
the ambient incident potential can just be added to the equation to obtain the
 
total water velocity potential,
 
<math>\phi=\phi^{\mathrm{I}}+\phi^{\mathrm{S}}</math>. Limiting the result to
 
the water surface leaves the integral equation for the water velocity
 
potential under the elastic plate,
 
<center><math>(43)
 
\phi(\mathbf{x}) = \phi^{\mathrm{I}}(\mathbf{x}) +
 
\int\limits_{\Delta} G (\mathbf{x};\mathbf{\xi}) \big( \alpha
 
\phi(\mathbf{\xi}) + \mathrm{i} \sqrt{\alpha} w(\mathbf{\xi}) \big)
 
\d\sigma_\mathbf{\xi}, \quad \mathbf{x} \in \Delta.
 
</math></center>
 
Since the surface displacement of the elastic plate appears in this
 
integral equation, it is coupled with the plate equation \eqref{plate_final}.
 
 
 
==The coupled elastic plate--water equations==
 
 
 
Since the operator <math>\nabla^4</math>, subject to the free-edge boundary
 
conditions, is self-adjoint, a thin plate must possess a set of modes <math>w^k</math>
 
which satisfy the free boundary conditions and the eigenvalue
 
equation
 
<center><math>
 
\nabla^4 w^k = \lambda_k w^k.
 
</math></center>
 
The modes which correspond to different eigenvalues <math>\lambda_k</math> are
 
orthogonal and the eigenvalues are positive and real. While the plate will
 
always have repeated eigenvalues, orthogonal modes can still be found and
 
the modes can be normalized. We therefore assume that the modes are
 
orthonormal, i.e.
 
<center><math>
 
\int\limits_\Delta w^j (\mathbf{\xi}) w^k (\mathbf{\xi})
 
\d\sigma_{\mathbf{\xi}} = \delta _{jk}.
 
</math></center>
 
 
 
The eigenvalues <math>\lambda_k</math>
 
have the property that <math>\lambda_k \rightarrow \infty</math> as </math>k \rightarrow
 
\infty<math> and we order the modes by increasing eigenvalue. These modes can be
 
used to expand any function over the wetted surface of the elastic
 
plate <math>\Delta</math>.
 
 
 
We expand the displacement of the plate in a finite number of modes <math>M</math>, i.e.
 
<center><math>(44)
 
w(\mathbf{x}) =\sum_{k=1}^{M} c_k w^k (\mathbf{x}).
 
</math></center>
 
>From the linearity of \eqref{int_eq_hs} the potential can be
 
written in the form
 
<center><math>(45)
 
\phi(\mathbf{x}) =\phi^0(\mathbf{x}) + \sum_{k=1}^{M} c_k \phi^k (\mathbf{x}),
 
</math></center>
 
where <math>\phi^0</math> and <math>\phi^k</math> respectively satisfy the integral equations
 
(46)
 
<center><math>(47)
 
\phi^0(\mathbf{x}) = \phi^{\mathrm{I}} (\mathbf{x}) +
 
\int\limits_\Delta \alpha G (\mathbf{x};\mathbf{\xi}) \phi^0
 
(\mathbf{\xi}) d\sigma_\mathbf{\xi}
 
</math></center>
 
and
 
<center><math> (48)
 
\phi^k (\mathbf{x}) = \int\limits_{\Delta} G (\mathbf{x};\mathbf{\xi})
 
\left( \alpha \phi^k (\mathbf{\xi}) + \mathrm{i} \sqrt{\alpha} w^k
 
(\mathbf{\xi})\right) \d\sigma_{\mathbf{\xi}}. 
 
</math></center>
 
 
 
The potential <math>\phi^0</math> represents the potential due to the incoming wave
 
assuming that the displacement of the elastic plate is zero. The potential
 
<math>\phi^k</math> represents the potential which is generated by the plate
 
vibrating with the <math>k</math>th mode in the absence of any input wave forcing.
 
 
 
We substitute equations \eqref{expansion} and \eqref{expansionphi} into
 
equation \eqref{plate_final} to obtain
 
<center><math>(49)
 
\beta \sum_{k=1}^{M} \lambda_k c_k w^k -\alpha \gamma
 
\sum_{k=1}^{M} c_k w^k = \mathrm{i} \sqrt{\alpha} \big( \phi^0 +
 
\sum_{k=1}^{M} c_k \phi^k \big) - \sum_{k=1}^{M} c_k w^k.
 
</math></center>
 
To solve equation \eqref{expanded} we multiply by <math>w^j</math> and integrate over
 
the plate (i.e.~we take the inner product with respect to <math>w^j</math>) taking
 
into account the orthonormality of the modes <math>w^j</math> and obtain
 
<center><math>(50)
 
\beta \lambda_k c_k + \left( 1-\alpha \gamma \right) c_k =
 
\int\limits_{\Delta} \mathrm{i} \sqrt{\alpha} \big( \phi^0 (\mathbf{\xi})
 
+ \sum_{j=1}^{N} c_j \phi^j (\mathbf{\xi}) \big) w^k (\mathbf{\xi})
 
\d\sigma_{\mathbf{\xi}},
 
</math></center>
 
which is a matrix equation in <math>c_k</math>.
 
 
 
Equation \eqref{final} cannot be solved without determining the modes of
 
vibration of the thin plate <math>w^k</math> (along with the associated
 
eigenvalues <math>\lambda_k</math>) and solving the integral equations
 
\eqref{phi}. We use the finite element method to
 
determine the modes of vibration \cite[]{Zienkiewicz} and the integral
 
equations \eqref{phi} are solved by a constant-panel
 
method \cite[]{Sarp_Isa}. The same set of nodes is used for the
 
finite-element method and to define the panels for the integral equation.
 
 
 
==The fixed and rigid body cases==
 
 
 
The fixed and rigid body cases can easily be solved by the method
 
outlined above since they can be considered special cases.
 
In the problem of a fixed body (dock), the displacement is always
 
zero, <math>w=0</math>, so we simply need to solve equation (46) for
 
<math>\phi=\phi^0</math>. For the case of a rigid body, we need to truncate
 
the sums in \eqref{expanded} to include the first three modes only
 
(which correspond to the three modes of rigid motion of the 
 
plate, namely the heave, pitch and roll). Note that for these modes the
 
eigenvalue is <math>\lambda_k=0</math> so that the term involving the stiffness
 
<math>\beta</math> does not appear in equation (50).
 
 
 
=Numerical calculations=
 
 
 
In this section, we present some numerical computations using the
 
theory developed in the previous sections. We are particularly
 
interested in comparisons with results from other methods as well as
 
using our method to compare the behaviour of different bodies. Besides
 
comparisons with results from other works, one way to
 
check the correctness of the implementation is to verify that energy
 
is conserved, i.e.~the energy of the incoming wave must be equal to
 
the sum of the energies of all outgoing waves. In terms of the
 
amplitudes of the scattered waves for each scattering angle </math>\pm
 
\chi_m<math>, </math>A^\pm_m<math>, (cf.~\eqref{scat_ampl}) this can be written as
 
<center><math>(51)
 
\sin \chi = \sum_{m\in \mathcal{M}} \left(\abs{A^-_m}^2 +
 
\abs{A^+_m + \delta_{0m}}^2 \right) \, \sin \chi_m
 
</math></center>
 
where we have assumed an ambient incident potential of unit amplitude.
 
 
 
In all calculations presented below, the absolute value of the
 
difference of both sides in \eqref{energy_cons} is at most
 
<math>10^{-3}</math>.
 
 
 
 
 
==Comparison with results from [[linton93]]== (52)
 
We first compare our results to those of [[linton93]] who
 
considered the acoustic scattering of a plane sound wave incident upon
 
a periodic array of identical rigid circular cylinders of radius <math>a</math>. It can
 
be noted that they also discussed the application of their theory to the
 
water-wave scattering by an infinite row of rigid vertical circular cylinders
 
extending throughout the water depth. Their method of solution was
 
based on a multipole expansion but they also included a separation of
 
variables method which can be viewed as a special case of our method.
 
 
 
As \citeauthor{linton93} considered circular cylinders, we need to
 
obtain the diffraction transfer matrix of rigid circular
 
cylinders. Due to the axisymmetry, they are particularly simple. In
 
fact, they are diagonal with diagonal elements
 
<center><math>
 
(\mathbf{B})_{pp} =  -I_p'(k_0 a) / K_p'(k_0 a)
 
</math></center>
 
 
 
\cite[cf.][p.~177, for example]{linton01}. Also, there are no evanescent modes
 
if the ambient incident wave does not contain evanescent modes (which is the
 
case in their considerations as well as ours).
 
 
 
We compare \citeauthor{linton93}' results to ours in terms of the
 
amplitudes of the scattered waves, <math>A^\pm_m</math>. In particular, we
 
reproduce their figures 1 (a) and (b) (corresponding to our figure
 
53) which show the absolute values of
 
the amplitudes of the scattered waves plotted against <math>ka</math> when </math>a/R =
 
0.2<math> for the cases </math>\chi = \pi/2<math> and </math>\pi/3<math>,
 
respectively. Note that they use different choices for defining the incident
 
angle and spacing of the cylinders. Since \citeauthor{linton93} only
 
give plotted data we also plot our results (shown in figure
 
53). A visual comparison of the plots shows that they
 
are in good agreement with \citeauthor{linton93}' results.
 
 
 
\begin{figure}
 
\begin{tabular}{p{.46\columnwidth}p{.02\columnwidth}p{.46\columnwidth}}
 
\includegraphics[width=.38\columnwidth]{linton_chi2} &&
 
\includegraphics[width=.38\columnwidth]{linton_chi3}
 
\end{tabular}
 
\caption{Absolute values of the amplitudes of the scattered waves
 
plotted against <math>ka</math> when <math>a/R = 0.2</math> for the two cases </math>\chi =
 
\pi/2<math> (left) and </math>\chi = \pi/3<math> (right).}(53)
 
\end{figure}
 
 
 
It is worth noting that for an ambient incident angle of </math>\chi =
 
\pi/2<math> (normal incidence) the scattered waves appear in pairs
 
of two corresponding to <math>\pm m</math>, i.e.~they travel in the directions
 
<math>\pm \chi_m</math> with respect to the array axis. Note that this is
 
generally true for normal incidence upon arrays of arbitrary bodies
 
and easily follows from the considerations at the beginning of \S
 
4. Moreover, as <math>ka</math> is increased, more scattered waves appear. From the
 
plots in figure 53, it seems that the amplitudes of
 
the scattered waves have poles in these points
 
of appearance but a careful consideration shows that they are
 
actually continuous at these points \cite[cf.][]{linton93}.
 
 
 
==Comparison with results from [[JEM05]]==  (54)
 
 
 
Next, we compare our results to those of [[JEM05]] who considered
 
the water-wave scattering by an infinite array of floating elastic
 
plates in water of infinite depth. The plates were modelled in exactly
 
the same way as our elastic plates in
 
\S 37. Their method of solution was based on the use of a
 
special periodic Green's function in \eqref{int_eq_hs}. As a way of
 
testing their method, \citeauthor{JEM05} also considered the
 
scattering from an array of docks (fixed bodies).
 
Therefore, we reproduce their results for the dock \cite[table 1
 
in][]{JEM05} and for elastic plates \cite[table 2 in][]{JEM05} in
 
tables 55 and 56, respectively. In
 
both cases, the plates are of square geometry with sidelength 4 and
 
spacing <math>R=6</math>. The ambient wave is of the same wavelength as the
 
sidelength of the bodies and the incident angle is <math>\chi = \pi/3</math> (in our
 
notation). In table
 
56, the elastic plates have non-dimensionalized stiffness
 
<math>\beta = 0.1</math> and mass <math>\gamma = 0</math>. We choose <math>d=4</math> in order to
 
simulate infinite depth. Since the elastic plates tend to
 
lengthen the wave it is necessary to choose a water depth greater than the
 
standard choice of half the ambient wavelength
 
\cite[cf.][]{FoxandSquire}.
 
 
 
As can be seen, each amplitude in table
 
55 has a relative difference of less than
 
<math>6 \cdot 10^{-2}</math> with respect to the values obtained by
 
\citeauthor{JEM05}. The analogue is true for table 56
 
with a relative error of less than <math>9 \cdot 10^{-2}</math> except for
 
<math>A^-_{-1}</math> where the relative error is <math>\approx 0.34</math> (note,
 
however, that the values in \citeauthor{JEM05} are only given up to
 
the third decimal place).
 
The results given in tables 55 and 56
 
were obtained using 23 angular
 
propagating modes, three roots of the dispersion relation
 
\eqref{eq_k_m} (not counting the zeroth root) and seven corresponding angular
 
evanescent modes each. Note that fewer modes also yield reasonably
 
good approximations. For example, taking 15 angular propagating
 
modes, one root of the dispersion relation and three corresponding
 
angular evanescent modes yields answers differing from those in
 
tables 55 and 56 only in the fourth decimal place.
 
 
 
 
 
\begin{table}
 
\begin{center}
 
\begin{tabular}
 
<math>m</math>  & <math>A^-_m</math>              & <math>A^+_m</math>\\
 
<math>-2</math> & <math>-0.2212 - 0.0493\i</math> & <math>+0.2367 + 0.0268\i</math> \\
 
<math>-1</math> & <math>+0.2862 - 0.2627\i</math> & <math>-0.2029 + 0.3601\i</math>\\
 
<math>0</math>  & <math>+0.6608 - 0.1889\i</math> & <math>-0.7203 - 0.1237\i</math>
 
\end{tabular}
 
\caption{Amplitudes of the scattered waves for the case of a dock.}
 
(55)
 
\end{center}
 
\end{table}
 
 
 
 
 
 
 
\begin{table}
 
\begin{center}
 
\begin{tabular}
 
<math>m</math>  & <math>A^-_m</math>              & <math>A^+_m</math>\\
 
<math>-2</math> & <math>+0.0005 + 0.0149\i</math> & <math>-0.0405 - 0.0138\i</math> \\
 
<math>-1</math> & <math>-0.0202 - 0.0125\i</math> & <math>-0.0712 - 0.1004\i</math> \\
 
<math>0</math>  & <math>-0.0627 - 0.0790\i</math> & <math>-0.2106 - 0.5896\i</math>
 
\end{tabular}
 
\caption{Amplitudes of the scattered waves for the case of a elastic plates.}
 
(56)
 
\end{center}
 
\end{table}
 
 
 
 
 
\subsection{Comparison of the scattering by an array of docks, rigid
 
plates and elastic plates}
 
In this section, we use our method to compare the behaviour of
 
arrays of docks, rigid plates and elastic plates. The equations describing
 
the different bodies have been derived in \S
 
37. In order to have a common setting, we choose all
 
bodies to be square with sidelength 2 and a body spacing of
 
<math>R=4</math>. The ambient wavelength is <math>\lambda = 1.5</math> and the water depth
 
is <math>d=0.5</math>.
 
 
 
 
 
In figures 58, 59 and 60 we show the
 
absolute values of the amplitudes of the scattering angles as
 
functions of incident angle as well as the solution of the scattering
 
problem for <math>\chi = \pi/5</math> for an array of docks, rigid plates and elastic
 
plates, respectively. The elastic plates are chosen to have non-dimensional
 
stiffness and mass <math>\beta=\gamma=0.02</math> while the rigid plates have
 
the same mass.  In the plots of the amplitudes of the
 
scattered waves, we plot <math>\abs{A^-_0}</math> and <math>\abs{1+A^+_0}</math> as solid
 
lines and the additional scattered waves with symbols as listed in
 
table 57. Note that the calculation of the amplitudes of the
 
scattered waves is fairly fast since the most difficult task -- the
 
calculation of the diffraction transfer matrix -- only needs to be
 
performed once for each type of body.
 
 
 
>From figures 58, 59 and
 
60, it can be seen that docks generally reflect the energy
 
much more than the flexible plates. From this point of view, the rigid plates
 
can be seen as a kind of intermediate setting.
 
 
 
For <math>\chi=\pi/5\approx 0.628</math>, the scattering angles are </math>\chi_{-4}
 
\approx 2.33<math>, </math>\chi_{-3} \approx 1.89<math>, </math>\chi_{-2} \approx 1.51<math>, </math>\chi_{-1}
 
\approx 1.12<math> (and their negative values). The docks particularly
 
reflect in the direction <math>-\chi_{-1}</math> (beside <math>-\chi</math>). It can also be
 
seen that the flexible plates already transmit most of the
 
energy for this incident angle. The strong decrease in the amplitudes
 
of their reflected waves appears at about <math>\chi\approx 0.58</math>.  The
 
decrease of the amplitudes of the reflected waves for the rigid plates
 
does not appear until a larger incident angle and is also not as
 
strong. For the docks, such a strong decrease is not observed at
 
all. Moreover, note that all three types of bodies reflect in the
 
direction <math>-\chi_1</math> fairly strongly for incident angles
 
around <math>0.91</math> (where we have </math>-\chi_1 \approx
 
-0.150<math> for </math>\chi = 0.91<math>).
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 +
For circular cylinders, i.e. cylinders which have a circular
 +
cross-section, this problem has been considered by [[Linton and Evans 1993]].
  
[[Category:Infinite Array]]</math>
+
[[Category:Infinite Array]]
 +
[[Category:Interaction Theory]]

Latest revision as of 08:21, 19 October 2009


Introduction

There are two approaches to solution for the Infinite Array, one is Infinite Array Green Function the other is based on Interaction Theory. We present here a solution based on the latter, using Kagemoto and Yue Interaction Theory to derive a system of equations for the infinite array. This is based on Peter, Meylan, and Linton 2006

System of equations

We start with the final system of equations of the Kagemoto and Yue Interaction Theory, namely

[math]\displaystyle{ 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]

[math]\displaystyle{ m \in \mathbb{N} }[/math], [math]\displaystyle{ \mu \in \mathbb{Z} }[/math], [math]\displaystyle{ l=1,\dots,N }[/math].

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by [math]\displaystyle{ R }[/math], we have [math]\displaystyle{ R_{jl} = |j-l| R }[/math] and

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

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients [math]\displaystyle{ A_{m\mu}^l }[/math] can be written as [math]\displaystyle{ A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu} }[/math], where the phase factor [math]\displaystyle{ P_l }[/math] is given by


[math]\displaystyle{ \ P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi}, }[/math]

where [math]\displaystyle{ \chi }[/math] is the angle which the direction of the ambient waves makes with the [math]\displaystyle{ x }[/math]-axis. The same can be done for the coefficients of the ambient wave, i.e. [math]\displaystyle{ \tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu} }[/math].

Therefore, the system simplifies to

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

Introducing the constants

[math]\displaystyle{ \sigma^n_\nu = \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_\nu(k_n|j|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR), }[/math]

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

[math]\displaystyle{ A_{m\mu} = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu \sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big], }[/math]

[math]\displaystyle{ m \in \mathbb{N} }[/math], [math]\displaystyle{ \mu \in \mathbb{Z} }[/math]. Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula [math]\displaystyle{ A_{m\mu}^l = P_l A_{m\mu} }[/math].

The far field

In this section, the far field is examined which describes the scattering far away from the array. The derivation is equivalent to that of Twersky 1962. First, we define the scattering angles which give the directions of propagation of plane scattered waves far away from the array. Letting [math]\displaystyle{ p=2\pi/R }[/math], define the scattering angles [math]\displaystyle{ \chi_m }[/math] by

[math]\displaystyle{ \chi_m = \cos^{-1} (\psi_m/k) \quad \mathrm{where} \quad \psi_m = k \cos \chi + m p }[/math]

and write [math]\displaystyle{ \psi }[/math] for [math]\displaystyle{ \psi_0 }[/math]. Also note that [math]\displaystyle{ \chi_0 = \chi }[/math] by definition. If [math]\displaystyle{ |\psi_m|\lt k }[/math], i.e. if

[math]\displaystyle{ -1 \lt \cos \chi +\frac{mp}{k}\lt 1, }[/math]

we say that [math]\displaystyle{ m\in \mathcal{M} }[/math] and then [math]\displaystyle{ 0\lt \chi_m\lt \pi }[/math]. It turns out (see below) that these angles ([math]\displaystyle{ \pm \chi_m }[/math] for [math]\displaystyle{ m \in \mathcal{M} }[/math]) are the directions in which plane waves propagate away from the array. If [math]\displaystyle{ |\psi_m|\gt k }[/math] then [math]\displaystyle{ \chi_m }[/math] is no longer real and the appropriate branch of the [math]\displaystyle{ \arccos }[/math] function is given by

[math]\displaystyle{ \cos^{-1} t = \begin{cases} \mathrm{i} \cosh^{-1} t, & t\gt 1,\\ \pi-\mathrm{i} \cosh^{-1} (-t) & t\lt -1, \end{cases} }[/math]

with [math]\displaystyle{ \cosh^{-1} t = \log \left(t+\sqrt{t^2-1}\right) }[/math] for [math]\displaystyle{ t\gt 1 }[/math].

For the total potential we have

[math]\displaystyle{ \begin{matrix} \phi &=\phi^\mathrm{In}+ \sum_{m=0}^{\infty} f_m(z) \sum_{j=-\infty}^{\infty} P_j \sum_{\mu = -\infty}^{\infty} A_{m\mu} K_\mu(k_m r_j)\mathrm{e}^{\mathrm{i} \mu\theta_j} \\ &\sim \phi^\mathrm{In}+ \frac{\pi}{2} f_0(z) \sum_{j=-\infty}^{\infty} P_j \sum_{\mu = -\infty}^{\infty} A_{0\mu} i^{\mu+1} H^{(1)}_\mu (kr_j) \mathrm{e}^{\mathrm{i} \mu\theta_j}, \end{matrix} }[/math]

as [math]\displaystyle{ kr\to\infty }[/math], away from the array axis [math]\displaystyle{ y=0 }[/math], where we have used the identity \eqref{H_K}.

The far field can be determined as follows. If we insert the integral representation

[math]\displaystyle{ H^{(1)}_\mu (kr) \mathrm{e}^{\mathrm{i} \mu \theta}= \frac{(-i)^{\mu+1}}{\pi} \int\limits_{-\infty}^{\infty} \frac{\mathrm{e}^{-k\gamma(t)|y|}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt}\,\mathrm{e}^{i \mu \sgn(y)\cos^{-1} t} \,\mathrm{d} t, }[/math]

in which [math]\displaystyle{ x=r\cos\theta }[/math], [math]\displaystyle{ y=r\sin\theta }[/math] and [math]\displaystyle{ \gamma(t) }[/math] is defined for real [math]\displaystyle{ t }[/math] by

[math]\displaystyle{ \gamma(t) = \begin{cases} -\mathrm{i} \sqrt{1-t^2} & |t| \leq 1 \\ \sqrt{t^2-1} & |t|\gt 1, \end{cases} }[/math]

we get

[math]\displaystyle{ \begin{matrix} \phi & \sim\phi^\mathrm{In}+ \frac{1}{2} f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty \int\limits_{-\infty}^{\infty} \frac{\mathrm{e}^{-k \gamma(t)|y|}}{\gamma(t)}\,\mathrm{e}^{\mathrm{i} kxt} \,\mathrm{e}^{i(\psi-kt) jR}\,\mathrm{e}^{i \mu \sgn(y) \cos^{-1} t} \,\mathrm{d} t \\ & =\phi^\mathrm{In}+ \frac{\pi}{kR} f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty \frac{\mathrm{e}^{-k\gamma(\psi_j/k)|y|}}{\gamma(\psi_j/k)} \,\mathrm{e}^{\mathrm{i} x\psi_j}\,\mathrm{e}^{i \mu\sgn(y)\cos^{-1} \psi_j/k} \\ & =\phi^\mathrm{In}+ \frac{\pi i}{kR} f_0(z) \sum_{\mu=-\infty}^\infty A_{0\mu} \sum_{j=-\infty}^\infty \frac{1}{\sin\chi_j} \,\mathrm{e}^{\mathrm{i} kr\cos(|\theta|-\chi_j)}\,\mathrm{e}^{i \mu \sgn(\theta)\chi_j}, \end{matrix} }[/math]

in which we have used the Poisson summation formula,

[math]\displaystyle{ \sum_{m=-\infty}^\infty \int_{-\infty}^{\infty} f(u)\, \mathrm{e}^{-\mathrm{i} mu} \,\mathrm{d} u= 2\pi \sum_{m=-\infty}^\infty f(2m\pi). }[/math]

The only terms which contribute to the far field are those for which [math]\displaystyle{ |\psi_m|\lt k }[/math]. Thus, as [math]\displaystyle{ y\to\pm\infty }[/math], the far field consists of a set of plane waves propagating in the directions [math]\displaystyle{ \theta=\pm\chi_m }[/math]:

[math]\displaystyle{ \phi\sim \phi^\mathrm{In}+ \frac{\pi i}{kR} f_0(z) \sum_{m\in\mathcal{M}} \frac{1}{\sin\chi_m} \,\mathrm{e}^{\mathrm{i} kr\cos(\theta\mp\chi_m)} \sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}. }[/math]

From \eqref{eqn:inffar} the amplitudes of the scattered waves for each scattering angle [math]\displaystyle{ \pm \chi_m }[/math] are given in terms of the coefficients [math]\displaystyle{ A_{0\mu} }[/math] by

[math]\displaystyle{ (31) A^\pm_m = \frac{\pi i}{kR} \frac{1}{\sin\chi_m} \sum_{\mu=-\infty}^\infty A_{0\mu} \,\mathrm{e}^{\pm\mathrm{i} \mu\chi_m}. }[/math]

Note that the primary reflection and transmission coefficients are recovered by [math]\displaystyle{ A^-_0 }[/math] and [math]\displaystyle{ 1 + A^+_0 }[/math], respectively.

It is implicit in all the above that [math]\displaystyle{ \sin\chi_m\neq 0 }[/math] for any [math]\displaystyle{ m }[/math]. If [math]\displaystyle{ \sin\chi_m=0 }[/math] then we have the situation where one of the scattered plane waves propagates along the array. We will not consider this resonant case here except for stating that then, the scattered field is dominated by waves travelling along the array, either towards [math]\displaystyle{ x = \infty }[/math] (if [math]\displaystyle{ \chi_m = 0 }[/math]) or towards [math]\displaystyle{ x=-\infty }[/math] (if [math]\displaystyle{ \chi_m = \pi }[/math]). Also, we will not consider the excitation of Rayleigh-Bloch Waves, which are waves which travel along the array with a phase difference between adjacent bodies greater than [math]\displaystyle{ Rk }[/math].

The efficient computation of the [math]\displaystyle{ \sigma_{\nu}^0 }[/math]

The constants [math]\displaystyle{ \sigma_{\nu}^0 }[/math] (cf.~\eqref{eq_op_sigma}) appearing in the system of equations for the coefficients of the scattered wavefield of the bodies cannot be computed straightforwardly. This is due to the slow decay of the modified Bessel function of the second kind for large imaginary argument as was discussed in \S 14. First, note that

[math]\displaystyle{ \sigma_{\nu}^0 = \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) K_{\nu} (-\mathrm{i} k j R) = \frac{\pi i^{\nu+1}}{2} \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR), }[/math]

where we have used \eqref{H_K}. Therefore, it suffices to discuss the computation of the constants [math]\displaystyle{ \tilde{\sigma}^0_\nu }[/math] defined via

[math]\displaystyle{ \tilde{\sigma}_{\nu}^0 = \sum_{j=1}^{\infty} (P_{-j}+ (-1)^\nu P_j) H^{(1)}_\nu (kjR) }[/math]

as the [math]\displaystyle{ \sigma^0_\nu }[/math] are then determined by [math]\displaystyle{ \sigma^0_\nu = \pi/2 \,\, i^{\nu+1} \, \tilde{\sigma}^0_\nu }[/math].

An efficient way of computing the [math]\displaystyle{ \tilde{\sigma}_{\nu}^0 }[/math] is given in Linton 1998 and the results are briefly outlined in our notation. Noting that [math]\displaystyle{ H^{(1)}_{-\nu} (\,\cdot\,)= (-1)^{\nu} H^{(1)}_{\nu} (\,\cdot\,) }[/math], it suffices to discuss the computation of the [math]\displaystyle{ \sigma_{\nu}^0 }[/math] for non-negative [math]\displaystyle{ \nu }[/math].

Referring to Linton 1998, the constants [math]\displaystyle{ \tilde{\sigma}_{\nu}^0 }[/math] can be written as

[math]\displaystyle{ \tilde{\sigma}_{0}^0 = -1 -\frac{2 i}{\pi} \left( C + \log \frac{k}{2p} \right) + \frac{2}{R k \sin \chi} - \frac{2 \mathrm{i} (k^2 + 2 \psi^2)}{p^3 R} \zeta(3) + \frac{2}{R} \sum_{m=1}^\infty \left( \frac{1}{k \sin \chi_{-m}} + \frac{1}{k \sin \chi_m} + \frac{2 i}{p m} + \frac{\mathrm{i} (k^2 + 2 \psi^2)}{p^3 m^3} \right) }[/math]

where [math]\displaystyle{ C \approx 0.5772 }[/math] is Euler's constant and [math]\displaystyle{ \zeta }[/math] is the Riemann zeta function and the terms in the sum converge like [math]\displaystyle{ O(m^{-4}) }[/math] as [math]\displaystyle{ m\rightarrow\infty }[/math] (by which we mean that the error in the sum is proportional to [math]\displaystyle{ m^{-4} }[/math] for large values of [math]\displaystyle{ m }[/math]) as well as

[math]\displaystyle{ \begin{matrix} \tilde{\sigma}_{2\nu}^0 &=& 2 (-1)^{\nu} \left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi} }{R k \sin \chi} - \frac{ i}{\pi} \left( \frac{k}{2 p} \right)^{2\nu} \zeta(2\nu +1) \right) + \frac{ i}{\nu \pi} \\ & + &2 (-1)^\nu \sum_{m=1}^\infty \left( \frac{\mathrm{e}^{2\mathrm{i} \nu \chi_m}}{R k \sin \chi_{m}} + \frac{\mathrm{e}^{-2 \mathrm{i} \nu \chi_{-m}}}{R k \sin \chi_{-m}} + \frac{ i}{m\pi} \left( \frac{k}{2 m p} \right)^{2\nu} \right)\\ & &+& \frac{ i}{\pi} \sum_{m=1}^\nu \frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m)! (\nu-m)!} \left( \frac{p}{k} \right)^{2m} B_{2m}(\psi/p), \end{matrix} }[/math]
[math]\displaystyle{ \begin{matrix} \tilde{\sigma}_{2\nu-1}^0 &=& - 2 (-1)^\nu \left( \frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1) \chi}}{R k \sin \chi} - \frac{ \psi R \nu}{\pi^2} \left( \frac{k}{2 p} \right)^{2\nu-1} \zeta(2\nu +1) \right)\\ & -& 2 (-1)^\nu \sum_{m=1}^\infty \left(\frac{\mathrm{i} \mathrm{e}^{\mathrm{i} (2\nu-1)\chi_m} }{R k \sin \chi_m} + \frac{\mathrm{i} \mathrm{e}^{-\mathrm{i} (2\nu-1) \chi_{-m}}}{R k \sin \chi_{-m}} + \frac{\psi R \nu}{m^2\pi^2} \left( \frac{k}{2 m p} \right)^{2\nu-1} \right)\\ & -& \frac{2}{\pi} \sum_{m=0}^{\nu-1} \frac{(-1)^m 2^{2m} (\nu+m-1)!}{(2m+1)! (\nu-m-1)!} \left( \frac{p}{k} \right)^{2m+1} B_{2m+1}(\psi/p), \end{matrix} }[/math]

for [math]\displaystyle{ \nu\gt 0 }[/math] where [math]\displaystyle{ B_m }[/math] is the [math]\displaystyle{ m }[/math]th Bernoulli polynomial. The slowest convergence in this representation occurs in [math]\displaystyle{ \tilde{\sigma}^0_1 }[/math] and [math]\displaystyle{ \tilde{\sigma}^0_2 }[/math] in which the terms converge like [math]\displaystyle{ O(m^{-5}) }[/math] as [math]\displaystyle{ m\rightarrow\infty }[/math].

Note that since [math]\displaystyle{ \sin \chi_m }[/math] is purely imaginary for [math]\displaystyle{ m \notin \mathcal{M} }[/math], the computation of the real part of [math]\displaystyle{ \tilde{\sigma}_{2\nu}^0 }[/math] and the imaginary part of [math]\displaystyle{ \tilde{\sigma}_{2\nu-1}^0 }[/math] is particularly simple. For [math]\displaystyle{ \nu \geq 0 }[/math], they are given by

[math]\displaystyle{ \begin{matrix} \Re \tilde{\sigma}_{2\nu}^0 &= -\delta_{\nu 0} + 2(-1)^\nu \sum_{m\in\mathcal{M}} \frac{\cos 2 \nu \chi_m}{R k \sin \chi_m}, \\ \Im \tilde{\sigma}_{2\nu+1}^0 &= 2\mathrm{i} (-1)^\nu \sum_{m\in\mathcal{M}} \frac{\cos (2 \nu-1) \chi_m}{R k \sin \chi_m}, \end{matrix} }[/math]

where [math]\displaystyle{ \delta_{mn} }[/math] is the Kronecker delta.

Acoustic scattering by an infinite array of identical generalized cylinders

The theory above has so far been developed for water-wave scattering of a plane wave by an infinite array of identical arbitrary bodies. It can easily be adjusted to the (simpler) two-dimensional problem of acoustic scattering. Namely, we consider the problem that arises when a plane sound wave is incident upon an infinite array of identical generalized cylinders (i.e.~bodies which have arbitrary cross-section in the [math]\displaystyle{ (x,y) }[/math]-plane but the cross-sections at any height are identical) in an acoustic medium.

For this problem, the [math]\displaystyle{ z }[/math]-dependence can be omitted and the above theory applies with the following modifications:

  1. The Dispersion Relation for a Free Surface is replaced by [math]\displaystyle{ k=\omega / c }[/math] where [math]\displaystyle{ c }[/math] is the speed of sound in the medium under consideration

and the Dispersion Relation for a Free Surface is omitted.

  1. All factors [math]\displaystyle{ \cos k_m(z+d) }[/math], [math]\displaystyle{ \cos k_m(c+d) }[/math], [math]\displaystyle{ \cos k_m d }[/math]

and [math]\displaystyle{ f_0 }[/math] are replaced by 1.

  1. The factor [math]\displaystyle{ N_0 }[/math] in \eqref{green_d} is [math]\displaystyle{ k/\pi }[/math].

Note that there are no evanescent modes in this problem, i.e. the sums over [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] in the eigenfunction expansions only contain the terms for [math]\displaystyle{ m=0 }[/math] and [math]\displaystyle{ n=0 }[/math]. Moreover, we have [math]\displaystyle{ k_0 = - \mathrm{i} \, \omega /c }[/math].

For circular cylinders, i.e. cylinders which have a circular cross-section, this problem has been considered by Linton and Evans 1993.