Difference between revisions of "Interaction Theory for Infinite Arrays"

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=Introduction =
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{{complete pages}}
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==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]],
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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
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<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.
 
 
 
 
 
 
=Formulation of the problem=
 
 
We consider the water-wave scattering of a plane wave by an
 
infinite array of identical vertically non-overlapping bodies, denoted by
 
<math>\Delta_j</math>. The mean-centre positions <math>O_j</math> of <math>\Delta_j</math> are assumed
 
to be <math>O_j = (jR, 0)</math> where the distance between the bodies, <math>R</math>, is
 
supposed sufficiently large so that there is no intersection of the
 
smallest cylinder which contains each body, with any other body. The
 
ambient plane wave is assumed to travel in the direction
 
<math>\chi \in (0,\pi/2]</math>
 
where <math>\chi</math> is measured with respect to the <math>x</math>-axis.
 
Let <math>(r_j,\theta_j,z)</math> be the local cylindrical coordinates of the
 
<math>j</math>th body, <math>\Delta_j</math>. Note that the zeroth body is centred at the
 
origin and its local cylindrical coordinates coincide with the
 
global ones, <math>(r,\theta,z)</math>. Figure 1 illustrates the setting.
 
 
\begin{figure}
 
\begin{center}
 
\includegraphics[width=.9\columnwidth]{floes}
 
\caption{Plan view of the relation between the bodies.}(1)
 
\end{center}
 
\end{figure}
 
 
The equations of motion for the water are derived from the linearized
 
inviscid theory. Under the assumption of irrotational motion the
 
velocity-vector field of the water can be written as the gradient
 
field of a scalar velocity potential <math>\Phi</math>. Assuming that the motion
 
is time-harmonic with radian frequency <math>\omega</math> the
 
velocity potential can be expressed as the real part of a complex
 
quantity,
 
<center><math>(2)
 
\Phi(\mathbf{y},t) = \Re \{\phi (\mathbf{y}) \mathrm{e}^{-\mathrm{i} \omega t} \}.
 
</math></center>
 
To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point
 
in the water, which is assumed to be of constant 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
 
(3)
 
<center><math>\begin{matrix}
 
\nabla^2 \phi &= 0, \; & & \mathbf{y} \in D,\\ 
 
(4)
 
\frac{\partial \phi}{\partial z} &= \alpha \phi, \; & &
 
{\mathbf{x}} \in \Gamma^\mathrm{f},\\
 
(5)
 
\frac{\partial \phi}{\partial z} &= 0, \; & & \mathbf{y} \in D, \ z=-d,
 
\intertext{where <math>D = (\mathds{R}^2 \times (-d,0)) \backslash \bigcup_{j}
 
\bar{\Delta}_j</math> 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 <math>\Delta_j</math>, the water velocity potential has to
 
equal the normal velocity of the body <math>\mathbf{v}_j</math>,}
 
(6)
 
\frac{\partial \phi}{\partial n} &= \mathbf{v}_j, \; && {\mathbf{y}}
 
\in \Gamma_j.
 
\end{matrix}</math></center>
 
Moreover, the Sommerfeld radiation condition is imposed
 
<center><math>
 
\lim_{\tilde{r} \rightarrow \infty} \sqrt{\tilde{r}} \, \Big(
 
\frac{\partial}{\partial \tilde{r}} - \mathrm{i} k
 
\Big) (\phi - \phi^{\mathrm{In}}) = 0,
 
</math></center>
 
 
where <math>\tilde{r}^2=x^2+y^2</math>, <math>k</math> is the wavenumber and
 
<math>\phi^\mathrm{In}</math> is the ambient incident potential. The
 
positive wavenumber <math>k</math>
 
is related to <math>\alpha</math> by the dispersion relation
 
<center><math>(7)
 
\alpha = k \tanh k d,
 
</math></center>
 
and the values of <math>k_m</math>, <math>m>0</math>, are given as positive real roots of
 
the dispersion relation
 
<center><math>(8)
 
\alpha + k_m \tan k_m d = 0.
 
</math></center>
 
For ease of notation, we write <math>k_0 = -\mathrm{i} k</math>. Note that <math>k_0</math> is a
 
(purely imaginary) root of \eqref{eq_k_m}.
 
 
 
==Eigenfunction expansion of the potential==
 
 
The scattered potential of a body
 
<math>\Delta_j</math> can be expanded in singular cylindrical eigenfunctions,
 
<center><math>(9)
 
\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>
 
The incident potential upon body <math>\Delta_j</math> can be also be expanded in
 
regular cylindrical eigenfunctions,
 
<center><math>(10)
 
\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 Bessel functions of the first and
 
second kind, respectively, both of order <math>\nu</math>.
 
Note that in
 
\eqref{basisrep_out_d} (and \eqref{basisrep_in_d}) the term for </math>m =
 
0<math> (</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. For
 
future reference, we remark that, for real <math>x</math>,
 
<center><math>(11)
 
K_\nu (-\mathrm{i} x) = \frac{\pi \i^{\nu+1}}{2} H_\nu^{(1)}(x) \quad
 
=and=  \quad
 
I_\nu (-\mathrm{i} x) = \i^{-\nu} J_\nu(x)
 
</math></center>
 
with <math>H_\nu^{(1)}</math> and <math>J_\nu</math> denoting the Hankel function and the
 
Bessel function, respectively, both of first kind and order <math>\nu</math>. 
 
 
 
\subsection{Representation of the ambient wavefield in the eigenfunction
 
representation}(12)
 
In what follows, it is necessary to represent the ambient wavefield in
 
the eigenfunction expansion \eqref{basisrep_in_d}. A short outline of
 
how this can be accomplished is given here.
 
 
In Cartesian coordinates centred at the origin, the ambient wavefield is
 
given by
 
<center><math>
 
\phi^{\mathrm{In}}(x,y,z) = \frac{A g}{\omega}
 
 
f_0(z) \mathrm{e}^{\mathrm{i} k (x \cos \chi + y \sin \chi)},
 
</math></center>
 
where <math>A</math> is the amplitude (in displacement) and <math>\chi</math> is the
 
angle between the <math>x</math>-axis and the direction in which the wavefield
 
travels (also cf.~figure 1).
 
This expression can be written in the eigenfunction expansion
 
centred at the origin as
 
<center><math>
 
\phi^{\mathrm{In}}(r,\theta,z) = \frac{A g}{\omega}
 
 
f_0(z)
 
\sum_{\nu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i} \nu (\pi/2 - \theta + \chi)} J_\nu(k r)
 
</math></center>
 
\cite[p.~169]{linton01}. 
 
The local coordinates of each body are centred at their mean-centre
 
positions <math>O_l = (l R,0)</math>.
 
In order to represent the ambient wavefield, which is
 
incident upon all bodies, in the eigenfunction expansion of an
 
incoming wave in the local coordinates of the body, a phase factor has to be
 
defined,
 
<center><math>(13)
 
P_l = \mathrm{e}^{\mathrm{i} l R k \cos \chi},
 
</math></center>
 
which accounts for the position from the origin. Including this phase
 
factor and
 
making use of \eqref{H_K}, the ambient wavefield at the <math>l</math>th body is given by
 
<center><math>
 
\phi^{\mathrm{In}}(r_l,\theta_l,z) = \frac{A g}{\omega}  \, P_l \,
 
f_0(z) \sum_{\nu = -\infty}^{\infty}
 
\mathrm{e}^{\mathrm{i} \nu (\pi -  \chi)} I_\nu(k_0 r_l) \mathrm{e}^{\mathrm{i} \nu \theta_l}.
 
</math></center>
 
We can therefore define the coefficients of the ambient wavefield in
 
the eigenfunction expansion of an incident wave,
 
<center><math>
 
\tilde{D}^l_{n\nu} =
 
\begin{cases}
 
\frac{A g}{\omega}  P_l \mathrm{e}^{\mathrm{i} \nu (\pi - \chi)}, & n=0,\\
 
0, & n > 0.
 
\end{cases}
 
</math></center>
 
Note that the evanescent coefficients are all zero due to the
 
propagating nature of the ambient wave.
 
 
 
=Derivation of the system of equations=(14)
 
Following the ideas of general interaction theories
 
\cite[]{kagemoto86,JFM04}, a system of equations for the unknown
 
coefficients (in the expansion \eqref{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>. From figure
 
1 we can see that this can be accomplished by using
 
Graf's addition theorem for Bessel functions given in
 
\citet[eq. 9.1.79]{abr_ste},
 
<center><math>(15)
 
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 \abs{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></center>
 
which is valid provided that <math>r_l < R</math>. The angles <math>\varphi_{n}</math>
 
account for the difference in direction depending if the <math>j</math>th body is
 
located to the left or to the right of the <math>l</math>th body and are
 
defined by
 
<center><math>
 
\varphi_n =
 
\begin{cases}
 
\pi, & n > 0,\\
 
0, & n < 0.
 
\end{cases}
 
</math></center>
 
The limitation <math>r_l < R</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
 
\eqref{transf}, the scattered potential
 
of <math>\Delta_j</math> (cf.~\eqref{basisrep_out_d}) can be expressed in terms of the
 
incident potential upon <math>\Delta_l</math> as
 
<center><math>\begin{matrix}
 
\phi_j^{\mathrm{S}} & (r_l,\theta_l,z) \\
 
&= \sum_{m=0}^\infty f_m(z) \sum_{\tau = -
 
\infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty}
 
(-1)^\nu K_{\tau-\nu} (k_m \abs{j-l} R) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i} \nu
 
\theta_l} \mathrm{e}^{\mathrm{i} (\tau-\nu) \varphi_{j-l}} \\
 
&= \sum_{m=0}^\infty f_m(z) \sum_{\nu =
 
-\infty}^{\infty}  \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j
 
(-1)^\nu K_{\tau-\nu} (k_m \abs{j-l} R) \mathrm{e}^{\mathrm{i} (\tau - \nu)
 
\varphi_{j-l}} \Big] I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i} \nu \theta_l}. 
 
\end{matrix}</math></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.~\S 12). The total
 
incident wavefield upon body <math>\Delta_j</math> can now be expressed as
 
<center><math>\begin{matrix}
 
\phi_l^{\mathrm{I}}(r_l,\theta_l,z) &= \phi^{\mathrm{In}}(r_l,\theta_l,z) +
 
\sum_{\genfrac{}{}{0pt}{}{j=-\infty}{j \neq l}}^{\infty} \, \phi_j^{\mathrm{S}}
 
(r_l,\theta_l,z)\\
 
&= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
 
\Big[  \tilde{D}_{n\nu}^{l} +
 
\sum_{\genfrac{}{}{0pt}{}{j=-\infty}{j \neq  l}}^{\infty} \sum_{\tau =
 
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
 
\abs{j-l}R)  \mathrm{e}^{\mathrm{i} (\tau - \nu) \varphi_{j-l}} \Big]\\
 
&\quad  \times I_\nu (k_n
 
r_l) \mathrm{e}^{\mathrm{i} \nu \theta_l}.
 
\end{matrix}</math></center>
 
The coefficients of the total incident potential upon <math>\Delta_l</math> are
 
therefore given by
 
<center><math>(16)
 
D_{n\nu}^l = \tilde{D}_{n\nu}^{l} +
 
\sum_{\genfrac{}{}{0pt}{}{j=-\infty}{j \neq  l}}^{\infty} \sum_{\tau =
 
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
 
\abs{j-l} R)  \mathrm{e}^{\mathrm{i} (\tau - \nu) \varphi_{j-l}}.
 
</math></center>
 
 
In general, it is possible to relate the total incident and scattered
 
partial waves for any body through the diffraction characteristics of
 
that body in isolation. There exist diffraction transfer operators
 
<math>B^l</math> that relate the coefficients of the incident and scattered
 
partial waves, such that
 
<center><math>
 
A^l = B^l (D^l), \quad l\in \mathds{Z},
 
</math></center>
 
where <math>A^l</math> are the scattered modes due to the incident modes
 
<math>D^l</math>. Note that since it is assumed that all bodies are identical in
 
this setting, only one diffraction transfer operator, <math>B</math>, is required.
 
In the case of a countable number of modes (i.e.~when
 
the water depth is finite), <math>B</math> is an infinite dimensional matrix. When
 
the modes are functions of a continuous variable (i.e.~infinite
 
depth), <math>B</math> is the kernel of an integral operator.
 
The scattered and incident potential can therefore be related by a
 
diffraction transfer operator acting in the following way,
 
<center><math>(18)
 
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
 
\mu \nu} D_{n\nu}^l.
 
</math></center>
 
If the diffraction transfer operator is known (its calculation
 
is discussed later), the substitution of
 
\eqref{inc_coeff} into \eqref{diff_op} gives the
 
required equations to determine the coefficients of the scattered
 
wavefields of all bodies,
 
<center><math>(19)
 
A_{m\mu}^l = \sum_{n=0}^{\infty}
 
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
 
\Big[ \tilde{D}_{n\nu}^{l} +
 
\sum_{\genfrac{}{}{0pt}{}{j=-\infty}{j \neq  l}}^{\infty} \sum_{\tau =
 
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
 
\abs{j-l} R) \mathrm{e}^{\mathrm{i} (\tau -\nu) \varphi_{j-l}} \Big],
 
</math></center>
 
<math>m \in \mathds{N}</math>, <math>l,\mu \in \mathds{Z}</math>.
 
 
 
 
 
Due to the periodicity of the geometry and of the incident wave, the
 
coefficients <math>A_{m\mu}^l</math> can be written as </math>A_{m\mu}^l = P_l
 
A_{m\mu}^0 = P_l A_{m\mu}<math>, say. The same can be done for the coefficients
 
of the incident ambient wave, i.e.~</math>\tilde{D}_{n\nu}^{l} = P_l
 
\tilde{D}_{n\nu}<math> (also cf.~\S
 
12). Noting that <math>P_l^{-1} = P_{-l}</math> and </math>P_j P_l =
 
P_{j+l}<math>, \eqref{eq_op} simplifies to 
 
<center><math>
 
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_{\genfrac{}{}{0pt}{}{j=-\infty}{j \neq
 
l}}^{\infty} P_{j-l}  K_{\tau - \nu}  (k_n \abs{j-l} R)  \mathrm{e}^{\i
 
(\tau - \nu) \varphi_{j-l}} \Big].
 
</math></center>
 
Introducing the constants
 
<center><math>(21)
 
 
 
 
 
 
\sigma_{\nu}^n = \sum_{\genfrac{}{}{0pt}{}{j=-\infty}{j \neq
 
l}}^{\infty}  P_{j-l}  K_{\nu} (k_n \abs{j-l} R) \mathrm{e}^{\i
 
\nu \varphi_{j-l}} =  \sum_{j=1}^{\infty}  (P_{-j}+ (-1)^\nu P_j)
 
K_{\nu} (k_n j R),
 
</math></center>
 
which can be evaluated separately since they do not contain any
 
unknowns, the problem reduces to
 
<center><math>(22)
 
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_{\tau - \nu}^n \Big].
 
</math></center>
 
The efficient computation of the constants <math>\sigma_{\nu}^0</math> is not
 
trivial as the sum in \eqref{sigma} is not absolutely convergent
 
due to the slow decay of the modified Bessel function of the second
 
kind for large imaginary argument (The terms in the sum
 
decay like <math>j^{-1/2} \mathrm{e}^{\mathrm{i} j \theta} </math> for some
 
<math>\theta</math>). Appropriate methods for the computation of the
 
<math>\sigma_{\nu}^0</math> are outlined in 
 
\S 36. The calculation of the constants <math>\sigma_{\nu}^n</math>,
 
<math>n\neq 0</math>, is easy, however, since the modified Bessel function of the
 
second kind decays exponentially for large real argument.
 
 
For numerical calculations, the infinite sums in \eqref{eq_op_sigma}
 
have to be truncated. Implying a suitable truncation, the
 
diffraction transfer operator can be represented by a
 
matrix <math>\mathbf{B}</math>,
 
the finite-depth diffraction transfer matrix.
 
Truncating the coefficients accordingly, defining <math>{\mathbf a}</math> to be the
 
vector of the coefficients of the scattered potential,
 
<math>\mathbf{d}^{\mathrm{In}}</math> to be the vector of
 
coefficients of the ambient wavefield, and making use of the matrix
 
<math>{\mathbf T}</math> given by
 
<center><math>
 
({\mathbf T})_{pq} = (-1)^{q} \, \sigma_{p-q}^n,
 
</math></center>
 
a linear system of equations
 
for the unknown coefficients follows from equations \eqref{eq_op_sigma},
 
<center><math>(24)
 
(\mathbf{Id} - \mathbf{B} \, \trans {\mathbf T}) {\mathbf a} = \mathbf{B} \,
 
{\mathbf d}^{\mathrm{In}}, 
 
</math></center>
 
where the left superscript <math>\mathrm{t}</math> indicates transposition and
 
<math>\mathbf{Id}</math> is the identity matrix of the same dimension as <math>\mathbf{B}</math>.
 
 
 
  
 
=The far field=
 
=The far field=
 
In this section, the far field is examined which describes the
 
In this section, the far field is examined which describes the
 
scattering far away from the array. The derivation is equivalent to that of
 
scattering far away from the array. The derivation is equivalent to that of
[[twersky62]]. First, we define the scattering angles
+
[[Twersky 1962]]. First, we define the scattering angles
 
which give the directions of propagation of plane scattered waves
 
which give the directions of propagation of plane scattered waves
 
far away from the array.   
 
far away from the array.   
 
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 = \arccos (\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 484: 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 504: 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 515: 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 525: 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 535: 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 557: 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 581: 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.
 
 
 
 
\section{Calculation of the diffraction transfer matrix for bodies
 
of arbitrary geometry}
 
  
Before we can solve the systems of equations for the coefficients in
+
==The efficient computation of the <math>\sigma_{\nu}^0</math> ==
the eigenfunction expansion of the scattered wavefield
 
\eqref{eq_tosolve}, we require the
 
diffraction transfer matrix <math>\mathbf{B}</math> which relates the incident and the
 
scattered potential for a body <math>\Delta</math> in isolation.
 
The elements of the diffraction transfer matrix, <math>({\mathbf B})_{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</math>.
 
 
 
An explicit method to calculate the diffraction transfer matrices
 
for bodies of arbitrary geometry in the case of finite depth is given by
 
[[goo90]]. We briefly recall their results in our notation.
 
Utilizing a Green's function the single diffraction boundary-value problem
 
can be transformed to an integral equation for the
 
source-strength-distribution function over the immersed surface of the
 
body. To obtain the potential in the cylindrical eigenfunction expansion,
 
the free-surface finite-depth Green's function given by [[black75]] and
 
[[fenton78]],
 
<center><math>(32)
 
G (r,\theta,z;s,\vartheta,c) = \sum_{m=0}^{\infty} N_m \, \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 - \vartheta)},
 
</math></center>
 
is then used allowing 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. The constants <math>N_m</math> are given by
 
<center><math>
 
N_m = \frac{1}{\pi} \frac{k_m^2+\alpha^2}{d(k_m^2+\alpha^2)-\alpha} =
 
\frac{1}{\pi} \left({d+ \frac{\sin 2 k_m d}{2 k_m}}\right)^{-1}
 
</math></center>
 
where the latter representation is often more favourable in numerical
 
calculations.
 
 
 
We assume that we have represented the scattered potential in terms of
 
the source-strength distribution <math>\varsigma</math> so that the scattered
 
potential can be written as
 
<center><math>(33)
 
\phi^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma} G
 
(\mathbf{y},\mathbf{\zeta}) \, \varsigma (\mathbf{\zeta})
 
\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</math> is the
 
immersed surface of body <math>\Delta</math>. The source-strength-distribution
 
function <math>\varsigma</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]].
 
Substituting the eigenfunction expansion of the Green's function
 
\eqref{green_d} into \eqref{int_eq_1}, the scattered potential can
 
be written as
 
\begin{multline*}
 
\phi^\mathrm{S}(r,\theta,z) =
 
\sum_{m=0}^{\infty} f_m(z)  \sum_{\nu = -
 
\infty}^{\infty} \bigg[ N_m \cos^2 k_md 
 
\int\limits_{\Gamma} \cos k_m(c+d) I_\nu(k_m s)
 
\mathrm{e}^{-\mathrm{i} \nu \vartheta} \varsigma({\mathbf{\zeta}})
 
\d\sigma_{\mathbf{\zeta}} \bigg]\\ \times K_\nu (k_m r) \mathrm{e}^{\mathrm{i} \nu \theta} \d\eta,
 
\end{multline*}
 
where
 
<math>\mathbf{\zeta}=(s,\vartheta,c)</math> and <math>r>s</math>.
 
This restriction implies that the eigenfunction expansion is only valid
 
outside the escribed cylinder of the body.
 
 
 
The columns of the diffraction transfer matrix are the coefficients of
 
the eigenfunction expansion of the scattered wavefield due to the
 
different incident modes of unit-amplitude. The elements of the
 
diffraction transfer matrix of a body of arbitrary shape are therefore given by
 
<center><math>(34)
 
({\mathbf B})_{pq} = N_m \cos^2 k_md
 
\int\limits_{\Gamma} \cos k_m(c+d) I_p(k_m s) \mathrm{e}^{-\mathrm{i} p
 
\vartheta} \varsigma_q(\mathbf{\zeta}) \d\sigma_\mathbf{\zeta}
 
</math></center>
 
where
 
<math>\varsigma_q(\mathbf{\zeta})</math> is the source strength distribution
 
due to an incident potential of mode <math>q</math> of the form
 
<center><math>(35)
 
\phi_q^{\mathrm{I}}(s,\vartheta,c) = f_m(c) I_q
 
(k_m s) \mathrm{e}^{\mathrm{i} q \vartheta}.
 
</math></center>
 
 
 
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 [[kashiwagi00a]]. 
 
 
 
 
 
=The efficient computation of the <math>\sigma_{\nu=^0</math>}(36)
 
  
 
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 693: 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 703: 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 733: 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 768: 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 782: 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 800: 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.