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\title[short title of paper] %% give here short title %%
{An interesting and seminal work on various phenomena in fluid mechanics}

\author[short author list]   %% give here short author list %%
{Alan N. Other$^1$%
  \thanks{Present address: Fluid Mech Inc., 24 The Street, Lagos, Nigeria.},
 H.-C. Smith$^1$ \break \and J.Q. Public$^2$}

\affiliation{$^1$Department of Chemical Engineering, University of America,
Somewhere, IN 12345, USA \break email: ...........\\[\affilskip]
$^2$Department of Aerospace and Mechanical Engineering, University of
Camford, \break Academic Street, Camford, CF3 5QL, UK \break email: ...........}

\pubyear{2009}
\volume{Volume 15}  %% insert here IAU Highlights of Astronomy Volume Number
\pagerange{119--126}
\date{?? and in revised form ??}
\setcounter{page}{119}
\jname{Highlights of Astronomy, Volume 14}
\editors{Ian F Corbett, ed.}
\begin{document}

\maketitle

\begin{abstract}
Using Stokes flow between eccentric counter-rotating cylinders as a prototype
for bounded nearly parallel lubrication flow, we investigate the effect of a
slender recirculation region within the flow field on cross-stream heat or mass
transport in the important limit of high P\'{e}clet number \Pen\ where the
`enhancement' over pure conduction heat transfer without recirculation is most
pronounced.  The steady enhancement is estimated with a matched asymptotic
expansion to resolve the diffusive boundary layers at the separatrices which
bound the recirculation region.  The enhancement over pure conduction is
shown to vary as $\epsilon^{1/2}$ at infinite \Pen, where $\epsilon^{1/2}$
is the characteristic width of the recirculation region. The enhancement decays
from this asymptote as $\Pen^{-1/2}$.
\keywords{Keyword1, keyword2, keyword3, etc.}
%% add here a maximum of 10 keywords, to be taken form the file <Keywords.txt>
\end{abstract}

\firstsection % if your document starts with a section,
              % remove some space above using this command.
\section{Introduction}

The use of integral equations to solve `exterior' problems in linear acoustics,
i.e.\ to solve the Helmholtz equation $(\nabla^2+k^2)\phi=0$ outside
a surface $S$ given that $\phi$ satisfies certain boundary conditions
on $S$, is very common. A good description is provided by \cite{Martin80}.
Integral equations have also been used to solve
the two-dimensional Helmholtz equation that arises in water-wave problems
where there is a constant depth variation.  The problem of wave
oscillations in arbitrarily shaped harbours using such techniques has
been examined (see for example \cite[Hwang \& Tuck 1970]{Hwang70};
\cite[Lee 1971]{Lee71}; \cite{Figer02}).

In a recent paper \cite{Linton92} have shown how radiation
and scattering problems for vertical circular cylinders placed on
the centreline of a channel of finite water depth can be solved
efficiently using the multipole method devised originally by
\cite{Ursell50}. This method was also used by \cite{Callan91}
to prove the existence of trapped modes in the vicinity of such a
cylinder at a discrete wavenumber $k<\upi/2d$ where $2d$ is the channel
width.

% NOTE use of \upi in above paragraph and subsequently throughout paper.
% The Greek constant character pi should be upright.

Many water-wave/body interaction problems in which the body is a
vertical cylinder with constant cross-section can be simplified by
factoring out the depth dependence. Thus if the boundary conditions
are homogeneous we can write the velocity potential
$\phi(x,y,z,t) =
 \hbox{Re}\{\phi(x,y)\cosh k(z+h)\mathrm{e}^{-\mathrm{i}\omega t}\}$,
where the $(x,y)$-plane corresponds to the undisturbed free surface and
$z$ is measured vertically upwards with $z=-h$ the bottom of the channel.

Subsequently \cite[Callan \etal\ (1991)]{Callan91} proved the existence of,
and computed the wavenumbers for, the circular cross-section case.
It should be noted however that experimental evidence for acoustic
resonances in the case of the circular cylinder is given by
\cite[Bearman \& Graham (1980, pp.~231--232)]{Bearman80}.

\mbox{}\cite{Koch83} provided a theory for determining the trapped-mode
frequencies for the thin plate, based on a modification of the Wiener--Hopf
technique. Further interesting results can be found in \cite{Williams64}
and \cite{Dennis85}.

The use of channel Green's functions allows the far-field behaviour to be computed in an extremely simple manner,
whilst the integral equation constructed in~\S\,\ref{sec:trapmode} enables
the trapped modes to be computed in~\S\,\ref{sec:torustrans} and the
scattering of an incident plane wave to be solved in~\S\,\ref{sec:concl}.
Appendix~\ref{ap:boundcon} contains comparisons with experiments. The Galactic Center region proves to be an area very rich in WR stars. The
VIIth Catalogue lists within 50 pc from the Galactic Center 15 WNL and 11
WCL stars, at near-IR wavelengths discovered by \cite{Krabbe95}
in the Galactic Center Cluster.


\section{Green's functions}\label{sec:greenfun}

\subsection{Construction of equations}

We are concerned with problems for which the solution, $\phi$, is
either symmetric or antisymmetric about the centreline of the
waveguide, $y=0$. The first step is the construction of a
symmetric and an antisymmetric Green's function, $\GsPQ$ and $\GaPQ$.
Thus we require
\begin{equation}
  (\nabla^2+k^2)G_s=(\nabla^2+k^2)G_a=0
  \label{Helm}
\end{equation}
in the fluid, where $\bnabla$ is a gradient operator,
\[
  \bnabla\bcdot\boldsymbol{v} = 0,\quad \nabla^{2}P=
    \bnabla\bcdot(\boldsymbol{v}\times \boldsymbol{w}).
\]
In (\ref{Helm})
\begin{equation}
  G_s,G_a\sim 1 / (2\upi)\ln r
  \quad \mbox{as\ }\quad r\equiv|P-Q|\rightarrow 0,
  \label{singular}
\end{equation}
\begin{equation}
  \frac{\p G_s}{\p y}=\frac{\p G_a}{\p y}=0
  \quad \mbox{on\ }\quad y=d,
  \label{wallbc}
\end{equation}
\begin{equation}
  \frac{\p G_s}{\p y}=0
  \quad \mbox{on\ }\quad y=0,
  \label{symbc}
\end{equation}
\begin{equation}
  G_a=0
  \quad \mbox{on\ }\quad y=0,
  \label{asymbc}
\end{equation}
and we require $G_s$ and $G_a$ to behave like outgoing waves as
$|x|\tti$.

One way of constructing $G_s$ or $G_a$ is to replace (\ref{Helm})
and (\ref{singular}) by
\begin{equation}
  (\nabla^2+k^2)G_s=(\nabla^2+k^2)G_a
  = -\delta(x-\xi)\delta(y-\eta)
\end{equation}
and to assume initially that $k$ has a positive imaginary part.


\subsection{Further developments}

Using results from \cite{Linton92} we see that this has the integral
representation
\begin{equation}
  -\frac{1}{2\upi} \int_0^{\infty} \gamma^{-1}[\mathrm{e}^{-k\gamma|y-\eta|}
   + \mathrm{e}^{-k\gamma(2d-y-\eta)}] \cos k(x-\xi)t\:\mathrm{d} t,
   \qquad 0<y,\quad \eta<d,
\end{equation}
where
\[
  \gamma(t) = \left\{
    \begin{array}{ll}
      -\mathrm{i}(1-t^2)^{1/2}, & t\le 1 \\[2pt]
      (t^2-1)^{1/2},         & t>1.
    \end{array} \right.
\]
In order to satisfy (\ref{symbc}) we add to this the function
\[
  -\frac{1}{2\upi}
   \pvi B(t)\frac{\cosh k\gamma(d-y)}{\gamma\sinh k\gamma d}
   \cos k(x-\xi)t\:\mathrm{d} t
\]
which satisfies (\ref{Helm}), (\ref{wallbc}) and obtain
\begin{equation}
  B(t) = 2\mathrm{e}^{-\kgd}\cosh k\gamma(d-\eta).
\end{equation}
Thus the function
\begin{equation}
  G = -\squart\mathrm{i} (H_0(kr)+H_0(kr_1))
    - \frac{1}{\upi} \pvi\frac{\mathrm{e}^{-\kgd}}%
    {\gamma\sinh\kgd} \cosh k\gamma(d-y) \cosh k\gamma(d-\eta)
\end{equation}
satisfies (\ref{Helm})--(\ref{symbc}). By writing this function
as a single integral which is even in $\gamma$, it follows that
$G$ is real. Similar ideas have been developed in a variety of ways
(\cite[Keller 1977]{Keller77}; \cite[Rogallo 1981]{Rogallo81};
\cite[van Wijngaarden 1968]{Wijngaarden68}).


\section{The trapped-mode problem}\label{sec:trapmode}

The unit normal from $D$ to $\p D$ is $\boldsymbol{n}_q=(-y^{\prime}(\theta),
x^{\prime}(\theta))/w(\theta)$.
Now $G_a=\squart Y_0(kr)+\Gat$ where
$r=\{[x(\theta)-x(\psi)]^2 + [y(\theta)-y(\psi)]^2\}^{1/2}$ and $\Gat$ is
regular as $kr\ttz$. In order to evaluate
$\p G_a(\theta,\theta)/\p n_q$ we note that
\[
  \ndq\left(\squart Y_0(kr)\right)\sim\ndq\left(\frac{1}{2\upi}
  \ln r\right)=\frac{1}{2\upi r^2 w(\theta)}[x^{\prime}(\theta)
  (y(\theta)-y(\psi))-y^{\prime}(\theta)(x(\theta)-x(\psi))]
\]
as $kr\ttz$.  Expanding $x(\psi)$ and $y(\psi)$ about the point
$\psi=\theta$ then shows that
\begin{eqnarray}
  \ndq\left(\squart Y_0(kr)\right) & \sim &
    \frac{1}{4\upi w^3(\theta)}
    [x^{\prime\prime}(\theta)y^{\prime}(\theta)-
    y^{\prime\prime}(\theta)x^{\prime}(\theta)] \nonumber\\
  & = & \frac{1}{4\upi w^3(\theta)}
    [\rho^{\prime}(\theta)\rho^{\prime\prime}(\theta)
    - \rho^2(\theta)-2\rho^{\prime 2}(\theta)]
    \quad \mbox{as\ }\quad kr\ttz . \label{inteqpt}
\end{eqnarray}


\subsection{Computation}

For computational purposes we discretize (\ref{inteqpt}) by dividing
the interval ($0,\upi$) into $M$ segments. Thus we write
\begin{equation}
  \thalf\phi(\psi) = \frac{\upi}{M} \sumjm\phi(\theta_j)\ndq
  G_a(\psi,\theta_j)\,w(\theta_j),
  \qquad 0<\psi<\upi,
  \label{phipsi}
\end{equation}
where $\theta_j=(j-\thalf)\upi/M$. Collocating at $\psi=\theta_i$ and
writing $\phi_i=\phi(\theta_i)$ \etc gives
\begin{equation}
  \thalf\phi_i = \frac{\upi}{M} \sumjm\phi_j K_{ij}^a w_j,
  \qquad i=1,\,\ldots,\,M,
\end{equation}
where
\begin{equation}
  K_{ij}^a = \left\{
    \begin{array}{ll}
      \p G_a(\theta_i,\theta_j)/\p n_q, & i\neq j \\[2pt]
      \p\Gat(\theta_i,\theta_i)/\p n_q
      + [\rho_i^{\prime}\rho_i^{\prime\prime}-\rho_i^2-2\rho_i^{\prime 2}]
      / 4\upi w_i^3, & i=j.
  \end{array} \right.
\end{equation}
For a trapped mode, therefore, we require the determinant of the
$M\times M$ matrix whose elements are
\[
  \delta_{ij}-\frac{2\upi}{M}K_{ij}^a w_j,
\]
to be zero.
%
\begin{table}\def~{\hphantom{0}}
  \begin{center}
  \caption{Values of $kd$ at which trapped modes occur when $\rho(\theta)=a$}
  \label{tab:kd}
  \begin{tabular}{lccc}\hline
      $a/d$  & $M=4$   &   $M=8$ & Callan \etal \\\hline%\\%[3pt]
       0.1~~   & 1.56905 & ~~1.56~ & 1.56904\\
       0.3~~   & 1.50484 & ~~1.504 & 1.50484\\
       0.55~   & 1.39128 & ~~1.391 & 1.39131\\
       0.7~~   & 1.32281 & ~10.322 & 1.32288\\
       0.913   & 1.34479 & 100.351 & 1.35185\\\hline
  \end{tabular}
 \end{center}
\end{table}
%
Table~\ref{tab:kd} shows a comparison of results
obtained from this method using two different truncation parameters
with accurate values obtained using the method of
\cite[Callan \etal\ (1991)]{Callan91}.

\begin{figure}
 \includegraphics{fig01.eps}
  \caption{Trapped-mode wavenumbers, $kd$, plotted against $a/d$ for
    three ellipses:\protect\\
    \hbox{{--}\,{--}\,{--}\,{--}\,{--}}, $b/a=0.5$; ---$\!$---,
    $b/a=1$; ---\,$\cdot$\,---, $b/a=1.5$.}\label{fig:wave}
\end{figure}
%
An example of the results that are obtained from our method is given
in figure~\ref{fig:wave}.
Figure~\ref{fig:contour}\,(\textit{a},\textit{b}) shows shaded contour plots of
$\phi$ for these modes, normalized so that the maximum value of $\phi$ on the
body is 1. Symmetric (figure~\ref{fig:contour}\textit{a}) modes are shown, while
the antisymmetric ones appear in figure~\ref{fig:contour}(\textit{b}).


\subsection{Basic properties}

Let
\refstepcounter{equation}
$$
  \rho_l = \lim_{\zeta \rightarrow Z^-_l(x)} \rho(x,\zeta), \quad
  \rho_{u} = \lim_{\zeta \rightarrow Z^{+}_u(x)} \rho(x,\zeta)
  \eqno{(\theequation{\mathit{a},\mathit{b}})}\label{eq35}
$$
be the fluid densities immediately below and above the cat's-eyes.  Finally
let $\rho_0$ and $N_0$ be the constant values of the density and the vorticity
inside the cat's-eyes, so that
\begin{equation}
  (\rho(x,\zeta),\phi_{\zeta\zeta}(x,\zeta))=(\rho_0,N_0)
  \quad \mbox{for}\quad Z_l(x) < \zeta < Z_u(x).
\end{equation}

The Reynolds number \Rey\ is defined by $u_\tau H/\nu$ ($\nu$ is the kinematic
viscosity), the length given in wall units is denoted by $(\;)_+$, and the
Prandtl number \Pran\ is set equal to 0.7. In (\ref{Helm}) and (\ref{singular}),
$\tau_{ij}$ and $\tau^\theta_j$ are
\begin{subeqnarray}
  \tau_{ij} & = &
    (\overline{\overline{u}_i \overline{u}_j}
    - \overline{u}_i\overline{u}_j)
    + (\overline{\overline{u}_iu^{SGS}_j
    + u^{SGS}_i\overline{u}_j})
    + \overline{u^{SGS}_iu^{SGS}_j},\\[3pt]
  \tau^\theta_j & = &
    (\overline{\overline{u}_j\overline{\theta}}
    - \overline{u}_j \overline{\theta})
    + (\overline{\overline{u}_j\theta^{SGS}
    + u^{SGS}_j \overline{\theta}})
    + \overline{u^{SGS}_j\theta^{SGS}}.
\end{subeqnarray}

\begin{figure}
%\includegraphics[height=2in,width=3in,angle=10]{fig02.eps}
\includegraphics{fig02.eps}
  \caption{Shaded contour plots of the potential $\phi$ for the two trapped
    modes that exist for an ellipse with $a/d=1.5$, $b/d=0.75$.
    (\textit{a}) Symmetric about $x=0$, $kd=0.96$;
    (\textit{b}) antisymmetric about $x=0$, $kd=1.398$.}\label{fig:contour}
\end{figure}

\subsubsection{Calculation of the terms}

The first terms in the right-hand side of (\ref{eq35}\textit{a}) and
(\ref{eq35}\textit{b}) are the Leonard terms explicitly calculated by
applying the Gaussian filter in the $x$- and $z$-directions in the
Fourier space.

The interface boundary conditions given by (\ref{Helm}) and (\ref{singular}),
which relate the displacement and stress state of the wall at the mean
interface to the disturbance quantities of the flow, can also be reformulated
in terms of the transformed quantities.  The transformed boundary conditions
are summarized below in a matrix form that is convenient for the subsequent
development of the theory:
%
% NOTE use of \slsQ below for the matrix Q. It has been defined in the
% preamble above to typeset the slanting sans serif Q that is available in
% Computer Modern. It should be the bold slanting sans serif and would be
% substituted by the printer.
%
\begin{equation}
\setlength{\arraycolsep}{0pt}
\renewcommand{\arraystretch}{1.3}
\slsQ_C = \left[
\begin{array}{ccccc}
  -\omega^{-2}V'_w  &  -(\alpha^t\omega)^{-1}  &  0  &  0  &  0  \\
  \displaystyle
  \frac{\beta}{\alpha\omega^2}V'_w  &  0  &  0  &  0  &  \mathrm{i}\omega^{-1} \\
  \mathrm{i}\omega^{-1}  &  0  &  0  &  0  &  0  \\
  \displaystyle
  \mathrm{i} R^{-1}_{\delta}(\alpha^t+\omega^{-1}V''_w)  &  0
    & -(\mathrm{i}\alpha^tR_\delta)^{-1}  &  0  &  0  \\
  \displaystyle
  \frac{\mathrm{i}\beta}{\alpha\omega}R^{-1}_\delta V''_w  &  0  &  0
    &  0  & 0 \\
  (\mathrm{i}\alpha^t)^{-1}V'_w  &  (3R^{-1}_{\delta}+c^t(\mathrm{i}\alpha^t)^{-1})
    &  0  &  -(\alpha^t)^{-2}R^{-1}_{\delta}  &  0  \\
\end{array}  \right] .
\label{defQc}
\end{equation}
$\biS^t$ is termed the displacement-stress vector and $\slsQ_C$
the flow--wall coupling matrix.  Subscript $w$ in (\ref{defQc}) denotes
evaluation of the terms at the mean interface.  It is noted that $V''_w = 0$
for the Blasius mean flow.


\subsubsection{Wave propagation in anisotropic compliant layers}

From (\ref{Helm}), the fundamental wave solutions to (\ref{inteqpt}) and
(\ref{phipsi}) for a uniformly thick homogeneous layer in the transformed
variables has the form of
\begin{equation}
\etb^t = \skew2\hat{\etb}^t \exp [\mathrm{i} (\alpha^tx^t_1-\omega t)],
\end{equation}
where $\skew2\hat{\etb}^t=\boldsymbol{b}\exp (\mathrm{i}\gamma x^t_3)$. For a
non-trivial wave, the substitution of (1.7) into (1.3) and
(1.4) yields the following determinantal equation:
\begin{equation}
\mbox{Det}[\rho\omega^2\delta_{ps}-C^t_{pqrs}k^t_qk^t_r]=0,
\end{equation}
where the wavenumbers $\langle k^t_1,k^t_2,k^t_3\rangle = \langle
\alpha^t,0,\gamma\rangle $ and $\delta_{ps}$ is the Kronecker delta.


\section{Torus translating along an axis of symmetry}\label{sec:torustrans}

Consider a torus with axes $a,b$ (see figure~\ref{fig:contour}), moving along
the $z$-axis. Symmetry considerations imply the following form for the stress
function, given in body coordinates:
\begin{equation}
\boldsymbol{f}(\theta,\psi) = (g(\psi)\cos \theta,g(\psi) \sin \theta,f(\psi)).
\label{eq41}
\end{equation}

Because of symmetry, one can integrate analytically in the $\theta$-direction
obtaining a pair of equations for the coefficients $f$, $g$ in (\ref{eq41}),
\begin{eqnarray}
f(\psi_1) = \frac{3b}{\upi[2(a+b \cos \psi_1)]^{{3}/{2}}}
  \int^{2\upi}_0 \frac{(\sin \psi_1 - \sin \psi)(a+b \cos \psi)^{1/2}}%
  {[1 - \cos (\psi_1 - \psi)](2+\alpha)^{1/2}},
\label{eq42}
\end{eqnarray}
\begin{eqnarray}
g(\psi_1) & = & \frac{3}{\upi[2(a+b \cos \psi_1)]^{{3}/{2}}}
  \int^{2\upi}_0 \left(\frac{a+b \cos \psi}{2+\alpha}\right)^{1/2}
  \left\{ \astrut f(\psi)[(\cos \psi_1 - b \beta_1)S + \beta_1P]
  \right. \nonumber\\
&& \mbox{}\times \frac{\sin \psi_1 - \sin \psi}{1-\cos(\psi_1 - \psi)}
  + g(\psi) \left[\left(2+\alpha - \frac{(\sin \psi_1 - \sin \psi)^2}
  {1- \cos (\psi - \psi_1)} - b^2 \gamma \right) S \right.\nonumber\\
&& \left.\left.\mbox{} + \left( b^2 \cos \psi_1\gamma -
  \frac{a}{b}\alpha \right) F(\thalf\upi, \delta) - (2+\alpha)
  \cos\psi_1 E(\thalf\upi, \delta)\right] \astrut\right\} \mathrm{d} \psi,
\label{eq43}
\end{eqnarray}
\begin{equation}
\alpha = \alpha(\psi,\psi_1) = \frac{b^2[1-\cos(\psi-\psi_1)]}%
  {(a+b\cos\psi) (a+b\cos\psi_1)};
  \quad
  \beta - \beta(\psi,\psi_1) = \frac{1-\cos(\psi-\psi_1)}{a+b\cos\psi}.
\end{equation}


\section{Conclusions}\label{sec:concl}

We have shown how integral equations can be used to solve a particular class
of problems concerning obstacles in waveguides, namely the Neumann problem
for bodies symmetric about the centreline of a channel, and two such problems
were considered in detail.


\appendix
\section{Boundary conditions}\label{ap:boundcon}

It is convenient for numerical purposes to change the independent variable in
(\ref{eq41}) to $z=y/ \tilde{v}^{{1}/{2}}_T$ and to introduce the
dependent variable $H(z) = (f- \tilde{y})/ \tilde{v}^{{1}/{2}}_T$.
Equation (\ref{eq41}) then becomes
\begin{equation}
  (1 - \beta)(H+z)H'' - (2+ H') H' = H'''.
\end{equation}

Boundary conditions to (\ref{eq43}) follow from (\ref{eq42}) and the
definition of \textit{H}:
\begin{equation}
\left. \begin{array}{l}
\displaystyle
H(0) = \frac{\epsilon \overline{C}_v}{\tilde{v}^{{1}/{2}}_T
(1- \beta)};\quad H'(0) = -1+\epsilon^{{2}/{3}} \overline{C}_u
+ \epsilon \skew5\hat{C}_u'; \\[16pt]
\displaystyle
H''(0) = \frac{\epsilon u^2_{\ast}}{\tilde{v}^{{1}/{2}}
_T u^2_P};\quad H' (\infty) = 0.
\end{array} \right\}
\end{equation}

\section{}

A simple sufficient condition for the method of separation of variables to hold
for the convection problem is derived.  This criterion is then shown to be
satisfied for the ansatz described by~(3.27), thus justifying the approach used
in~\S\,\ref{sec:trapmode}.  The basic ingredient of our argument is contained
in the following estimate for a Rayleigh--Ritz ratio:
\begin{lemma}
Let $f(z)$ be a trial function defined on $[0,1]$.  Let $\varLambda_1$ denote
the ground-state eigenvalue for $-\mathrm{d}^2g/\mathrm{d} z^2=\varLambda g$,
where $g$ must satisfy $\pm\mathrm{d} g/\mathrm{d} z+\alpha g=0$ at $z=0,1$
for some non-negative constant~$\alpha$.  Then for any $f$ that is not
identically zero we have
\begin{equation}
\frac{\displaystyle
  \alpha(f^2(0)+f^2(1)) + \int_0^1 \left(
  \frac{\mathrm{d} f}{\mathrm{d} z} \right)^2 \mathrm{d} z}%
  {\displaystyle \int_0^1 f^2\mathrm{d} z}
\ge \varLambda_1 \ge
\left( \frac{-\alpha+(\alpha^2+8\upi^2\alpha)^{1/2}}{4\upi} \right)^2.
\end{equation}
\end{lemma}

Before proving it, we note that the first inequality is the standard
variational characterization for the eigenvalue~$\varLambda_1$.

\begin{corollary}
Any non-zero trial function $f$ which satisfies the boundary condition
$f(0)=f(1)=0$ always satisfies
\begin{equation}
  \int_0^1 \left( \frac{\mathrm{d} f}{\mathrm{d} z} \right)^2 \mathrm{d} z.
\end{equation}
\end{corollary}

\begin{acknowledgments}
We would like to acknowledge the useful comments of a referee concerning
the solution procedure used in \S\,\ref{sec:concl}. A.\,N.\,O. is supported
by SERC under grant number GR/F/12345.
\end{acknowledgments}

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\begin{discussion}

\discuss{Massey}{I’m wondering if you have considered the expected intrinsic dispersion in absolute
magnitude of WRs --— if you consider the (large) mass range that becomes an
early WN or late WC according to the evolutionary models, wouldn’t you expect a large
dispersion in M$_v$?}

\discuss{van der Hucht}{Indeed, we will be always left with some intrinsic scatter in M$_v$ due
to mass differences within the same spectral subtype. But in my opinion, the current
large dispersion is for a large fraction due to incertainties of the adopted distances of
open clusters and OB associations.}

\discuss{Walborn}{I think that the scatter in WNL absolute magnitudes is dominated by intrinsic
spread rather than errors. In the LMC, one finds a range of $–5$ to nearly $–8$. This
in turn likely reflects different formation channels: mass-transfer binaries, post-RSG,
and extremely massive stars in giant H {\scshape ii} regions.}

\discuss{van der Hucht}{As said above, there is likely to be intrinsic scatter. But, I wonder
whether a scatter of 3 magnitudes perhaps reflects undetected
multiplicity.}

\discuss{Ma\'{\i}z-Apell\'{a}niz}{I could not agree more with your comment on the need for an updated
catalogue of O-type stars (as a follow up of that of Garmany
{\itshape et al.} 1982). We
are currently working on precisely that (see our poster, these Proceedings) and we will
soon make it available.}

\discuss{van der Hucht}{Wonderful.}

\discuss{Koenigsberger}{Is the ratio WR/O-stars in clusters similar or different from this ratio
for the field stars?}

\discuss{van der Hucht}{I think it is different because incompleteness among field stars is even
larger than that among cluster stars. But perhaps it should also be different because
WR stars are older and could have drifted away from clusters, more
than O-type stars.}

\discuss{Gies}{How many of the WR stars in your catalogue might be low mass objects?}

\discuss{Walborn}{Comment: PN central stars in the WR sample would be only [WC].}

\discuss{van der Hucht}{Among the WR stars in our VIIth Catalogue we doubt only one:
WR109 (V617 Sgr), which is a peculiar object (not even a [WR] central star of a PN).
All other stars in our cataloge are true massive Population I WR stars, and properly
classified as such. We have not listed known Population II [WC] objects, as we did
separately in our VIth Catalogue (van der Hucht {\itshape et al.} 1981). [WN] objects are not
known to exist, see the comment by Nolan.}

\discuss{Zinnecker}{Are all Galactic WR stars in open clusters and OB association or are there
many WR stars in the field?}

\discuss{van der Hucht}{See the VIIth WR catalogue (van der Hucht 2001): of the listed 227
Galactic WR stars, only 53 are in open culsters and OB associations, or believed to be.
The other 184 are supposedly field stars.}
\end{discussion}

\end{document}


