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\begin{document}
\section*{Appendix to `Variational Principles for slender fluxtubes I'
(A. Achterberg)}
\section*{Main paper to be published in Astronomy \& Astrophysics}
\vskip \baselineskip
\appendix
\section{Pressure perturbations in a stratified, rotating flow}

In this Appendix I briefly derive the equation describing the small
pressure perturbations which are induced by the motion of a flux tube
in a (differentally) rotating, stratified flow.
I will assume that the unperturbed fluid is rotating on cylinders 
\begin{equation}
\label{unpvelo}
	{\bf V}_{\rm e} = \Omega(R) \: R \: \hat{\btheta} \:, 
\end{equation}
and is cylindrically symmetric ($\partial / \partial \theta = 0$). I will 
employ cylindrical coordinates $R$, $\theta$ and $Z$. Unperturbed
quantities will be denoted with a subscript $0$.

The equation of motion describing the unperturbed flow reads in this case
\begin{eqnarray}
\label{upflow}
	\frac{1}{\rho_{\rm e}} \: \frac{\partial P_{\rm e}}{\partial R} 
	& = & 
	\Omega^{2}(R) \: R - \frac{\partial \Phi}{\partial R} \; ,
	\nonumber \\
	& & \\
	\frac{1}{\rho_{\rm e}} \: \frac{\partial P_{\rm e}}{\partial Z} 
	& = & 
	- \frac{\partial \Phi}{\partial Z} \: . \nonumber
\end{eqnarray}
Here $\Phi(R,Z)$ is the gravitational potential. I will neglect the
effect of self-gravity of the material,  assuming that $\Phi$ is given.
The assumption that the fluid is rotating on cylinders implies that it
is barotropic:
\begin{equation}
\label{barotrop}
	\grad P_{\rm e} \btimes \grad \rho_{\rm e} = 0 \; .
\end{equation}

Consider small perturbations, which displace fluid elements according to
${\bf x} \Longrightarrow {\bf x} + \bxi$.
Small perturbations $\delta {\bf V}$ in velocity are given in terms of the 
Lagrangian displacement vector $\bxi$ as (e.g. Frieman \& Rotenberg, 1960;
Lynden-Bell \& Ostriker, 1967)
\begin{equation}
\label{velopert}
	\delta {\bf V} = D_{0} \: \bxi - \xigr {\bf V} \: .
\end{equation}	
Here $D_{0}$ is the comoving time derivative of the unperturbed flow, 
in this
case given by
\begin{equation}
\label{Doperator}
	D_{0} \equiv \frac{\partial}{\partial t} + 
	\Omega(R) \: \frac{\partial}{\partial \theta} 
 	\; .
\end{equation}
The linearized equation of motion reads in component form 
(compare for instance Shu, 1992, p. 137, or Chandrasekhar, 1961, Ch. VII):
\begin{eqnarray}
\label{lineom}
	D_{0} \delta V_{\rm R} - 2 \Omega \: \delta V_{\theta} & = & 
	- \frac{1}{\rho_{\rm e}} \frac{ \partial P_{\rm r}}{\partial R}
	+ \left( \frac{\delta \rho}{\rho_{\rm e}} \right) \: 
	\left( \frac{1}{\rho_{\rm e}} \: 
	\frac{\partial P_{\rm e}}{\partial R} \right)
	\; , \nonumber \\
	& & \nonumber \\
	D_{0} \delta V_{\theta} + 
	\frac{\kappa^{2}}{2 \Omega} \: \delta V_{\rm R} & = & 	
	- \frac{1}{\rho_{\rm e} R} 
	\frac{ \partial P_{\rm r}}{\partial \theta} \; , \\
	& & \nonumber \\
	D_{0} \delta V_{\rm Z} & = & 
	- \frac{1}{\rho_{\rm e}} \frac{ \partial P_{\rm r}}{\partial Z}
	+ \left( \frac{\delta \rho}{\rho_{\rm e}} \right) \: 
	\left( \frac{1}{\rho_{\rm e}} \: 
	\frac{\partial P_{\rm e}}{\partial Z} \right)
	\; . \nonumber 
\end{eqnarray} 	
Here $\kappa^{2}$ is the epicyclic frequency defined in 
Eqn. (61) of the main paper.
 
The Eulerian pressure variation $\delta P \equiv P_{\rm r}$ and 
density variation
$\delta \rho$ are, for isentropic disturbances, given by
\begin{eqnarray}
\label{pressdens}
	P_{\rm r} & = & 
	- \gamma P_{\rm e} \: \xidi - \xigr P_{\rm e} \: , 
	\nonumber \\
	& & \\
	\delta \rho & = & 
	- \rho_{\rm e} \: \xidi - \xigr \rho_{\rm e} \: . 
	\nonumber
\end{eqnarray}
In the case of the pressure induced by the motion of a small object through
the flow one can introduce a major simplification: compressibility
effects can be neglected to lowest order. This corresponds to the
Boussinesq approximation, where one puts
\begin{equation}
\label{Boussi}
	(\grad \bdot \delta {\bf V}) \approx 0 \; \; \; , \; \; \; 
	\xidi \approx 0 \; .
\end{equation}
The second equation follows from the first and relation (\ref{velopert}).
Formally, this corresponds to taking the limit of infinite sound speed: 
$s_{\rm e}^{2} \equiv \gamma P_{\rm e} / \rho_{\rm e} \rightarrow \infty$.
By eliminating $\xidi$ between the two equations (\ref{pressdens}) one finds
\begin{equation}
\label{denspert}
	\frac{\delta \rho}{\rho_{\rm e}} \approx
	\frac{1}{\gamma} \: \xigr 
	\left[ \ln{\left( P_{\rm e} \rho_{\rm e}^{-\gamma} \right)} \right]
\end{equation}
where, again in the Boussinesq limit, I neglected a term 
$P_{\rm r} / \gamma P_{\rm e} = P_{\rm r} / (\rho_{\rm e} s_{\rm e}^{2})$.

Taking the time derivative $D_{0}$ of the $R$ and $Z$ components 
of the equation of motion, and substituting (\ref{denspert}) in the 
resulting expressions one finds the following
set of equations:
\begin{eqnarray}
\label{lineom2}
	D_{0}^{2} \: \delta V_{\rm R} - 2 \Omega \: D_{0} \: \delta V_{\theta} 
	+ N^{2}_{\rm RR} \: \delta V_{\rm R} & + & 
	N^{2}_{\rm RZ} \:  \delta V_{\rm Z}
	\nonumber \\ 
	& & \nonumber \\
	& = & -  
	\frac{D_{0}}{\rho_{\rm e}}  \: 
	\left[ \frac{ \partial P_{\rm r}}{\partial R} \right]
	\; , \nonumber \\
	& &  \\
	D_{0}^{2} \: \delta V_{\rm Z} 
	+ N^{2}_{\rm ZZ} \: \delta V_{\rm Z} & + & 
	N^{2}_{\rm ZR} \:  \delta V_{\rm R}
	\nonumber \\ 
	& & \nonumber \\
	& = & 
	- \frac{1}{\rho_{\rm e}} 
	\left[\frac{ \partial (D_{0}P_{\rm r})}{\partial Z} \right] 
	\; . \nonumber 
\end{eqnarray} 	
Here I have defined the tensor $({\bf N}^{2})$ (not the square of ${\bf N}$!) 
which has the components 
\begin{equation}
\label{BVtensor}
	\left( N^{2} \right)_{ij} = 
	- \left[ \frac{1}{\gamma \rho_{\rm e}} \: 
	\frac{\partial P_{\rm e}}{\partial x^{i}} \right] \: 
	\frac{\partial}{\partial x^{j}} \:
	\left[ \ln{\left( P_{\rm e} \rho_{\rm e}^{- \gamma} \right)} 
	\right] \; .
\end{equation}
The Brunt-V\"ais\"al\"a frequency is defined as:
\begin{eqnarray}
\label{BVdef}
	N^{2} & \equiv & - 
	\left[ \frac{1}{\gamma \rho_{\rm e}} \: \grad P_{\rm e} \right]
	\bdot \grad \left[ \ln{\left( P_{\rm e} \rho_{\rm e}^{- \gamma} 
	\right)} \right] 
	\nonumber \\
	& & \\
	& = &  N^{2}_{\; \rm RR} + N^{2}_{\; ZZ} \; , \nonumber
\end{eqnarray}
which motivates the notation employed here.
The barotropic condition (\ref{barotrop}) implies 
$N^{2}_{\; RZ} = N^{2}_{\; ZR}$,
and $N^{2}_{\; RR}N^{2}_{\; ZZ} = \left( N^{2}_{\; RZ} \right)^{2}$.

For notational purposes I introduce the acceleration vector due to 
the pressure perturbation:
\begin{equation}
\label{avector}
	{\bf a} = - \frac{1}{\rho_{\rm e}} \: \grad P_{\rm r} \; .
\end{equation} 
By eliminating $D_{0} \: \delta V_{\theta}$ from these equations using 
(\ref{lineom}) one finds, after some straightforward algebra, 
the following wave equation which links the velocity perturbation
to the pressure force:
\begin{equation}
\label{wave-eqn}
	\left( D_{0}^{4} + \left[ \kappa^{2} + N^{2} \right] D_{0}^{2} +
	\kappa^{2} N^{2}_{\; \rm ZZ} \right) \: \delta {\bf V} = {\bf Q}
	\; .
\end{equation}
The vector ${\bf Q}$ has the components:
\begin{eqnarray}
\label{Qvector}
	Q_{\rm R} & = & \left(D_{0}^{2} + N^{2}_{\rm \; ZZ} \right)
	\left[ D_{0}a_{\rm R} + 2 \Omega \: a_{\theta} \right] -
	N^{2}_{\rm \; RZ} D_{0} a_{\rm Z} \; , \nonumber \\
	& & \nonumber \\
	Q_{\theta} & = & 
	\left( D_{0}^{2} + N^{2} \right) D_{0} a_{\theta}
	- 
	\nonumber \\ 
	& & \; \; \; \;  - \frac{\kappa^{2}}{2 \Omega} \: 
	\left[\left( D_{0}^{2} + N^{2}_{\rm \; ZZ} \right) a_{\rm R} -
	N^{2}_{\rm \; RZ} a_{\rm Z} \right]  \; , \\
	& & \nonumber \\
	Q_{\rm Z} & = & 
	\left( D_{0}^{2} + \kappa^{2} + N^{2}_{\rm \; RR} \right)
	D_{0} a_{\rm Z} - N^{2}_{\rm ZR} \:
	\left[ D_{0}a_{\rm R} + 2 \Omega \: a_{\theta} \right] \; .
	\nonumber
\end{eqnarray}
I now employ the slender flux tube approximation, 
assuming that the radius of
the tube is much less than the cylindrical radius, or the pressure and
density gradient scale:
\begin{equation}
\label{sftapprox}
	a \ll {\rm min} \left( R 
	, \; L_{\rm P} \equiv \| \grad \ln{P_{\rm e}} \|^{-1} 
	, \; L_{\rho} \equiv \| \grad \ln{\rho_{\rm e}} \|^{-1} 
	\right) \; .
\end{equation}
This means that the induced pressure $P_{\rm r}$ and the associated velocity
perturbation $\delta {\bf V}$ will generally vary on a small scale $\sim a$. 
In the Boussinesq approximation this means that one can then neglect 
gradients in $\rho_{\rm e}$, $\Omega$, $\kappa$ and $ N^{2}_{\; ij}$ 
with respect to gradients in $\delta {\bf V}$ and $P_{\rm r}$
{\em at this stage of the calculation}. The leading order effects of the
stratification are already taken into account by the appearance of the
terms $\propto N^{2}_{\; ij}$.
 
Taking the divergence of Eqn. (\ref{wave-eqn})
yields because of the Boussinesq approximation (\ref{Boussi}):
\begin{equation}
\label{presseqn}
	\grad \bdot {\bf Q} \approx 0.
\end{equation}
This equation, together with the `impenetrable boundary' condition  
\begin{equation}
\label{normalvelo}
	\left( \delta {\bf V} - {\bf V}_{\rm t} \right) 
	\bdot {\bf n}_{\rm t} = 0
\end{equation}
at the surface (with normal ${\bf n}_{\rm t}$) of the tube moving with 
velocity ${\bf V}_{\rm t}$, determines the induced pressure.

I will consider a number of limiting cases. Let a section of the tube be 
located at a cylindrical radius 
$R = R_{0}$, $\theta = \theta_{0}$ and $Z = Z_{0}$.  
Introducing local cartesian coordinates $x$, $y$ and $z$, 
corotating with the
local angular frequency $\Omega_{0} = \Omega(R_{0})$.
\begin{equation}
\label{localco}
	x = R - R_{0} ,  \; y = R_{0}(\theta - \theta_{0}) 
	, \;  z = Z - Z_{0} \; ,
\end{equation}
one can approximate $\grad \bdot {\bf Q}$ by its cartesian version, and
(\ref{presseqn}) becomes
\begin{equation}
\label{cartpres}
	\frac{\partial Q_{x}}{\partial x} + 
	\frac{\partial Q_{y}}{\partial y} 
	+ \frac{\partial Q_{z}}{\partial z} = 0 \; .
\end{equation}

The first case is that of a non-rotating fluid ($\Omega = \kappa = 0$), 
stratified 
in the $Z-$direction so that $N^{2}_{\; RR} = N^{2}_{\; RZ} = 0$.
Using (\ref{Qvector}) and (\ref{cartpres}) one then finds:
\begin{equation}
\label{stratif}
	\grad_{\perp}^{2} \:
	\left( \frac{\partial^2 P_{\rm r}}{\partial t^2} +
	N^{2} P_{\rm r} \right) 
	+ \frac{\partial^2}{\partial z^2}
	\left( \frac{\partial^2 P_{\rm r}}{\partial t^2} \right) = 0.
\end{equation}
Here I have defined the Laplace operator in the $x-y$ plane:
\begin{equation}
\label{vertLapl}
	\grad_{\perp}^{2} \equiv 	
	\frac{\partial^{2}}{\partial x^{2}} +
	\frac{\partial^{2}}{\partial y^{2}}  
	\; .
\end{equation}
Another simple case is that of a radially stratified 
($N^{2}_{\; RZ} = N^{2}_{\; ZZ} = 0$), rotating fluid. 
In that case the equation for the induced pressure reads  	
\begin{eqnarray}
\label{purerot}
	\left[ \grad_{\perp}^{2} \:
	\frac{\partial}{\partial t} \right.  &-& \left.
	\left( R \: \frac{{\rm d} \Omega}{{\rm d}R} \right) \: 
	\frac{\partial^{2}}{\partial x \partial y} \: \right]
	\left( \frac{\partial P_{\rm r}}{\partial t} \right) +
	\nonumber \\
	& & \\
 	& + & 
	\frac{\partial^{2}}{\partial z^{2}} \: 
	\left( \frac{\partial^{2}P_{\rm r}}{\partial t^{2}} + 
	\left[\kappa^{2} + N^{2} \right] \: P_{\rm r} \right) 
	= 0 \; .
	\nonumber
\end{eqnarray}
Here I have written simply $\partial / \partial t$ for the
time derivative $D_{0}$ in the corotating frame, and used
$N^{2} = N^{2}_{\; RR}$.

Finally, in a rotating fluid stratified in the $Z-$direction, the
induced pressure satisfies 
\begin{eqnarray}
\label{combi}
	\left[ \grad_{\perp}^{2} \:  
	\frac{\partial}{\partial t} \right.  
	& - & 
	\left.  \left( R \: \frac{{\rm d} \Omega}{{\rm d}R} \right)  
	\frac{\partial^{2}}{\partial x \partial y} \right]
	\left( \frac{\partial^2 P_{\rm r}}{\partial t^2} +
	N^{2} P_{\rm r} \right) + \nonumber \\
	& & \\ 
	& + & \frac{\partial^{2}}{\partial z^{2}}  
	\left( \frac{\partial^{2}}{\partial t^{2}} + 
	\kappa^{2}  \right) 
	\left( \frac{\partial P_{\rm r}}{\partial t} \right) = 0 \; .
	\nonumber
\end{eqnarray}
The last two equations simplify considerably in the case of uniform
rotation (${\rm d} \Omega / {\rm d}R = 0$), 
or when the tube axis is oriented
along the radial ($x-$) or azimuthal ($y-$)direction so that, 
in the slender flux tube approximation, 
one can put $\partial P_{\rm r}/ \partial x \approx 0$ or 
$\partial P_{\rm r} / \partial y \approx 0$ respectively.

In the case where the tube motion is harmonic with frequency $\omega$
so that $P_{\rm r} \propto e^{-i \omega t}$ one sees that there
are resonances at the two natural frequencies of the external medium:
the epicyclic frequency $\kappa$ and the Brunt V\"ais\"al\"a frequency $N$.
In a uniform, non-rotating medium these equations all reduce to the
`simple' case where the pressure satisfies Laplace's equation
$\grad^{2} P_{\rm r} = 0$, the case considered in some detail in the
main paper.

The equations derived in this Section for the pressure response are
closely related to the occurence of low-frequency internal waves.
Consider the simple case of uniform rotation where 
${\rm d} \Omega / {\rm d}R = 0$ and $\kappa = 2 \Omega$.
Assuming a pressure perturbation varying in time as 
$P_{\rm r} \propto e^{-i \omega t}$ in the corotating frame, 
the equation for the pressure in the local approximation
is
\begin{eqnarray}
\label{presswave}
	& & \left(\omega^{2} - N^{2}_{\; \rm ZZ} \right) \: 
	\frac{\partial^{2} P_{\rm r}}{\partial x^{2}} 
	+ \left(\omega^{2} - N^{2} \right) \: 
	\frac{\partial^{2} P_{\rm r}}{\partial y^{2}}
	+ \nonumber \\
	& & \\
	& & 
	+ \: 2 N^{2}_{\; \rm RZ} 
	\: \frac{\partial^{2} P_{\rm r}}{\partial x \partial z} +
	\left( \omega^{2} - \left[ \kappa^{2} + N^{2}_{\; \rm RR} 
	\right] \right)
	\: \frac{\partial^{2} P_{\rm r}}{\partial z^{2}} = 0 \; . 
	\nonumber
\end{eqnarray}
Assuming plane waves, varying in space as
\begin{equation}
\label{planewaves}
	P_{\rm r} \propto 
	e^{ik_{\rm x}x + ik_{\rm y}y + ik_{\rm z}z} \; ,
\end{equation}
this equation yields a dispersion relation which can be written in the form:
\begin{equation}
\label{pwavedisp}
	\omega^{2} - \frac{k_{\rm y}^{2}}{k^{2}} \: N^{2} -
	\frac{k_{\rm z}^{2}}{k^{2}} \left[
	\kappa^{2} + \left( \frac{k_{\rm x}}{k_{\rm z}} 
	\: N^{2}_{\; \rm ZZ}
	- N^{2}_{\; \rm RR} \right)^{2} \right] = 0 \; .
\end{equation}
In the case of axisymmetric waves ($k_{\rm y} = 0$) 
this equation is also valid
when $d \Omega / dR \ne 0$. In deriving this relation I have used 
$\left( N^{2}_{\; \rm RZ} \right)^{2} = N^{2}_{\; \rm RR} N^{2}_{\; \rm ZZ}$.
Note that the wave-vector in the $y-$direction is formally given by
$k_{\rm y} = m/R$, with $m$ an integer. In a number of limiting cases this
dispersion relation reduces to that of known internal (inertial) waves
(e.g. Priest, 1982, Ch. 4).
In a homogeneous medium ($N^{2}_{\; ij} = 0$) this corresponds to the
dispersion relation for inertial waves 
\begin{equation}
\label{inertwave}
	\omega = \pm \frac{k_{\rm z}}{k} \: \kappa = \pm 
	2 \: \frac{ {\bf k} \bdot \bOmega}{k} \; ,
\end{equation}
where the last relation is valid for uniform rotation.
For a non-rotating ($\kappa = 0$) fluid stratified in the $Z-$direction
($N^{2} = N^{2}_{\; {\rm ZZ}}, N^{2}_{\; \rm RR} = 0$) 
one finds the dispersion relation for internal
gravity waves 
\begin{equation}
\label{igravwaves}
	\omega = \pm N \sin{\theta_{\rm g}}
\end{equation}
where $\theta_{\rm g} = {\rm cos}^{-1}(k_{\rm z}/k)$.

\section{Calculating the reaction force for harmonic motion}

In the rest of the Appendix, I will concentrate on the case of the
a tube undergoing a harmonic motion with frequency $\omega$ in the
corotating frame. To simplify the mathematics, I will neglect the
cross-terms such as $\partial^{2} P_{\rm r} / \partial x \partial y$,
assuming for instance uniform rotation, a tube oriented in the 
$x-$ or $y-$direction,
and/or $(N^{2})_{ij} \ll N^{2}$ for $i \ne j$.
The general case can in principle be solved by a transformation to 
principal axes. 

Without cross terms, the equation of motion for the medium 
surrounding the tube can be written in the form
\begin{equation}
\label{simpleom}
	\left( \frac{\partial^{2}}{\partial t^{2}} + 
	\sigma_{i}^{2} \right) 
	\delta V_{i} = 
	- \frac{1}{\rho_{\rm e}} \: 
	\frac{\partial^{2} P_{\rm r}}{\partial x_{i} \partial t} \; ,
\end{equation}
where the index $i$ takes the values $i = 1, 2$.
Here I have written $\partial / \partial t$ for $D_{0}$ and introduced 
two cartesian coordinates $x_{1}$ and $x_{2}$ perpendicular to 
the axis of the tube.
For harmonic motion with $\delta V_{i}, \: P_{\rm r} \propto e^{-i \omega t}$
one can solve this equation for the velocity amplitude:
\begin{equation}
\label{veloampl}
	\delta V_{i} = - i \: \frac{\omega}{D_{i}(\omega)} \: 
	\left(\frac{1}{\rho_{\rm e}} \: 
	\frac{\partial P_{\rm r}}{\partial x_{i}} \right)
	\; .
\end{equation}
The functions $D_{1}(\omega)$ and $D_{2}(\omega)$ are given by
\begin{equation}
\label{Dcoeff}
	D_{1,2}(\omega) = \omega^{2} - \sigma_{1,2}^{2} \; ,
\end{equation}
with $\sigma_{1}$ and $\sigma_{2}$ the natural frequencies of the system.
The equation for the induced pressure follows from the Boussinesq condition
$\partial \delta V_{1} / \partial x_{1} + 
\partial \delta V_{2} / \partial x_{2} = 0$
and, in the local approximation where one neglects large-scale gradients
in the density,  can be written in the form
\begin{equation}
\label{standform}
	\frac{1}{D_{1}(\omega)} \: 
	\frac{\partial^{2} P_{\rm r}}{\partial x_{1}^{2}}
	+ \frac{1}{D_{2}(\omega)} \: 
	\frac{\partial^{2} P_{\rm r}}{\partial x_{2}^{2}}
	= 0 \; .
\end{equation}
For instance, for a tube oriented along the $x-$axis Eqn. (\ref{stratif}) 
reduces to this form with $x_{1} = y$,  $x_{2} = z$, 
$\sigma_{1} = 0$ and $\sigma_{2} = N$. I will assume that
$D_{1}$ and $D_{2}$ are both positive so that this equation is of the
elliptic type.

This equation must be solved subject to two boundary conditions. 
At the tube surface condition (\ref{normalvelo}) applies. 
Using (\ref{veloampl}) this condition 
can be seen to correspond to:
\begin{equation}
\label{tubebc}
	\left[ \frac{\omega \: \cos{\theta}}{D_{1}(\omega)} \right] \: 
	\frac{\partial P_{\rm r}}{\partial x_{1}} +
	\left[ \frac{\omega \: \sin{\theta}}{D_{2}(\omega)} \right] \: 
	\frac{\partial P_{\rm r}}{\partial x_{2}} =
	i \rho_{\rm e} V_{\perp} \: \cos{(\theta - \beta)} \; .
\end{equation}
Here $V_{\perp}$ is the magnitude of the velocity-component of the 
slender tube perpendicular to its axis, and $\beta$ the position angle 
of that velocity in the $x_{1}-x_{2}$ plane, i.e.  
${\bf V}_{\perp} = V_{\perp} \: (\cos{\beta} \: , \: \sin{\beta})$.
The angle $\theta$ is the polar angle in that plane.
The second boundary condition is that the induced pressure vanishes 
at infinity:
\begin{equation}
\label{vanishP}
	P_{\rm r}(x_{1} , \: x_{2}) \rightarrow 0 \; \; 
	\mbox{when $\| x_{1} \|$, $\| x_{2}\| \rightarrow \infty$.}
\end{equation}
 
By introducing scaled coordinates 
\begin{equation}
\label{scaledco}
	\xx = 
	\frac{\omega}{\sqrt{D_{2}(\omega)}} \: x_{1} \; \; , \; \; 
	\xz = 	
	\frac{\omega}{\sqrt{D_{1}(\omega)}} \: x_{1} \;  , 
\end{equation}
(note the juxtaposition of indices)
this equation reduces to the standard form
\begin{equation}
\label{standard}
	\omega^{2} \: 
	\left[ \frac{\partial^{2} P_{\rm r}}{\partial \xx^{2}} +  
	\frac{\partial^{2} P_{\rm r}}{\partial \xz^{2}} \right] 
	= 0 \; .
\end{equation}
In these coordinates, the surface of a cylindrical tube 
with radius $a$ corresponds to an ellipse 
\begin{equation}
\label{tubesurf}
	\frac{\xx^{2}}{a_{1}^{2}} + \frac{\xz^{2}}{a_{2}^{2}} = 1 \; ,
\end{equation}
with pricipal axes
\begin{equation}
\label{ellipse}
	a_{1} = \frac{\omega}{\sqrt{D_{2}(\omega)}} \: a 
	\; \; , \; \; 
	a_{2} = \frac{\omega}{\sqrt{D_{1}(\omega)}} \: a \; .
\end{equation}  
It is therefore advantageous to use elliptical cylindrical coordinates 
($u$, $v$) which are defined by (see for instance Morse \& Feshbach, 
1953, Ch. 5 ; Arfken, 1970, Ch. 2)  
\begin{equation}
\label{ellcylco}
	\xx = \alpha \: \cosh{u} \: \cos{v} 
	\; \; \; , \; \; 
	\xz = \alpha \: \sinh{u} \: \sin{v} \; .
\end{equation}
In these coordinates Laplace's equation remains separable, and the
induced pressure satisfies
\begin{equation}
\label{indpressep}
	\frac{\partial^{2} P_{\rm r}}{\partial u^{2}} 
	+ \frac{\partial^{2} P_{\rm r}}{\partial v^{2}} = 0 \; .
\end{equation} 
By choosing
\begin{equation}
\label{choice}
	\alpha =   \sqrt{ a_{1}^{2} - a_{2}^{2}} =  a \: 
	\left[ \frac{\omega^{2}(\sigma_{2}^{2} - \sigma_{1}^{2})}
	{D_{1}(\omega) D_{2}(\omega)} \right]^{1/2} \; ,
\end{equation}
one immediately sees that the surface of the tube corresponds to
the coordinate surface $u = {\rm constant} = u_{\rm t}$ with
\begin{equation}
\label{tubesurf2}
	\coth{u_{\rm t}} =  \frac{a_{1}}{a_{2}} =  
	\left[ \frac{D_{1}(\omega)}{D_{2}(\omega)} \right]^{1/2}
	\; 
\end{equation}
Here I have assumed, without loss of generality, 
that $\sigma_{2} > \sigma_{1}$.

This choice of coordinates also
implies that on the surface of the tube (and only there) 
the angle variable $v$
coincides with the polar angle $\theta$. 
From (\ref{scaledco}) one has
\begin{equation}
\label{angles}
	\cot{\theta} = \frac{x_{1}}{x_{2}} = 
	\left[ \frac{D_{2}(\omega)}{D_{1}(\omega)} \right]^{1/2} \: 
	\coth{u} \: \cot{v} \; .
\end{equation}
From condition (\ref{tubesurf2}) it immediately follows 
that $\cot{\theta} = \cot{v}$ on
the tube surface $u = u_{\rm t}$.

The boundary conditions (\ref{tubebc}) and (\ref{vanishP}) 
take the simple form:
\begin{eqnarray}
\label{newbc}
	\left(\frac{\partial P_{\rm r}}{\partial u} \right)_{u = u_{\rm t}} 
	& = &  
	i \frac{\sqrt{D_{1}(\omega) D_{2}(\omega)}}{\omega} \:
	\rho_{\rm e} V_{\perp} a \: \cos{(v - \beta)} \; , 
	\nonumber \\
	& & \\
	P_{\rm r}(u \: , \: v) & \rightarrow&  0 \; \;  
	\mbox{when $u \rightarrow \infty$.}
	\nonumber
\end{eqnarray}
Note that $u \approx \ln{(2 \tilde{r} / \alpha)}$ at large distances 
(so that $\tilde{\varpi} \equiv \sqrt{\xx^{2} + \xz^{2}} \gg a$) 
from the tube. The solution of Eqn. (\ref{indpressep}) satisfying 
both boundary conditions reads
\begin{equation}
\label{solution}
	P_{\rm r} = - i \omega 
	\left( \frac{D_{1}D_{2}}{\omega^{4}} \right)^{1/2} 
	e^{(u_{\rm t} - u)} \rho_{\rm e} 
	V_{\perp} a \cos{(v - \beta)} \; .
\end{equation}
Since $v = \theta$ on the tube surface, the angular dependence of the 
induced pressure on the tube surface is the same as in the unstratified case 
discussed in Section 4 of the main paper (see Eqn. (52) of the main paper), 
where the coordinate system was chosen such that $\beta = 0$). 
The influence of the stratification and
rotation of the external medium, as parametrized by the frequencies 
$\sigma_{1}$ and $\sigma_{2}$, is given by the factor
\begin{equation}
\label{corrfact} 
	\left( \frac{D_{1}(\omega) D_{2}(\omega)}
	{\omega^{4}} \right)^{1/2} = \sqrt{ 
	\left( 1 - \frac{\sigma_{1}^{2}}{\omega^{2}} \right) 
	\left( 1 - \frac{\sigma_{2}^{2}}{\omega^{2}} \right)}
\end{equation} 
in this expression.

The force on a section of tube can be immediately calculated 
by integration 
over the tube surface $u = u_{\rm t}, \; v = \theta$
(compare Eqn. (53) of the main paper, with $\rad = {\bf n}_{\rm t}$
and 
$\partial {\bf V}_{\perp} / \partial t = - i \omega {\bf V}_{\perp}$):
\begin{eqnarray}
\label{extforce2}
	\delta {\bf F} & = &  
	- a \: \delta \ell \int_{0}^{2 \pi} \: d \theta \:
	P_{\rm r}(\varpi = a, \theta) \; {\bf n}_{\rm t} 
	\nonumber \\
	& & \\
	& = & 
	\left( \pi a^2 \: \delta \ell \: \rho_{\rm 0} \right) \btimes 
	i \omega \: 
	\left( \frac{D_{1}D_{2}}{\omega^{4}} \right)^{1/2} \:
	{\bf V}_{\perp} \; .
	\nonumber
\end{eqnarray}
This force vanishes at the resonant frequencies 
$\omega = \pm \sigma_{1}$, $\omega = \pm \sigma_{2}$ 
where $D_{1}(\omega)= 0 $ and $D_{2}(\omega) = 0$ respectively.
In the high-frequency limit $\| \omega \| \gg \sigma_{1,2}$
one has $D_{1} \sim D_{2} \sim \omega^{2}$, and one recovers the
result (53), valid for the force on the tube in a 
homogeneous external medium at rest. 

In this derivation I have assumed that $D_{1}$ and $D_{2}$ are both positive
so that $\| \omega \| \ge \sigma_{2} \ge \sigma_{1})$.
The same result applies in the low-frequency limit as long as $D_{1}$ and
$D_{2}$ have the same sign ($\|\omega \| \le \sigma_{1} \le \sigma_{2}$), 
provided one replaces $D_{i}(\omega)$ by $\| D_{i}(\omega)\|$.
In the frequency regime $\sigma_{1} < \|\omega \| < \sigma_{2}$ the
functions $D_{1} > 0$ and $D_{2} < 0$ have opposite signs, and the equation
for the induced pressure becomes hyperbolic. 
Using scaled coordinates, with $D_{2}$
replaced by $\| D_{2} \|$ in Eqn. (\ref{scaledco}), 
the equation for the induced
pressure becomes a wave equation:
\begin{equation}
\label{indpreswave}
	\frac{\partial^{2} P_{\rm r}}{\partial u^{2}} 
	- \frac{\partial^{2} P_{\rm r}}{\partial v^{2}} = 0 \; .
\end{equation} 
This equation has no solution where $P_{\rm r}$ decays at infinity, 
signaling a breakdown in this case of the validity of the 
approximations used
to derive the equation governing the induced pressure.

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\end{document}
