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$\begin{figure} \par\includegraphics[width=8.5cm,clip]{5227f1.eps}\end{figure}$ Figure 1: Sketch of the two-slab system. The shaded area represents the density enhancement of the two slabs while the hatched area represents the photospheric medium, that fixes the feet of the slabs and produces the line-tying effect.
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In the text

$\begin{figure} \par\mbox{\hspace{-0.cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspace{... ...7f4.eps}\hspace*{5mm}\includegraphics[width=4.25cm,clip]{5227f5.eps}\end{figure}$ Figure 2: Velocity profile, v_x(x), for the fundamental symmetric mode ( upper row) and the fundamental antisymmetric mode ( lower row) for a slab half-width a=0.05 Land a density enhancement $\rho _{\rm i}/\rho _{\rm e} = 3$ . The left and right columns correspond to a distance between slabs d=0.5 L and d=2 L, respectively. a) and b) show that the symmetric mode is trapped for the two separations, but c) and d) indicate that the antisymmetric mode becomes leaky for small distances. The shaded surface corresponds to the density enhancement of the slabs.
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In the text

$\begin{figure} \par\hspace{-0.15cm}\includegraphics[width=5.05cm,clip]{5227f6.eps}\par\includegraphics[width=4.85cm,clip]{5227f7.eps}\end{figure}$ Figure 3: a) Real part, $\omega _{\rm R}$ , and b) imaginary part, $\omega _{\rm I}$ , of the frequency as functions of the separation, d, for a density enhancement $\rho _{\rm i}/\rho _{\rm e} = 3$ and a half-width a=0.05 L. The line styles correspond to the fundamental symmetric mode (solid line), the fundamental antisymmetric mode (dashed line), the first symmetric harmonic (dot-dashed line) and the first antisymmetric harmonic (three-dot-dashed line). The dotted line is the external cut-off frequency, $\omega _{\rm ce}$ . Thick curves represent trapped modes while thin lines correspond to leaky modes. $\omega _{\rm R} L /v_{\rm Ae}$ and $\omega _{\rm I} L /v_{\rm Ae}$ are normalised frequencies. In the top panel the calculated frequency from the time-dependent results for the symmetric (triangles) and antisymmetric (diamonds) modes is also represented.
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In the text

$\begin{figure} \par\mbox{\hspace{-0.cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspace{... ...10.eps}\hspace*{4mm}\includegraphics[width=4.35cm,clip]{5227f11.eps}\end{figure}$ Figure 4: Time-evolution of the velocity, v_x, for a distance between slabs d=0.5 Land a symmetric initial impulse.
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In the text

$\begin{figure} \par\includegraphics[width=5.15cm,clip]{5227f12.eps}\par\includegraphics[width=5.1cm,clip]{5227f13.eps}\end{figure}$ Figure 5: a) Measured velocity at the centre of the right slab, x=d/2, for the symmetric initial perturbation of Fig. 4. After a short transient the system oscillates in a trapped mode, with period close to $2\tau _{\rm A}$ , and the oscillatory amplitude remains unchanged. b) As expected, the periodogram of the signal in a) features a large power peak at a period around $2\tau _{\rm A}$ . There is an excellent agreement between the period of this peak and the period of the normal mode obtained from Eq. (8) (dotted line). The periodogram lacks other power peaks.
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In the text

$\begin{figure} \par\mbox{\hspace{-0.cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspace{... ...16.eps}\hspace*{4mm}\includegraphics[width=4.25cm,clip]{5227f17.eps}\end{figure}$ Figure 6: Time-evolution of the velocity, v_x, for a distance between slabs d=0.5 Land an antisymmetric initial impulse.
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In the text

$\begin{figure} \par\includegraphics[width=5.15cm,clip]{5227f18.eps}\par\includegraphics[width=5.1cm,clip]{5227f19.eps}\end{figure}$ Figure 7: a) Measured velocity at the centre of the right slab, x=d/2, for the antisymmetric initial perturbation of Fig. 6. After a short transient the system oscillates in a leaky mode and so the perturbation attenuates exponentially. b) The periodogram of the signal in a) has a power peak whose period is in excellent agreement with that of the normal mode obtained from Eq. (9) (dotted line). The periodogram lacks other power peaks.
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In the text

$\begin{figure} \par\mbox{\hspace{-0.cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspace{... ...22.eps}\hspace*{4mm}\includegraphics[width=4.25cm,clip]{5227f23.eps}\end{figure}$ Figure 8: Time-evolution of v_x for d=0.5 L and a non-symmetric initial excitation.
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In the text

$\begin{figure} \par\includegraphics[width=5.25cm,clip]{5227f24.eps}\end{figure}$ Figure 9: v_x measured at the centres of the slabs for the simulation shown in Fig. 8. The solid and dashed lines correspond to the right and left slabs, respectively.
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In the text

$\begin{figure} \par\mbox{\hspace{-0.cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspace{... ...27.eps}\hspace*{4mm}\includegraphics[width=4.25cm,clip]{5227f28.eps}\end{figure}$ Figure 10: Time evolution of v_x for d=2L and a non-symmetric initial disturbance. Note the interchange of energy between the two slabs in the last two frames.
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In the text

$\begin{figure} \par\includegraphics[width=5.15cm,clip]{5227f29.eps}\par\includegraphics[width=5.2cm,clip]{5227f30.eps}\end{figure}$ Figure 11: a) v_x measured at the centre of the slabs for the simulation shown in Fig. 10. The solid and dashed lines correspond to the right and left slabs, respectively. b) Power spectrum of the previous signal (solid line). The two vertical dashed lines indicate the periods of the fundamental antisymmetric mode (at $2.04821 \tau _{\rm A}$ ) and the fundamental symmetric mode ( $2.10633 \tau _{\rm A}$ ).
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In the text

$\begin{figure} \par\includegraphics[width=5.25cm,clip]{5227f31.eps}\par\includegraphics[width=5.25cm,clip]{5227f32.eps}\end{figure}$ Figure 12: The solid lines are a rescaled close-up view of Figs. 10d and c, respectively. The dashed lines correspond to f₁(x) and f₂(x), respectively. The analytical approximation then reproduces the velocity profile obtained in the time-dependent simulation when a substantial amount of energy is concentrated in a single slab. The difference among both curves to the right and left of the slabs arises from the system not having reached the stationary state. For greater times the difference becomes smaller.
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In the text

$\begin{figure} \par\includegraphics[width=5.15cm,clip]{5227f33.eps}\par\includegraphics[width=5.15cm,clip]{5227f34.eps}\end{figure}$ Figure 13: a) Superposition of v_x measured at the centre of the right slab from the time-dependent numerical simulation (solid line) and the analytical approximation (dashed line). b) The same for the left slab.
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In the text

$\begin{figure} \par\mbox{\hspace{-0.02cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspac... ...39.eps}\hspace*{4mm}\includegraphics[width=4.28cm,clip]{5227f40.eps}\end{figure}$ Figure 14: Plots of $\omega _{\rm R}$ ( left column) and $\omega _{\rm I}$ ( right column) as functions of the slab separation for different values of the density enhancement, $\rho _{\rm i}/\rho _{\rm e}$ . a) and b) $\rho _{\rm i}/\rho _{\rm e} = 5$ ; c) and d) $\rho _{\rm i}/\rho _{\rm e} = 8$ ; e) and f) $\rho _{\rm i}/\rho _{\rm e} = 10$ .
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In the text

$\begin{figure} \par\includegraphics[width=5cm,clip]{5227f41.eps}\par\includegrap... ...,clip]{5227f42.eps}\par\includegraphics[width=5cm,clip]{5227f43.eps}\end{figure}$ Figure 15: a) Oscillatory period, b) beating period and c) beating factor, for the density contrasts $\rho _{\rm i}/\rho _{\rm e} = 3$ (solid line), $\rho _{\rm i}/\rho _{\rm e} = 5$ (dashed line), $\rho _{\rm i}/\rho _{\rm e} = 8$ (dot-dashed line) and $\rho _{\rm i}/\rho _{\rm e} = 10$ (three-dot-dashed line). Note that the curves start at the slab separation for which the fundamental antisymmetric mode transforms from leaky into trapped.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{5227f1.eps}\end{figure}$	Figure 1: Sketch of the two-slab system. The shaded area represents the density enhancement of the two slabs while the hatched area represents the photospheric medium, that fixes the feet of the slabs and produces the line-tying effect.
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$\begin{figure} \par\mbox{\hspace{-0.cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspace{... ...10.eps}\hspace*{4mm}\includegraphics[width=4.35cm,clip]{5227f11.eps}\end{figure}$	Figure 4: Time-evolution of the velocity, v_x, for a distance between slabs d=0.5 Land a symmetric initial impulse.
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$\begin{figure} \par\mbox{\hspace{-0.cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspace{... ...16.eps}\hspace*{4mm}\includegraphics[width=4.25cm,clip]{5227f17.eps}\end{figure}$	Figure 6: Time-evolution of the velocity, v_x, for a distance between slabs d=0.5 Land an antisymmetric initial impulse.
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$\begin{figure} \par\mbox{\hspace{-0.cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspace{... ...22.eps}\hspace*{4mm}\includegraphics[width=4.25cm,clip]{5227f23.eps}\end{figure}$	Figure 8: Time-evolution of v_x for d=0.5 L and a non-symmetric initial excitation.
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$\begin{figure} \par\includegraphics[width=5.25cm,clip]{5227f24.eps}\end{figure}$	Figure 9: v_x measured at the centres of the slabs for the simulation shown in Fig. 8. The solid and dashed lines correspond to the right and left slabs, respectively.
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$\begin{figure} \par\mbox{\hspace{-0.cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspace{... ...27.eps}\hspace*{4mm}\includegraphics[width=4.25cm,clip]{5227f28.eps}\end{figure}$	Figure 10: Time evolution of v_x for d=2L and a non-symmetric initial disturbance. Note the interchange of energy between the two slabs in the last two frames.
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$\begin{figure} \par\includegraphics[width=5.15cm,clip]{5227f33.eps}\par\includegraphics[width=5.15cm,clip]{5227f34.eps}\end{figure}$	Figure 13: a) Superposition of v_x measured at the centre of the right slab from the time-dependent numerical simulation (solid line) and the analytical approximation (dashed line). b) The same for the left slab.
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$\begin{figure} \par\mbox{\hspace{-0.02cm}{\bf (a)}\hspace{4.25cm}{\bf (b)}\hspac... ...39.eps}\hspace*{4mm}\includegraphics[width=4.28cm,clip]{5227f40.eps}\end{figure}$	Figure 14: Plots of $\omega _{\rm R}$ ( left column) and $\omega _{\rm I}$ ( right column) as functions of the slab separation for different values of the density enhancement, $\rho _{\rm i}/\rho _{\rm e}$ . a) and b) $\rho _{\rm i}/\rho _{\rm e} = 5$ ; c) and d) $\rho _{\rm i}/\rho _{\rm e} = 8$ ; e) and f) $\rho _{\rm i}/\rho _{\rm e} = 10$ .
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