All Figures

$\begin{figure} \par\includegraphics[width=6.9cm,clip]{8385f1a.eps}\vspace*{3mm} \includegraphics[width=6.9cm,clip]{8385f1b.eps}\end{figure}$ Figure 1: Dispersion diagram plotting non-dimensional frequency $\omega L/c_{\rm Ai}$ against the non-dimensional loop radius a/L for a uniform loop of length 2L and a uniform environment, with a) $c_{\rm Ae}=2.5 c_{\rm Ai}$ and b) $c_{\rm Ae}=4.62 c_{\rm Ai}$ . Sausage modes are shown as dashed curves and kink modes as solid curves; the cutoffs are shown as dotted horizontal lines.
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In the text

$\begin{figure} \par\includegraphics[width=6.9cm,clip]{8385f2a.eps}\vspace*{3mm} \includegraphics[width=6.9cm,clip]{8385f2b.eps}\end{figure}$ Figure 2: Dispersion diagram plotting non-dimensional frequency $\omega L/c_{\rm Ai}$ against the non-dimensional loop radius a/L for an exponentially structured loop and homogeneous environment with $c_{\rm Ae}=2.5 c_{\rm Ai}$ , and a) $\rho _{\rm i}(L)/\rho _{\rm i}(0)=1.6$ or b) $\rho _{\rm i}(L)/\rho _{\rm i}(0)=8$ . Sausage modes are shown as dashed curves and kink modes as solid curves; cutoffs are shown as dotted horizontal lines.
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In the text

$\begin{figure} \par\includegraphics[width=6.9cm,clip]{8385f3.eps}\end{figure}$ Figure 3: Dispersion diagram plotting $\omega L/c_{\rm Ai}$ against a/L for models having uniform environment with $c_{\rm Ae}=2.5 c_{\rm Ai}$ . Solid lines correspond to the uniform loop of Fig. 1a (kink and sausage modes), while dashed lines correspond to the exponentially structured loop with $\rho _{\rm i}(L)/\rho _{\rm i}(0)=8$ , which has been rescaled by a factor 1.43 for comparison purposes.
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In the text

$\begin{figure} \par\includegraphics[width=7.1cm,clip]{8385f4.eps}\end{figure}$ Figure 4: Dispersion diagram plotting as solid lines the non-dimensional frequency $\omega L/c_{\rm Ai}(0)$ against the longitudinal density ratio $\rho _{\rm i}(L)/\rho _{\rm i}(0)$ for an internal density with $c_{\rm Ae}(z=0)=2.5c_{\rm Ai}(z=0)$ and a/L=0.01. Overplotted in a dashed line is the result of a uniform loop (Eq. (24)) with the averaged density given in Eq. (33).
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In the text

$\begin{figure} \par\includegraphics[width=6.9cm,clip]{8385f5.eps}\end{figure}$ Figure 5: Dispersion diagram plotting $\omega L/c_{\rm Ai}$ against a/L for homogeneous loops in a structured environment, with $c_{\rm Ae}$ (z = 0) = $2.5c_{\rm Ai}(z=0)$ . Solid lines are the frequencies for a homogeneous environment ( $\rho _{\rm e}(L)/\rho _{\rm e}(0)=1$ ), dotted lines for an exponentially structured environment with $\rho _{\rm e}(L)/\rho _{\rm e}(0)=10$ , and dot-dashed lines for an exponentially structured environment with $\rho _{\rm e}(L)/\rho _{\rm e}(0)=50$ . The cutoff for each model has been overplotted as a horizontal dashed line.
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{8385f6.eps}\end{figure}$ Figure 6: Dispersion diagram for a uniform loop in a exponentially structured environment, plotting $\omega L/c_{\rm Ai}(0)$ against the external density structuring $\rho _{\rm e}(L)/\rho _{\rm e}(0)$ , for $c_{\rm Ae}(0)=2.5c_{\rm Ai}(0)$ and a loop radius of a/L=0.01.
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{8385f7.eps}\end{figure}$ Figure 7: Dispersion diagram plotting $\omega L/c_{\rm Ai}$ against the a/L for a) an exponentially structured loop and environment with $c_{\rm Ae}(z=0)=2.5c_{\rm Ai}(z=0)$ , $c_{\rm Ae}(z=L)=c_{\rm Ai}(z=L)$ and $\rho _{\rm e}(L)/\rho _{\rm e}(0)=100$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{8385f8.eps}\end{figure}$ Figure 8: Dispersion diagram plotting $\omega L/c_{\rm Ai}$ against density structuring $\rho _{\rm e}(L)/\rho _{\rm e}(0)$ for a/L=0.01 and $c_{\rm Ae}(z=0)=2.5c_{\rm Ai}(z=0)$ , with an uniform base ( $\rho _{\rm e}(L)=\rho _{\rm i}(L)$ ).
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{8385f9.eps}\end{figure}$ Figure 9: Dispersion diagram plotting $\omega L/c_{\rm Ai}$ against the fractional coronal depth W/L for a/L=0.01, $\rho _{\rm i}(L)/\rho _{\rm i}(0) = 16$ and $c_{\rm Ae}(0)=2.5c_{\rm Ai}(0)$ . The solid lines are the modes of a loop with chromospheric layers embedded in a uniform environment (whose cutoff is the horizontal straight line); the dashed line is the only propagating mode of a uniform loop embedded in an environment with chromospheric layers, while the dot-dashed line corresponds to a loop with chromospheric layers both inside and outside. The cutoff of these last two models has been overplotted as a dotted line.
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In the text

$\begin{figure} \par\includegraphics[width=6.9cm,clip]{8385f10.ps}\vspace*{4mm} \end{figure}$ Figure 10: P₁/2P₂ as a consequence of longitudinal structuring. The density is exponentially stratified along the loop, the solid line has a base density that is 8 times the density at the apex, $\rho _{\rm i}(L)/\rho _{\rm i}(0)=8$ , and the dotted line has a base density that is 16 times the density at the apex, $\rho _{\rm i}(L)/\rho _{\rm i}(0) = 16$ . The tube is structured radially so that $c_{\rm ae}(0)=2.5 c_{\rm Ai}(0)$ , corresponding to a tube density enhancement at the apex of 25/4 times the environment density at the apex.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{8385f11.eps}\end{figure}$ Figure 11: P₁/(2P₂) as a function of the density contrast $\rho _{\rm i}(L)/\rho _{\rm i}(0)$ for a coronal loop structured exponentially in density. Here we have taken a/L=10^-3 and $c_{\rm Ae}(0)=2.5c_{\rm Ai}(0)$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{8385f12.eps}\end{figure}$ Figure 12: P₁/(3P₃) as a function of a/L for a structured loop with $\rho _{\rm i}(L)/\rho _{\rm i}(0)=8$ for two different surrounding media: a homogeneous corona with $c_{\rm Ae}(0)=2.5c_{\rm Ai}(0)$ (dashed line) and a structured corona with $c_{\rm Ae}(0)=2.5c_{\rm Ai}(0)$ and $\rho _{\rm e}(L)/\rho _{\rm e}(0)=8$ (solid line). The point where the second kink harmonic has a cutoff has been labelled in the plot.
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In the text

$\begin{figure} \par\includegraphics[width=6.9cm,clip]{8385f1a.eps}\vspace*{3mm} \includegraphics[width=6.9cm,clip]{8385f1b.eps}\end{figure}$	Figure 1: Dispersion diagram plotting non-dimensional frequency $\omega L/c_{\rm Ai}$ against the non-dimensional loop radius a/L for a uniform loop of length 2L and a uniform environment, with a) $c_{\rm Ae}=2.5 c_{\rm Ai}$ and b) $c_{\rm Ae}=4.62 c_{\rm Ai}$ . Sausage modes are shown as dashed curves and kink modes as solid curves; the cutoffs are shown as dotted horizontal lines.
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$\begin{figure} \par\includegraphics[width=7.1cm,clip]{8385f4.eps}\end{figure}$	Figure 4: Dispersion diagram plotting as solid lines the non-dimensional frequency $\omega L/c_{\rm Ai}(0)$ against the longitudinal density ratio $\rho _{\rm i}(L)/\rho _{\rm i}(0)$ for an internal density with $c_{\rm Ae}(z=0)=2.5c_{\rm Ai}(z=0)$ and a/L=0.01. Overplotted in a dashed line is the result of a uniform loop (Eq. (24)) with the averaged density given in Eq. (33).
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{8385f6.eps}\end{figure}$	Figure 6: Dispersion diagram for a uniform loop in a exponentially structured environment, plotting $\omega L/c_{\rm Ai}(0)$ against the external density structuring $\rho _{\rm e}(L)/\rho _{\rm e}(0)$ , for $c_{\rm Ae}(0)=2.5c_{\rm Ai}(0)$ and a loop radius of a/L=0.01.
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{8385f7.eps}\end{figure}$	Figure 7: Dispersion diagram plotting $\omega L/c_{\rm Ai}$ against the a/L for a) an exponentially structured loop and environment with $c_{\rm Ae}(z=0)=2.5c_{\rm Ai}(z=0)$ , $c_{\rm Ae}(z=L)=c_{\rm Ai}(z=L)$ and $\rho _{\rm e}(L)/\rho _{\rm e}(0)=100$ .
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{8385f8.eps}\end{figure}$	Figure 8: Dispersion diagram plotting $\omega L/c_{\rm Ai}$ against density structuring $\rho _{\rm e}(L)/\rho _{\rm e}(0)$ for a/L=0.01 and $c_{\rm Ae}(z=0)=2.5c_{\rm Ai}(z=0)$ , with an uniform base ( $\rho _{\rm e}(L)=\rho _{\rm i}(L)$ ).
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{8385f11.eps}\end{figure}$	Figure 11: P₁/(2P₂) as a function of the density contrast $\rho _{\rm i}(L)/\rho _{\rm i}(0)$ for a coronal loop structured exponentially in density. Here we have taken a/L=10^-3 and $c_{\rm Ae}(0)=2.5c_{\rm Ai}(0)$ .
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