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$\begin{figure} \par\includegraphics[scale=0.48]{8016f1.eps} \end{figure}$ Figure 1: The period ratio P₁/2P₂ for a thin tube with exponential longitudinal structuring. The dashed curve corresponds to the numerically determined solution of the dispersion relations (9) and (10). The solid curve corresponds to the analytical approximation Eq. (46).
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In the text

$\begin{figure} \par\includegraphics[scale=0.48]{8016f2.eps} \end{figure}$ Figure 2: The period ratio P₁/2P₂ as a function of the density ratio $\rho _{\rm apex}/\rho _{\rm base}$ of internal density $\rho _{\rm i}(z=0)$ at the loop apex to density $\rho _{\rm i}(L)$ at the loop base. The solid curve corresponds to the exponentially structured density profile, numerically determined by the solution of dispersion relations (9) and (10). Also shown for comparison are the results for a selection of other profiles, considering the linear profile (dashed line), the profile discussed by Dymova & Ruderman (2006a), Eq. (48) (dotted line), the $\tanh$ profile (Eq. (50)) with kL=1 (dot-dashed line) and the profile discussed by Andries et al. (2005b) (Eq. (49), dot-dot-dashed line); for these profiles we have used the variational formulation.
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In the text

$\begin{figure} \par\includegraphics[scale=0.48]{8016f1.eps} \end{figure}$	Figure 1: The period ratio P₁/2P₂ for a thin tube with exponential longitudinal structuring. The dashed curve corresponds to the numerically determined solution of the dispersion relations (9) and (10). The solid curve corresponds to the analytical approximation Eq. (46).
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