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We study axial density wave propagation along self-gravitating fluid filaments, such as those found in molecular cloud complexes. We also study the linear stability of such filaments. Using the inhomogeneous nonlinear equilibrium of a self-gravitating cylindrically symmetric fluid, we perturb the equilibrium and solve the linearized Euler equations for an inviscid fluid. Numerical solutions and dispersion relations are found for various barotropic equations of state. Observed velocity dispersions of molecular clouds suggest adding a ``turbulence'' term to the normal isothermal term, i.e.\ the total pressure can be written as $p = c_{s}^{2} \rho + p_{0}{\rm log}(\rho/\rho_{0})$. Finally, we discuss observable characteristics of periodic structure in filamentary molecular clouds modelled in this fashion. In particular, we consider the core spacing and the spatial profiles of the waves.
