Motivated by the filamentary structure of solar magnetic fields, this paper considers the propagation of quasi-longitudinal waves along a slender magnetic tube in pressure equilibrium with the ambient medium. The velocity of such waves is expressed in terms of the Alven speed and the sound speed inside the tube. The divergence of a vertical magnetic tube in a stratified atmosphere introduces a critical frequency below which propagation is impossible.
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Richard J. Defouw (1976) studied this question.