shear alfv n wave slow alfv n wave torsional alfv n wave magnetosonic wave longitudinal compressional alfv n wave fast alfv n wave magnetoacoustic wave phase velocity depends angle wave vector magnetic field mhd wave propagating arbitrary angle respect time independent bulk field satisfy dispersion relation math frac omega k v cos theta math math v frac b sqrt mu rho math alfv n speed branch corresponds shear alfv n mode
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