A　spectral　approach　was　presented　in　the　computation　of　dispersion　curves　for　the　general　anisotropic　hollow　cylinders.　The　derivation　is　based　on　the　hybrid　method　of　the　state-vector　formalism　and　Legendre　polynomials　expansion,　which　was　previously　adopted　for　the　anisotropic　plates.　This　method　will　lead　to　an　eigenvalue/eigenvector　problem　for　the　calculation　of　wavenumbers　and　displacement　profiles.　This　hybrid　method　avoids　solving　the　transcendental　dispersion　equation.　A　closed-form　solution　for　the　hollow　cylinder,　involving　multiple　integral　expressions,　is　demonstrated.　A　stable　scheme　for　the　integration　expansion　was　established　by　re-expanding　the　expansion　operators　from　the　first　round　Legendre　polynomial　expansion　versus　the　displacements.　Usually,　the　traditional　matrix　methods　are　based　on　root-finding　algorithms,　which　is　difficult　to　implement　in　anisotropic　tubes.　In　this　research,　the　hybrid　approach　we　proposed　provides　a　reliable　mathematical　solution　of　wave　propagations　in　an　anisotropic　hollow　cylinder.　Applications　will　be　illustrated　using　isotropic　and　orthotropic　hollow　cylinders,　in　which　the　isotropic　case　agrees　well　with　the　results　by　global　matrix　method.　The　dispersion　curves　of　orthotropic　hollow　cylinders,　when　the　out　radius　set　to　approximate　infinity,　are　compared　to　its　corresponding　anisotropic　plate,　which　is　obtained　from　our　previous　work.　Furthermore,　the　displacement　and　stress　profiles　will　be　given　and　analyzed　for　an　orthotropic　tube,　which　has　10　mm　thickness　with　an　out　radius　of　50　mm.　©　2018　Elsevier　Ltd