Based　on　linear　three-dimensional　elasticity,　the　wave　motion　equations　for　coupled　longitudinal　guided　waves　are　derived　by　using　a　Legendre　orthogonal　polynomial　approach,　and　the　coupled　equations　of　wave　motion　are　solved　numerically.　Firstly,　to　determine　the　applicability　and　accuracy　of　this　method,　the　dispersion　curves　of　longitudinal　guided　waves　in　isotropic　hollow　cylinders　were　solved　by　using　Disperse　software.　The　results　obtained　by　Disperse　calculation　are　compared　with　those　by　Legendre　orthogonal　polynomial　approach　method.　The　calculation　results　of　the　two　methods　are　consistent.　Then,　the　influence　of　the　Legendre　polynomial　cutoff　value　on　the　convergence　of　the　axisymmetric　guided　wave　dispersion　curves　is　discussed.　The　reason　of　the　influence　is　analyzed　by　the　numerical　calculation.　Finally,　for　the　composite　hollow　cylinder　wound　by　carbon　fiber,　the　dispersion　curves　of　three　different　modes　of　longitudinal　guided　waves　are　calculated　separately.　The　phase　velocity　dispersion　curves　for　different　ratios　of　outer　radius　to　thickness　are　calculated　to　analyze　the　influence　of　the　ratio　on　the　dispersion　curves.　©　2018,　Engineering　Mechanics　Press.　All　right　reserved.