In　order　to　inspect　longitudinal　cracks　of　anisotropic　pipes　more　efficiently,　guided　circumferential　waves　in　infinitely　long　orthotropic　hollow　cylinders　are　studied　by　an　orthogonal　polynomial　approach.　The　formulation　is　based　on　linear　three-dimensional　elasticity.　Results　are　compared　with　the　finite　element　simulation　to　check　up　the　accuracy　and　range　of　applicability　of　this　polynomial　approach.　The　group　velocity　dispersion　curves　and　displacements　distributions　of　guided　circumferential　waves　for　orthotropic　cylindrical　curved　plate　are　calculated　and　appropriate　frequency　and　modes　that　can　be　used　to　detect　a　longitudinal　defect　are　found.　Finally,　finite　element　methods　are　used　to　simulate　the　detection　and　location　of　a　longitudinal　defect.