The　early　transient　responses　of　multi-span　stepped　single　walled　carbon　nanotubes　(SWCNTs)　under　impact　loadings　are　studied　by　the　method　of　reverberation　ray　matrix　(MRRM).　The　dynamics　model　of　the　carbon　nanotubes　is　established　in　the　Fourier　phase　space　on　the　basis　of　the　nonlocal　Timoshenko　beam　model.　The　wave　solutions　of　SWCNTs　with　arbitrary　boundary　conditions　are　obtained　by　the　wave　method.　The　reverberation　ray　matrix　of　the　multi-span　stepped　SWCNTs　is　the　product　of　scattering,　phase　and　permutation　matrices,　which　can　be　determined　by　the　impact　loadings,　continuous　conditions　and　boundary　conditions.　The　early　transient　responses　can　be　calculated　by　the　inverse　Fourier　transform　of　the　sum　of　initial　ray　groups.　It　can　be　found　that　the　early　transient　displacement　response　in　the　very　short　time　subjected　to　the　impact　loading　is　very　small,　while　the　transient　transverse　shear　strain　becomes　large　in　the　very　short　time.　The　influences　of　nanotubes　span　number,　nanotubes　type　and　boundary　conditions　on　the　early　transient　responses　of　multi-span　stepped　SWCNTs　are　investigated.　(C)　2013　Elsevier　B.V.　All　rights　reserved.