The　traditional　vibrational　standing-wave　modes　of　beams　and　strings　show　static　overall　contour　with　finite　number　of　fixed　nodes.　The　travelling　wave　modes　are　investigated　in　this　study　of　axially　moving　string　and　beam　although　the　solutions　have　been　obtained　in　the　literature.　The　travelling　wave　modes　show　time-varying　contour　instead　of　static　contour.　In　the　model　of　an　axially　moving　string,　only　backward　travelling　wave　modes　are　found　and　verified　by　experiments.　Although　there　are　n　-　1　fixed　nodes　in　the　n(th)　order　mode,　similar　to　the　vibration　of　traditional　static　strings,　the　presence　of　travelling　wave　phenomenon　is　still　spotted　between　any　two　adjacent　nodes.　In　contrast　to　the　stationary　nodes　of　string　modes,　the　occurrence　of　galloping　nodes　of　axially　moving　beams　is　discovered:　the　nodes　oscillate　periodically　during　modal　motions.　Both　forward　and　backward　travelling　wave　phenomena　are　detected　for　the　axially　moving　beam　case.　It　is　found　that　the　ranges　of　forward　travelling　wave　modes　increase　with　the　axially　moving　speed.　It　is　also　concluded　that　backward　travelling　wave　modes　can　transform　to　the　forward　travelling　wave　modes　as　the　transport　speed　surpasses　the　buckling　critical　speed.