As　to　an　ideally　layered　structure　with　a　functionally　graded　piezoelectric　substrate　(material　parameters　change　continuously　along　the　thickness　direction)　and　a　hard　dielectric　layer,　the　existence　and　propagation　behavior　of　transverse　surface　waves　is　studied　by　analytical　technique.　The　dispersion　equations　for　the　existence　of　the　transverse　surface　waves　with　respect　to　phase　velocity　are　obtained　for　electrically　open　and　short　circuit　conditions,　respectively.　A　detailed　investigation　of　the　effect　of　gradient　coefficient　on　dispersion　relation,　electromechanical　coupling　factor　and　penetration　depth　is　carried　out.　It　is　found　by　numerical　examples　that　adjusting　gradient　coefficient　makes　the　electromechanical　coupling　factor　of　the　transverse　surface　waves　achieve　quite　high　values　at　some　appropriate　ratio　values　of　the　layer　thickness　to　the　wavelength,　and　at　the　same　time,　the　penetration　depth　can　be　reduced　to　the　same　order　as　the　wavelength.　(C)　2008　Elsevier　B.　V.　All　rights　reserved.