A　rigorous　solution　is　proposed　for　the　kinematic　transverse　response　of　a　single　floating　pile　in　the　two-layered　soil　by　vertically　propagating　S-waves.　The　soil　displacements　of　the　two　layers　are　solved　by　Helmholtz　decomposition　and　the　method　of　separation　of　variables.　The　soil　mode　functions　with　a　series　of　eigenvalues　are　calculated　and　satisfy　the　weighted　orthogonality.　The　unknown　coefficients　in　the　solution　functions　are　obtained　in　light　of　the　boundary　conditions.　An　infinite　series　of　algebraic　equations　with　unknown　coefficients　can　be　solved　by　truncating　the　series　number.　The　cutoff　frequency,　soil　reaction　factor,　and　pile　curvature　along　the　shaft　are　investigated　for　different　properties　of　the　adjacent　soil.　For　the　pile　in　the　soil　with　soft　upper　and　stiff　lower　layers,　the　dimensionless　cutoff　frequency　is　smaller　than　that　of　homogeneous　soil,　but　it　follows　opposite　rules　when　the　upper　layer　is　rigid.　©　2023　Elsevier　Ltd