A　mathematical　model　is　proposed　to　investigate　the　dynamic　behavior　of　an　end-bearing　pile　under　vertical　seismic　loading　in　a　water-pile-saturated　soil　system.　The　model　is　based　on　the　Euler-Bernoulli　beam　theory　and　takes　into　account　the　stress　balance　between　pile　sections,　representing　the　pile　as　a　one-dimensional　rod.　The　overall　response　of　site　is　divided　into　two　components:　the　free-field　response　and　the　scattered-field　response.　The　pile　and　soil　are　considered　as　linearly　viscoelastic　materials　with　hysteresis-based　damping,　while　water　is　modeled　as　a　linear　acoustic　medium.　By　accounting　for　boundary　conditions　involving　force　equilibrium　and　displacement　continuity　at　the　pile-soil　and　water-soil　interfaces,　an　analytical　solution　for　the　system　response　is　derived.　Analytical　expressions　of　pore　water　pressure　and　displacements　are　obtained,　on　the　basis　of　considering　water-soil　interaction　in　both　fields.　A　parameterization　study　is　then　conducted　to　evaluate　the　impact　of　key　parameters　on　the　vibration　characteristics　of　piles　in　a　water-pile-saturated　soil　system.　©　2024　Elsevier　Ltd