This paper employs an analytical approach to study vibration problems in continuous systems. For the riser in deep-water drilling， a second-order partial differential equation for the vibration of its continuous elastic system is established. This equation considers the forces and constraints acting on the riser as a homogeneous continuous elastic body along its axial direction， including constraints at the seabed wellbore， the influence of the riser tensioner， and the vertical heave motion of the drilling platform due to waves. Based on this， a differential equation for the longitudinal vibration of the rod member is further developed， and the frequency equation for the longitudinal vibration of the riser is solved. The research results indicate that the longitudinal vibration frequency of the deep-water drilling riser is related to six main parameters： the cross-sectional area of the riser， the Young's modulus of the pipe body， the volumetric mass， the spring coefficient of the riser tensioner， the wave height of wave motion， and the water depth. This study has important practical significance for gaining a deeper understanding of the longitudinal vibration characteristics of the riser and scientifically guiding drilling operations.