By using a new method based on perturbation method and Fredholm alternative theorem, the eigenvalues and eigenfunctions of nonuniform Euler-Bernoulli beam vibration are solved analytically. The eigenvalues and eigenfunctions under boundary conditions are given for the tension beam problem which considers the variation of tension in vertical marine riser from top to bottom caused by gravity effect. The asymptotic analytical results are compared with classical results to verify the effectiveness of the new method based on perturbation method. The influence of the change of the bottom tension on the perturbation method is analyzed. The results show that: for each order of natural frequency of riser, the natural frequency will gradually increase with the increase of bottom tension, and the obtained natural frequency will become more and more accurate; when the bottom tension is constant, the new method based on perturbation method in this paper can give good accuracy for higher order natural frequency rather than lower order.