[en] Improved formulation are presented for the Finite Element Analysis of the linearized coupled structure-fluid with free surface interaction problem, during earthquake or flow induced vibrations. Eulerian coordinates are used for the compressible or incompressible fluid, and Lagrangian coordinates for the shell/solid containment and free surface waves. Assuming small displacements in the fluid, modes shapes and frequencies of the coupled system are obtained. The improvements reside in a systematic way of removing zero frequency modes by Lagrange multipliers, and the treatment of free surface waves in Lagrangian coordinates, resulting in a symmetric matrix equation system solvable with standard structural finite element programs. Examples are presented with PAM-AX3D, a finite element program for axisymmetric geometries and Fourier decomposition of non-axisymmetric earthquake loads