A finite element technique (FEM) is proposed for solving the two-point ordinary differential equation (ODE) boundary value problem for the wavefield in a layered horizontally homogeneous fluid-solid medium at fixed horizontal wavenumber (k). The medium may consist of an arbitrary mixture of fluid and solid layers, bounded at both ends by homogeneous half-spaces. In homogeneous layers the FEM basis functions are chosen to be exact local solutions of the governing source-free ODE's, and in a medium composed of homogeneous layers only the overall solution, at fixed horizontal wavenumber (k), is exact without subdivision of the layers. In layers with non-constant material parameters, standard second order accurate linear elements are used. The FEM stiffness matrix is symmetric with dimension and bandwidth essentially half of those obtained with a parallel shooting global matrix technique. The stiffness matrix remains bounded and well conditioned as the lengths of exact elements grow, and it can, in general, be decomposed into triangular factors without scaling or pivoting, which contributes significantly to computational efficiency. The use of the finite element technique for wavefield synthesis by Hankel transform integration or by normal mode expansion is discussed. In both cases, general purpose numerical quadrature and zero-finding techniques can be used if proper care is taken of the poles of the stiffness matrix in the complex (k) plane. A pole-free dispersion function, efficiently computable from the FEM stifness matrix, is derived. A simple but useful general result in the theory of non-linear eigenvalue problems for matrices is pointed out and used for computing the excitation coefficients of the modes of the discretized system. The usefulness of the method is illustrated by presenting numerical results from a computationally intensive application example.