We analyze the convergence of finite element discretizations of time-harmonic wave propagation problems. We propose a general methodology to derive stability conditions and error estimates that are explicit with respect to the wavenumber. This methodology is formally based on an expansion of the solution in powers of k, which permits to split the solution into a regular, but oscillating part, and another component that is rough, but behaves nicely when the wavenumber increases. The method is developed in its full generality and is illustrated by two particular cases: the elastic and convected sound waves. Numerical experiments are provided which confirm that the stability conditions and error estimates are sharp.