A wavelet-based spectral finite element method (WSFEM) is presented that may be used for an accurate and efficient analysis of elastic wave propagation in two-dimensional (2D) structures. The approach is characterised by a temporal transformation of the governing equations to the wavelet domain using a wavelet-Galerkin approach, and subsequently performing the spatial discretisation in the wavelet domain with the finite element method (FEM). The final solution is obtained by transforming the nodal displacements computed in the wavelet domain back to the time domain. The method straightforwardly eliminates artificial temporal edge effects resulting from the discrete wavelet transform and allows for the modelling of structures with arbitrary geometries and boundary conditions. The accuracy and applicability of the method is demonstrated through (i) the analysis of a benchmark problem on axial and flexural waves (Lamb waves) propagating in an isotropic layer, and (ii) the study of a plate subjected to impact loading. The wave propagation response for the impact problem is compared to the result computed with standard FEM equipped with a direct time-integration scheme. The effect of anisotropy on the response is demonstrated by comparing the numerical result for an isotropic plate to that of an orthotropic plate, and to that of a plate made of two dissimilar materials, with and without a cut-out at one of the plate corners. The decoupling of the time-discretised equations in the wavelet domain makes the method inherently suitable for parallel computation, and thus an appealing candidate for efficiently studying high-frequency wave propagation in engineering structures with a large number of degrees of freedom.