The multiresolution finite wavelet domain method has been meticulously studied in wave propagation simulations. The multiresolution procedure always starts with the coarse solution, and then finer solutions can be superimposed on the coarse solution, until convergence is achieved. Based on remarkable observations on the multiple resolution components of the method, a residual-based convergence indicator that reveals convergence at the coarse solution is developed. This convergence metric is rapidly applicable and straightforward and can also divulge the spatial and temporal ranges/domains that the already obtained solution needs to be enhanced. In that way, an automatic adaptive refinement technique is proposed for the local enrichment of the solution, only in the specific grid points and time-steps that it is needed. A numerical case study regarding wave propagation in an inhomogeneous rod manifests the effectiveness and accuracy of the proposed automatic refinement methodology, as also the performance of the suggested convergence indicator.