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, two indices have been developed: a residual-based convergence indicator that reveals convergence at the coarse solution, and a displacement-based convergence metric that indicates convergence on the next step of the multiresolution process. Those convergence/error metrics are both 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, targeted and adaptive refinement techniques are proposed for the specific enrichment of the solution, only in the specific regions and time-steps that it is needed. Numerical case studies regarding wave propagation in rod structures manifest the effectiveness and accuracy of the proposed automatic refinement techniques, as also the excellent performance of the suggested convergence indicators.