The sequential Bayesian reconstruction of sparse source waveforms from sensor array data is analyzed. Acoustic waves are observed by a sensor array. The waves are emitted by a spatially-sparse set of sources. A weighted Laplace-like prior is assumed for the sources such that the maximum a posteriori source estimate at the current time step can be approximated from the weighted LASSO. The new weighting for time step $k+1$ is defined from a fit to the approximated posterior distribution at the previous time step $k$. Thus, a sequence of weighted LASSO problems is solved for estimating the temporal evolution of a sparse source field. Finally, we explore M. E. Tipping's approach to fast marginal likelihood maximization for sparse Bayesian models for sequential source waveform reconstruction.