We consider the sequential reconstruction of source waveforms under a sparsity constraint from a Bayesian perspective. We assume that the wave field that is observed by a sensor array is from a spatially sparse source distribution. A spatially weighted Laplace-like prior is assumed for the source distribution and the corresponding weighted LASSO cost function is derived. We demonstrate the sequential sparse sampling using a line array and track the direction of arrival. In a real world example we track a source using a 2D array.