Summary: We consider numerical techniques for a nonsmooth acoustic pressure system arising in electromagnetic interrogation of dielectric materials. We first describe several formulations of coupled electromagnetic/acoustic systems that arise naturally when traveling acoustic interfaces are used as reflectors for pulsed microwave input signals for dielectric property and geometry identification. We then develop fully Galerkin schemes in a somewhat nonstandard variational formulation of only the acoustic system with distributional inputs. Sample computational findings using the resulting large algebraic system are given.