Electromagnetic interrogation techniques have numerous useful applications, including locating mines or bunkers beneath the ground, and detecting abnormal tissue noninvasively within the body. Several recent successful such techniques involve using some type of interface, such as a superconductive metal backing or a standing acoustic wave grating, to reflect an oncoming electromagnetic wave. These electromagnetic wave reflections are then used to identify dielectric properties (conductivity and polarization) of the target materials. Many useful polarization models (for example, Debye, Lorentz, and higher order models) result in a hysteretic term in Maxwell’s equations. Thus, a wide class of electromagnetic interrogation problems involve identification of hysteresis mechanisms. We wish to examine a technique in which an acoustic wave traveling toward the oncoming electromagnetic wave acts as a virtual interface. As a first step in assessing this interrogation technique, we consider the equations describing an acoustic pressure wave produced by a windowed sine wave pulse traveling through a layered medium and develop computational methods for solving these equations. After considering several other approaches, we suggest that an adequate way to solve this system with the finite element method is to use a fully Galerkin scheme in a nonstandard weak formulation. Our approximation methods, resulting in large algebraic systems, are explained and computational findings are presented which support the efficacy of the approach.