When investigating thin materials with pulse echo ultrasound, multiple reflections (reverberations) from the layer(s) will overlap. It is therefore difficult to deduce information about speed of sound, thickness, density, etc. from the raw data. In order to extract this information, the overlapping pulses must be either decoupled or we must find some model of the material sample describing the wave propagation. It is, however, often reasonable to assume that the the number of reflections is small relative to the number of samples in the record signal of interest. In other words, the system describing the reverberations is sparse. In this paper we investigate, with simulations and with experiments on a 4.8 and 2.2 mm thick glass plate, respectively, how the framework of compressed sensing can be adopted in order to retrieve the impulse response of the material specimen