The present study outlines a new approach to collecting shock Hugoniot data in foams using photonic Doppler velocimetry to perform mid-plane measurements of the foam deformation. Plate impact experiments were carried out to investigate wave propagation in a closed-cell polymeric foam and an open-cell aluminum foam. Dual-wave structures were observed in both materials with the leading precursor wave determined to be an elastic wave. The discussion of the results focuses on the nature of foam compression under high-rate loading, particularly the difference between the strain history in a foam undergoing uniform stress compaction and uniaxial strain compression. These results are discussed in reference to the current interpretations of Taylor-Hopkinson bar experiments on similar metallic foams. The importance of gas-filtration driven flows in the wave dynamics of open-cell foams is discussed in relation to the nature of the precursor waves.