## 1 Introduction  
Closed cell aluminum foams are known to absorb a significant amount of energy during quasi-static compression by undergoing a large amount of plastic deformation at low transmitted stress [1]. Plastic deformation occurs in bands of cells by cell wall bending and buckling, causing cell collapse and foam densification [2]. [PERSON] et al. [3] and [PERSON] et al. [4] carried out mathematical analyses and limited experiments of head-on collisions of planar shock waves of Mach 1.08-1.41 in a rectangular shock tube with ultra-low density, (0.03 g/cc) polyurethane foams placed against the end-wall of a shock tube. They found that the reflected pressure from the front face of the foam was less than that from the end-wall of the empty shock tube, which was termed as \"shock attenuation\". However, the pressure generated by the wave after transmission through the foam, reflection from the end-wall, transmission back through the foam, and re-emergence from the front face was found to be more than the reflected pressure off the end-wall of the empty shock tube, which was termed as \"shock amplification\". According to the authors, shock amplification occurs while the foam deformation is restricted to the plateau domain, i.e., the region in the compressive stress-strain relationship of foams in which large strains occur at low and near-constant stress, since under such conditions compression waves inside the foam can transform into shock waves. [PERSON] and [PERSON] [5] studied the compaction behavior of high density (0.7 to 1.4 g/cc) porous aluminum and found the permissible momentum and minimum areal density of porous aluminum required to prevent the plastic waves from reaching the anvil. [PERSON] and [PERSON] [6] studied the behavior of high relative density (0.4-0.8, i.e., in the absolute density range 1.1-2.2 g/cc) porous aluminum under shock loading and calculated the Hugoniot conditions, which are shown to be dependent on strength and coherence of the solid matrix. The porous aluminum was shown to condense to its full density after exposure to about 10 kbar of shock pressure, generated using impact of projectiles on targets. [PERSON] et al. [7] studied the elastic-plastic behavior of polyurethane foams on blast wave and shock tube loading. They observed that a compaction wave induced in the foam due to shock loading exerted a greater stress on the end-wall of the shock tube, than the incident shock wave in the absence of the foam (the baseline case). In other words, a shock-exposed polyurethane foam amplified the stress on the posterior object. They noted that if several interfaces with increasing impedances exist in the path of a compression wave, the stress generated at each interface gets added, and results in stress amplification. Thus, in an air-foam-steel system, where the foam has an impedance value intermediate to those of air and steel, a higher amount of stress is expected on the steel when a shock wave generated in air impacts the foam. Gas filtration through granular layers during weak shock wave impact was studied by [PERSON] et al. [8]. Sensors placed in front of the granular medium showed an initial steep rise in pressure, the amplitude of which was inversely proportional to the permeability of the medium, followed by gradual rise reaching up to a steady value, which is attributed to gas filtration and reflection of the compaction wave from the rigid wall at the end of the shock tube. The pressure at the rigid wall, measured by an unprotected sensor inserted into the granular medium, shows an unsteady steep rise in pressure, attributed to the effect of compaction wave on the gas pressure, followed by a drop and attainment of a steady value due to gas filtration. Effect of length of a granular material medium on attenuation of blasts was studied by [PERSON] et al. [9]. They showed that the relative exit overpressure decreased with increase in length of the granular medium and decrease in Mach number of the shock wave.