Structured shock waves and the fourth-power law

A fourth-power law relating the stress jump through a steady structured shock wave and the maximum strain rate within the shock wave has received recognition as a unifying relation over a sensibly wide range of materials and shock compression amplitudes. Less widely recognized is the complementary applicability of the invariance of the product of the energy dissipated and the time over which this energy is dissipated through a structure shock wave—a property with the dimensions of action. While the latter invariance implies the fourth-power behavior for steady shock waves, this invariance also has an apparent application to unsteady structured waves under certain conditions. A brief history of the origin of the fourth-power law is provided. Some commentary is offered on the physical principles of solid viscosity in the shock wave and the underlying invariance of the energy-time product in the shock wave event. The results for steady structured waves in porous and composite solids, which do not in general exhibit fourth-power behavior, are examined. Observations of systematic behaviors with component cell size and impedance disparities suggest acoustic scattering may contribute to wave structuring in selected materials.