Fracture of a liquefied crack and the physics of Rayleigh waves

The standard free-surface boundary conditions for in-plane crack dynamics are shown to be identical to the conditions for crack dynamics on a liquefied crack. The surfaces of both the free and liquefied cracks do not separate during faulting and hence the static normal stress is not relaxed by the faulting. A crack with either free or liquid boundary conditions deforms in the transverse direction during slip. It follows that both the free and liquefied cracks may represent solutions to the heat-flow paradox. As an application of the proof, we derive a physical understanding of the properties of harmonic Rayleigh waves on a uniform elastic half-space without solving a cubic equation.