On the Eshelby-Kostrov property for the wave equation in the plane

Abstract

Abstract. This work deals with the linear wave equation considered in the whole plane R 2 except for a rectilinear moving slit, represented by a curve Γ (t) = {(x 1 , 0) : −∞ < x 1 < λ (t)} with t ≥ 0. Along Γ (t) , either homogeneous Dirichlet or Neumann boundary conditions are imposed. We discuss existence and uniqueness for these problems, and derive explicit representation formulae for solutions. The latter have a simple geometrical interpretation, and in particular allow us to derive precise asymptotic expansions for solutions near the tip of the curve. In the Neumann case, we thus recover a classical result in fracture dynamics, namely the form of the stress intensity factor in crack propagation under antiplane shear conditions.

Gusev’s Stochastic Model for the Seismic Source: High-Frequency Behavior in the Far Zone

“…As to the active rupture phase, the classical models by Kostrov [1964] and Madariaga [1977Madariaga [ , 1983 demonstrate singularities of the 'inverse square root' type,  / 1 . A rigorous justification of the singularity in the framework of elasticity theory can be found in [Herrero et al, 2006]. Combining the two cases, we shall consider the source function around , and the spectral amplitude depends on radiation directivity via the stationary point.…”