The ray-path method is used to analyze the stress-strain state near mine workings acted upon by discontinuous waves. A dynamic failure criterion is proposed for analyzing the stability of mine workings. The efficiency of the approach is demonstrated with a specific example Keywords: mine workings, discontinuous waves, ray-path method Introduction. Problems of theoretical modeling of propagating discontinuous waves are of importance for analyzing the strength and stability of the rock near and the ground support of mine workings subjected to the action of seismic waves and waves generated by blasting operations and rock bursts [4][5][6][18][19][20][21]. The dynamic theory of elasticity solves such problems by studying the propagation and evolution of discontinuity fronts of derivatives of field functions. The discontinuity of a solution in the theory of elasticity is called strong if the discontinuous derivatives are of low order. The nonlinear theory of elasticity terms moving fronts of strong discontinuities shock waves [3,11] because the stresses are zero ahead of their fronts, are finite behind the fronts, and discontinue on the fronts. The linear theory of elasticity calls moving fronts of stress discontinuities weak shock waves. They carry pressure discontinuity with speed of sound [13]. Such dynamic models can be used to describe, say, the propagation and transformation of waves caused by seismic effects, rock bursts, or blasts.Weak shock waves undergo the most complex transformations at interfaces between elastic media with different mechanical properties in which one incident discontinuous wave decomposes into two triples (pairs) of differently reflected and refracted discontinuous waves [7,12,[14][15][16][17]. These waves can be focused or scattered. The discontinuous behavior of the functions describing these waves, evolution of their fronts, and high variability (in time and space) of field functions in its neighborhood makes it very difficult to describe them both in the class of analytic functions and numerically. Therefore, a special role in describing discontinuous waves is played by the ray-path method, which is used in wave theory of geometrical optics [1,9,12,13].1. Problem Formulation. For propagating nonstationary perturbations and discontinuities (jumps) of derivatives of the field function described by the wave equation for a scalar potential j