This paper presents the results from a Lagrangian numerical scheme of calculating cylindrically symmetric, elastic‐plastic, transient disturbances. The numerical technique involves transforming the Eulerian equations of motion containing a cylindrically symmetric stress tensor into Lagrangian coordinates. These transformed equations are then differenced in the Lagrangian coordinate system. Therefore, a given instantaneous stress field determines the accelerations of the points in the Lagrangian mesh. These accelerations are allowed to act over a small time‐step so that the mesh becomes distorted. This distortion determines the change in strain at a point; the strain change is related to a stress change by Hooke’s law. This new stress is used to determine new accelerations. If the material behaves plastically, then the stresses are adjusted so that a von Mises yield condition is satisfied. Five numerical solutions of various elastic‐plastic wave propagation problems are presented; these solutions agree extremely well with their corresponding analytic and experimental solutions. A shear and tensile failure mechanism is presented that is consistent with the continuum hypothesis. This mechanism gives good results when applied to NTS alluvium. Data for hard rock are not yet available.