SUMMARYThe transient deformation of an elastic half-space under a line-concentrated impulsive vector shear load applied momentarily is disclosed in this paper. While in an earlier work, the author gave an analyticalnumerical method for the solution to this transient boundary-value problem, here, the resultant response of the half-space is presented and interpreted. In particular, a probe is set up for the kinematics of the source signature and wave fronts, both explicitly revealed in the strained half-space by the solution method. The source signature is the imprint of the spatiotemporal configuration of the excitation source in the resultant response. Fourteen wave fronts exist behind the precursor shear wave S: four concentric cylindrical, eight plane, and two relativistic cylindrical initiated at propagating centres that are located on the stationary boundaries of the solution domain. A snapshot of the stressed half-space reveals that none of the 14 wave fronts fully extend laterally. Instead, each is enclosed within point bounds. These wave arresting points and the two propagating centres of the relativistic waves constitute the source signature. The obtained 14 wave fronts are further combined into 11 disparate wave fronts that are grouped into four categories: an axis of symmetry wave}so named here by reason of being a wave front that is contiguous to the axis of symmetry, three body waves, five surface waves and two inhibitor waves}so named here by reason that beyond them the material motion dies out. Of the three body waves, the first is an unloading shear wave, the second is a diffracted wave and the third is a reflected longitudinal two-branch wave. Of the two inhibitor waves, the first is a two-joint relativistic wave, while the second is a two-branch wave. The wave system, however, is not the same for all the dependent variables; a wave front that appears in the behaviour of one dependent variable may not exist in the behaviour of another. It is evident from this work that Saint-Venant's principle for wave propagation problems cannot be formulated. Therefore, the above results are valid for the particular proposed model for the momentary line-concentrated shear load. The formulation of the source signature, the wave system, and their role in the half-space transient deformation are presented here.

SUMMARYFirst, the response of an ideal elastic half-space to a line-concentrated impulsive normal load applied to its surface is obtained by a computational method based on the theory of characteristics in conjunction with kinematical relations derived across surfaces of strong discontinuities. Then, the geometry is determined of the obtained waves and the source signature}the latter is the imprint of the spatiotemporal configuration of the excitation source in the resultant response. Behind the dilatational precursor wave, there exists a pencil of three plane waves extending from the vertex at the impingement point of the precursor wave on the stress-free surface of the half-space to three points located on the other two boundaries of the solution domain. These four wave-arresting points (end points) of the three plane waves constitute the source signature. One wave is an inhibitor front in the behaviour of the normal stress components and the particle velocity, while in the behaviour of the shear stress component, it is a surface-axis wave. The second is a surface wave in the behaviour of the horizontal components of the dependent variables, while the third is an inhibitor wave in the behaviour of the shear stress component. An inhibitor wave is so named, since beyond it, the material motion is dying or becomes uniform. A surface-axis wave is so named, since upon its arrival, like a surface wave, the dependent variable in question features an extreme value, but unlike a surface wave, it exists in the entire depth of the solution domain. It is evident from this work that SaintVenant's principle for wave propagation problems cannot be formulated; therefore, the above results are a consequence of the particular model proposed here for the line-concentrated normal load.

The half-space elastic response to a source of a nonuniformly spatially distributed impulsive load is presented in this paper. Specifically, an infinite semicircular canyon of a finite radius is embedded in the surface of the half-space where an impulsive load is radially distributed over the entire surface of the canyon. The spatial distribution of this load is such that the load is maximum at the axis of symmetry of the half-space and then descends smoothly to zero at the stress-free surface of the half-space. In an earlier work, the author gave an analytical-numerical method for transient multicurvilinear dimensional boundary-value problems and its employment to the present problem. Here, the resultant transient deformation of the half-space is described and interpreted. In particular, a detailed discussion is devoted to the appearance of spatially stationary strong discontinuity fronts in the interior of the deformed half-space. These fronts, which disclose the nature of the prescribed source of disturbance, are called here the source signature. A complete account is then given of the waves emitted from the source signature and those generated by boundary conditions, all revealing themselves explicitly by the solution method.

Geometrical, strong discontinuity formulas are obtained here in order to accommodate multidimensional wave characteristic equations which are the interior differential equations on wave surfaces. As they stand, the multidimensional wave characteristic equations are integrable only in a weak material motion where discontinuities across waves are allowed in first partial derivatives of continuous dependent variables. This paper deals with the extension of these equations in order to make them also compatible with strong material motion where the dependent variables are themselves discontinuous across the wave surfaces. In particular, the discontinuity formulas are obtained for the characteristics computational scheme on Huygen's wavelets. For this purpose, strong discontinuity formulas are derived which do not depend on the motion of a surface. The derivation is guided by the intrinsic properties of the wave characteristic equations and is based on concepts of differential geometry in conjunction with Hadamard's basic proposition of a jump across a surface of discontinuity. The known concepts 0f differential geometry needed for the analysis are obtained directly from the propagation derivative'of the two-point parallel propagator tensor by a unified approach in non-Riemannian geometry.