Scattering of stress waves by a circular elastic cylinder embedded in an elastic medium is investigated. The axis of the scatterer is perpendicular to the propagation vector of the incident plane compressional stress pulse wave. Making use of modified Kirchhoff’s integral formulas developed for elastodynamics by Ko [1], wave-front stresses and displacements during the early stage of interaction are obtained for both interior and exterior fields, and for the scatterer-medium interface. The solutions are valid for the whole spectrum of material properties of the scatterer ranging from void to infinitely dense materials. It is found that Kirchhoff’s method of retarded potentials predicts singular wave-front response at caustics as does the geometrical acoustics. The basic integral equations presented are applicable to a scatterer of arbitrary shape and do not only give the wave-front solution, but also the solutions after the arrival of the wave front.

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