A half space composed of two joined elastic quarter spaces of different material properties is subjected to time-varying shear tractions, which are applied parallel to the plane of juncture. The method of homogeneous solutions is used to solve the transient wave-propagation problem for the case that the surface tractions vary in time as Dirac delta functions. For surface tractions varying in time as Heaviside step functions, Duhamel superposition is employed to derive closed-form expressions for the shear stress in the plane of juncture. The interface shear stress shows a logarithmic singularity at the free surface.