The classical sharp-blow wave in one-dimensional gas dynamics portraying disturbances following a surface impact or explosion forecasts the severest possible power-law decay in the strength of a leading shock wave entering a uniform medium. In a Lagrangian formulation, the present work establishes, for the same decay law, a new class of solutions for which the existence of an interior trailing shock discontinuity in response to a boundary back-pressure presence is a central thesis. It is determined that if the back pressure lies below a certain critical level relative to the frontal pressure, the strength of the trailing discontinuity is invariant but is inversely related to the magnitude of the relative back pressure. At the critical pressure level, the solution is at its weak-interior-shock limit, recognizable as the continuous wave previously discovered by Adamskii and Popov ( Appl. Math. PMM Vol. 23, pp. 793–806). With vanishing boundary pressure for the extreme case, the solution formally reaches its strong-interior-shock limit while physically reducing to the sharp-blow streaming description. In between the two limits, representative wave domains have been evaluated for γ = 7 5 , where solutions are partially analytic. In each of the cases considered, the wave domain lying beyond the trailing shock wave is compressive and piston-like medium termination is achieved.