The problem of maximizing or minimizing the amplitude of stress waves propagating through a one-dimensional elastic layered structure is investigated. The properties of layers in series, situated between free and fixed surfaces, are used in deriving difference equations that relate the applied stress wave form at the free surface to the transmitted stress wave form at the fixed surface along characteristic paths. Optimal material requirements are determined for the first transmitted stress wave, which strongly influences the subsequent propagation. Similarity parameters are derived by transform methods which provide optimization criteria for the two-layer case. Materials are systematically selected that can provide stress amplitude reductions of more than 99 percent.