We propose a new type of multilayered resonant structure that is named as tail-resonant structure. It is composed of two components, i.e., a tail component and a resonance component. Compared with traditional monolayered resonant structures, tail-resonant structures have a unique advantage, i.e., it can realize the effective coupling of waves from one medium with large impedance to the other medium with extremely small impedance. By using analytical and numerical methods, we discuss propagation properties of elastic waves in two kinds of tail-resonant structures. In these resonant structures, the tail component can amplify the amplitude of waves; the resonant component can choose proper frequencies of waves for realizing the effective coupling of waves between two media with large impedance mismatching.