||The locally resonant material is a composite made with statistically distributed locally resonant unit. It can completely block the elastic wave in certain frequency ranges. It provides a new and promising method for noise and vibration control. In this research, theoretical and experimental investigation has been carried out to characterize the nature of such materials. In theoretical investigation, a new approach was developed to calculate the effective mass density for a sphere-type locally resonant unit, which has a hard sphere core, a soft shell layer surrounding, and a stiff host medium. The composite was developed as low-frequency sound shielding material. Different from the model in the original paper, which was based on wave propagation theory, the new approach was based on the vibration theory. The natural frequency of local resonance unit was calculated and then the composite behavior under harmonic load was investigated. It was shown that the effective mass density of the composite could turn to negative and infinite at the frequency close to resonance. In order to examine the influence of important parameters on the resonant frequency and effective mass density, parameter study has been carried out. It was found that a thicker soft layer with smaller modules could lead to lower resonant frequency. The predictions of wave spectral gap from the new approach and from the model in the literature approach were almost identical. A new membrane-type of locally resonant unit was fabricated and tested. It was consisted of hard circular plate adhered on the soft membrane that fixed in a rigid frame. It exhibited completely sound transmission gap. An analytical model was developed based on the wave propagation theory to describe the low-frequency effective mass per unit area for the composite material. The model provided a direct explanation of nature of the system and an accurate method to calculate effective mass per unit area of such a system. It was shown that the effective mass per unit area could turn to negative at certain frequency ranges, which led to complete sound attenuation. In order to measure the sound transmission gap position and verify the theoretical model, experimental work has been carried out. It was found that the theoretical predictions agreed well with experimental results.