Sound reduction of side-branch resonators: An energy-based theoretical perspective

For over a century, side-branch resonators have served as effective acoustic filters, yet the explanation for their sound reduction capability has varied. This paper introduces a novel theory applicable to all types of side-branch resonators from an energy perspective and explains sound reduction as a consequence of acoustic energy redistribution. Our theory posits that a standing wave inside the resonator induces air vibration at the opening, which then acts as a secondary sound source, emitting acoustic energy predominantly in the form of kinetic energy. Due to the formation process of the standing wave, the sound wave generated by the resonator undergoes a phase shift relative to the original sound wave in the main pipe. Consequently, this generated sound wave, while matching the amplitude, possesses an opposite phase compared to the original noise wave within the main pipe. This antiphase relationship results in the cancellation of sound waves when they interact post-resonator in the main pipe. Our theory, grounded in an energy perspective, is derived from the principles of standing wave vibration and energy conservation.