Alternative sum rules and waterbed effects of Lorentz resonator system for sound absorption and transmission in a unidimensional waveguide

We investigate fundamental constraints on passive linear time-invariant acoustic systems through the developing alternative linear sum rules for sound absorption and transmission. Our approach, based on the Herglotz function method, yields integral identities without non-linear logarithmic terms or frequency weightings, providing clearer physical insights into system performance limits. The study focuses on unidimensional waveguides with Lorentz resonators, encompassing various practical acoustic structures. The developed sum rules are found to be particularly effective in predicting constraints on the average sound absorption coefficient for broadband absorbers operating in deep-subwavelength structures. Based on these rules, we demonstrate the waterbed effect in such systems, highlighting the inherent compromises between absorption efficiency, bandwidth, and device thickness. Through case studies of resonator arrays and membranes, we illustrate the practical implications of these new sum rules for designing optimal sound absorbers and isolators. The work concludes with a discussion on the challenges and future prospects in passive noise control, suggesting potential pathways to surpass current performance boundaries.