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              <p>We investigate the design of passive acoustic liners for targeted absorption in a one-dimensional (1D) transmission problem. The liner consists of an array of Helmholtz resonators. It can be demonstrated that, fundamental limitations imposed by causality and stability leads to sum rules of the 1D system and establish a direct relationship between the transmission spectrum and the total volume of the resonators. Unlike the 1D reflection problem, where the reflection spectrum determines the theoretical minimum total volume, sum rules derived for the transmission problem reveal that the total volume is fixed when the transmission spectrum of the liner is specified. This finding highlights that among all liner designs achieving the same targeted absorption spectrum, the reflectionless design (characterized by minimum reflectance and maximum transmittance) minimizes the total volume. The concept is validated through specific examples involving both single-resonator and multi-resonator configurations.</p>
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