  Spatially exponential distributions of material properties are ubiquitous in
many natural and engineered systems, from the vertical distribution of the
atmosphere to acoustic horns and anti-reflective coatings. These media
seamlessly interface different impedances, enhancing wave transmission and
reducing internal reflections. This work advances traditional transfer matrix
theory by integrating analytical solutions for acoustic exponential materials,
which possess exponential density and/or bulk modulus, offering a more accurate
predictive tool and revealing the physical mechanism of broadband
anti-reflection for sound propagation in such non-uniform materials. Leveraging
this method, we designed an acoustic dipole array that effectively mimics
exponential mass distribution. Through experiments with precisely engineered
micro-perforated plates, we demonstrate an ultra-low reflection rate of about
0.86% across a wide frequency range from 420 Hz to 10,000 Hz. Our modified
transfer matrix approach underpins the design of exponential materials, and our
layering strategy for stacking acoustic dipoles suggests a pathway to more
functional gradient acoustic metamaterials.
