Retrieving Effective Acoustic Impedance and Refractive Index for Size Mismatch Samples
Mohammad Javad Khodaei, Amin Mehrvarz, Reza Ghaffarivardavagh, Nader Jalili
TL;DR
This work tackles the problem of extracting effective acoustic properties, specifically the refractive index $n$ and impedance $z$, of metamaterials when the sample size does not match the impedance-tube cross-section. It models the setup as a bilayer metamaterial inside a duct, including an air gap, and derives an analytical retrieval framework based on the transfer matrix method and Helmholtz solution to obtain $n_1$ and $z_1$ by solving eight linear equations from measured transmission $T$ and reflection $R$. The method computes averaged pressures and velocities, leading to explicit expressions for $z_1$ and $n_1$, and is validated by FEM simulations showing high fidelity (approximately $99 ext{%}$ accuracy) in two representative samples. This provides a practical analytical tool for characterizing size-mismatch acoustic metamaterials without requiring custom impedance tubes for each sample.
Abstract
In this paper, we have presented an analytical solution to extract the effective properties of acoustic metamaterials from the measured complex transmission and reflection coefficients when the metamaterial and impedance tube have different sizes. We have first modeled this problem as a bilayer metamaterial located inside a duct and treated the air gap as a separate domain. Then we have mathematically proved that the effective properties of acoustic metamaterial can be obtained by solving a set of eight linear equations when the dimensions are known. Finally, we have evaluated the proposed method with results from numerical simulations. It is shown that the proposed method can calculate the effective refractive index and impedance with an error of below 1\%. This method provides an efficient approach to analyzing the effective properties of acoustic metamaterials of various sizes.