  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.
