M. Bednarik

Acoustic wave propagation in a one-dimensional periodic and asymmetric duct is studied theoretically and numerically to derive the effective properties. Closed form expressions for these effective properties, including the asymmetric Willis coupling, are derived through truncation of the Peano–Baker series expansion of the matricant (which links the state vectors at the two sides of the unit-cell) and Padé's approximation of the matrix exponential. The results of the first-order and second-order homogenization (with Willis coupling) procedures are compared with the numerical results. The second-order homogenization procedure provides scattering coefficients that are valid over a much larger frequency range than the usual first-order procedure. The frequency well below which the effective description is valid is compared with the lower bound of the first Bragg bandgap when the profile is approximated by a two-step function of identical indicator function, i.e., two different cross-sectional areas over the same length. This validity limit is then questioned, particularly with a focus on impedance modeling. This article attempts to facilitate the engineering use of Willis materials.

There is only a limited amount of known analytical solutions to the Pridmore-Brown equation, mostly employing asymptotic behavior for a certain frequency limit and specifically chosen flow profiles. In this paper, we show the possibility of transformation of the Pridmore-Brown equation into the Schrödinger-like equation for the case of two-dimensional homentropic mean flow without critical layers. The corresponding potential that depends on the mean flow profile can then be approximated by a quartic polynomial, leading to a triconfluent Heun equation whose solution based on the triconfluent Heun functions is generally known. The quality of this approximation procedure is presented for the case of symmetric polynomial flow profiles for various values of polynomial order and the Mach number. A more detailed example is then shown for a quadratic mean flow profile, where the solution is accurate up to the third order of the Mach number.