Willis couplings in continuously varying cross-sectional area duct

Area of Science:

• Acoustics
• Wave Propagation
• Materials Science

Background:

• Understanding acoustic wave behavior in complex media is crucial for designing advanced acoustic devices.
• Periodic structures offer unique wave manipulation properties, but asymmetry introduces complexities.
• Effective medium theories simplify the analysis of wave propagation in heterogeneous materials.

Purpose of the Study:

• To theoretically and numerically derive effective properties for acoustic wave propagation in a one-dimensional periodic and asymmetric duct.
• To develop closed-form expressions for effective properties, including asymmetric Willis coupling.
• To compare the validity and performance of first-order and second-order homogenization procedures.

Main Methods:

• Theoretical analysis using Peano-Baker series expansion and Padé's approximation.
• Numerical simulations of acoustic wave propagation.
• First-order and second-order homogenization procedures, including Willis coupling.
• Comparison of homogenization results with numerical data.

Main Results:

• Closed-form expressions for effective properties, including asymmetric Willis coupling, were derived.
• The second-order homogenization procedure yielded scattering coefficients valid over a significantly larger frequency range compared to the first-order method.
• The frequency limit for the validity of the effective description was investigated and compared to the Bragg bandgap.

Conclusions:

• Second-order homogenization provides a more accurate and broader frequency description of acoustic wave propagation in asymmetric periodic ducts.
• The study advances the understanding and engineering application of Willis materials.
• Further investigation into the validity limit, especially for impedance modeling, is warranted.