Explicit approximation of the wavenumber for lined ducts

TL;DR

This paper develops highly accurate explicit formulas for the wavenumber in lined ducts, simplifying the analysis of acoustic wave propagation by avoiding complex numerical solutions.

Contribution

It introduces improved explicit approximations for the fundamental mode wavenumber in lined ducts, surpassing previous Eckart-based expressions in accuracy.

Findings

Eckart expression has 5% accuracy for the wavenumber.

New approximations achieve relative errors less than 10^-8.

Approximations are valid for low dissipation and axisymmetric ducts.

Abstract

For acoustic waves in lined ducts, at given frequencies, the dispersion relation leads to a transcendental equation for the wavenumber that has to be solved by numerical methods. Based on Eckart explicit expression initially derived for water waves, accurate explicit approximations are proposed for the wavenumber of the fundamental mode in lined ducts. While Eckart expression is 5 % accurate, some improved approximations can reach maximum relative error of less than 10 raised to -8. The cases with small dissipation part in the admittance of the liner and/or axisymmetric ducts are also considered.