  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.
