  The inverse medium problem for a circular cylindrical domain is studied using
low-frequency acoustic waves as the probe radiation. It is shown that to second
order in $k_{0}a$ ($k_{0}$ the wavenumber in the host medium, $a$ the radius of
the cylinder), only the first three terms (i.e., of orders 0, -1 and +1) in the
partial wave representation of the scattered field are non-vanishing, and the
material parameters enter into these terms in explicit manner. Moreover, the
zeroth-order term contains only two of the unknown material constants (i.e.,
the real and imaginary parts of complex compressibility of the cylinder
$\kappa_{1}$) whereas the $\pm 1$ order terms contain the other material
constant (i.e., the density of the cylinder $\rho_{1}$). A method, relying on
the knowledge of the totality of the far-zone scattered field and resulting in
explicit expressions for $\rho_{1}$ and $\kappa_{1}$, is devised and shown to
give highly-accurate estimates of these quantities even for frequencies such
that $k_{0}a$ is as large as 0.1.
