  Measurements of the ac response represent a widely-used method for probing
the properties of superconductors. In the surface superconducting state (SSS),
increase of the current beyond the surface critical current $I_c$ leads to
breakdown of SSS and penetration of external magnetic field into the sample
bulk. An interesting free-of-bulk system in SSS is offered by thin-walled
superconducting cylinders. The critical state model (CSM) asserts the ac
susceptibility $\chi$ to exhibit jumps as a function of the external ac field
amplitude $H_{ac}$, because of the periodic destruction and restoration of SSS
in the cylinder wall. Here, we investigate experimentally the low-frequency
(128-8192\,Hz) ac response of thin-walled superconducting cylinders in
superimposed dc and ac magnetic fields applied parallel to the cylinder axis.
Distinct from the CSM predictions, experiments reveal that $\chi$ is a smooth
function of $H_{ac}$. For the explanation of our observations we propose a
phenomenological model of partial penetration of magnetic flux (PPMF). The PPMF
model implies that after a restoration of the superconducting state, the
magnetic fields inside and outside the cylinder are not equal, and the value of
the penetrating flux is random for each penetration. This model fits very well
to the experimental data on the temperature dependence of the first-harmonic
$\chi_1$ at any $H_{ac}$ and dc field magnitude. However, in a certain
temperature range the values of physical parameters deduced within the
framework of the PPMF model are questionable.
