  We present results of non-linear, 2D, numerical simulations of
magneto-acoustic wave propagation in the photosphere and chromosphere of
small-scale flux tubes with internal structure. Waves with realistic periods of
three to five minutes are studied, after applying horizontal and vertical
oscillatory perturbations to the equilibrium model. Spurious reflections of
shock waves from the upper boundary are minimized thanks to a special boundary
condition. This has allowed us to increase the duration of the simulations and
to make it long enough to perform a statistical analysis of oscillations. The
simulations show that deep horizontal motions of the flux tube generate a slow
(magnetic) mode and a surface mode. These modes are efficiently transformed
into a slow (acoustic) mode in the vA < cS atmosphere. The slow (acoustic) mode
propagates vertically along the field lines, forms shocks and remains always
within the flux tube. It might deposit effectively the energy of the driver
into the chromosphere. When the driver oscillates with a high frequency, above
the cut-off, non-linear wave propagation occurs with the same dominant driver
period at all heights. At low frequencies, below the cut-off, the dominant
period of oscillations changes with height from that of the driver in the
photosphere to its first harmonic (half period) in the chromosphere. Depending
on the period and on the type of the driver, different shock patterns are
observed.
