  We analyse the non-linear propagation and dissipation of axisymmetric waves
in accretion discs using the ZEUS-2D hydrodynamics code. The waves are
numerically resolved in the vertical and radial directions. Both vertically
isothermal and thermally stratified accretion discs are considered. The waves
are generated by means of resonant forcing and several forms of forcing are
considered. Compressional motions are taken to be locally adiabatic ($\gamma =
5/3$). Prior to non-linear dissipation, the numerical results are in excellent
agreement with the linear theory of wave channelling in predicting the types of
modes that are excited, the energy flux by carried by each mode, and the
vertical wave energy distribution as a function of radius. In all cases, waves
are excited that propagate on both sides of the resonance (inwards and
outwards). For vertically isothermal discs, non-linear dissipation occurs
primarily through shocks that result from the classical steepening of acoustic
waves. For discs that are substantially thermally stratified, wave channelling
is the primary mechanism for shock generation. Wave channelling boosts the Mach
number of the wave by vertically confining the wave to a small cool region at
the base of the disc atmosphere. In general, outwardly propagating waves with
Mach numbers near resonance ${\cal M}_{\rm r} \ga 0.01$ undergo shocks within a
distance of order the resonance radius.
