  We here analyse numerical simulations of supersonic, hypersonic and
magnetohydrodynamic turbulence that is free to decay. Our goals are to
understand the dynamics of the decay and the characteristic properties of the
shock waves produced. This will be useful for interpretation of observations of
both motions in molecular clouds and sources of non-thermal radiation.
  We find that decaying hypersonic turbulence possesses an exponential tail of
fast shocks and an exponential decay in time, i.e. the number of shocks is
proportional to t exp (-ktv) for shock velocity jump v and mean initial
wavenumber k. In contrast to the velocity gradients, the velocity Probability
Distribution Function remains Gaussian with a more complex decay law.
  The energy is dissipated not by fast shocks but by a large number of low Mach
number shocks. The power loss peaks near a low-speed turn-over in an
exponential distribution. An analytical extension of the mapping closure
technique is able to predict the basic decay features. Our analytic description
of the distribution of shock strengths should prove useful for direct modeling
of observable emission. We note that an exponential distribution of shocks such
as we find will, in general, generate very low excitation shock signatures.
