  One can define several properties of wave equations that correspond to the
absence of tails in their solutions, the most common one by far being Huygens'
principle. Not all of these definitions are equivalent, although they are
sometimes assumed to be. We analyse this issue in detail for linear scalar
waves, establishing some relationships between the various properties. Huygens'
principle is almost always equivalent to the characteristic propagation
property, and in two spacetime dimensions the latter is equivalent to the
zeroth order progressing wave propagation property. Higher order progressing
waves in general do have tails, and do not seem to admit a simple physical
characterisation, but they are nevertheless useful because of their close
association with exactly solvable two-dimensional equations.
