Relationships between various characterisations of wave tails

One can define several properties of wave equations that correspond to the absence of tails in their solutions, the most common, 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 characterization, but they are nevertheless useful because of their close association with exactly solvable two-dimensional equations.