Relationships between various characterisations of wave tails

TL;DR

This paper examines different definitions of wave tails in scalar wave equations, clarifying their relationships and highlighting the special cases where these properties are equivalent, especially in two-dimensional spacetimes.

Contribution

It provides a detailed analysis of the relationships between various properties related to wave tails, including Huygens' principle and progressing wave properties, in linear scalar waves.

Findings

Huygens' principle is nearly always equivalent to the characteristic propagation property.

In two spacetime dimensions, the characteristic propagation property is equivalent to the zeroth order progressing wave property.

Higher order progressing waves generally have tails and lack simple physical characterizations.

Abstract

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.