Finite propagation is not the Huygens principle

Assume the Axiom of Countable Choice. For homogeneous waves with compactly supported initial data, the strong Huygens principle implies finite propagation, but the converse is false. The interior tails in dimension $1$ and every even dimension give witnesses to this failure of the converse.

Finite propagation (Compact support expands at speed at most c) holds in every spatial dimension $n\ge 1$: data supported in a compact $K$ (with a source supported in the corresponding cone) have $\mathrm{supp}u(\cdot ,t)\subseteq K+{\bar{B}}_{ct}(0)$ at time $t$, the full solid $ct$-neighbourhood of the data support. It bounds the outer front and nothing more.

The strong Huygens principle (The strong Huygens principle in the homogeneous Cauchy setting) is the stronger shell statement: the value at $(x_0,t_0)$ is carried by the sphere $S(x_0,t_0)=\mathrm{\partial }{B}_{c{t}_0}(x_0)$, so data supported strictly inside the base ball have no effect and, for compactly supported data, the disturbance is carried by the shell rather than by the solid cone. It holds for odd $n\ge 3$ (The strong Huygens principle in odd spatial dimensions) and fails for $n=1$ and every even $n$ (Wave tails in one and even spatial dimensions: strong Huygens fails), where a lasting interior tail remains after the front has passed; the failure witnesses are compactly supported and obey the finite-speed bound.

Two consequences deserve care. First, an interior tail is not a violation of finite speed: the tail stays inside the cone $\mathrm{\mid }x-x_0\mathrm{\mid }\le ct$ — it is the interior of that cone, not its exterior, that remains affected. Second, the informal shorthand "the value depends only on the values of the initial data on $\mathrm{\partial }{B}_t(x)$" is not a correct reading of strong Huygens: the functional is a finite linear combination of sphere integrals of radial derivatives, as The strong Huygens principle in odd spatial dimensions proves; neighbourhood agreement preserves those derivatives, whereas bare restriction agreement need not. This is consistent with the germ form (iii) of The strong Huygens principle in the homogeneous Cauchy setting. The positive theorem above is the sharp statement, and this remark records the exact distinction fixed by the drift review of this page.

Explicit compactly supported failure witnesses are recorded in Finite speed of propagation does not imply strong Huygens ↗.