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A three-dimensional spherical pulse leaves a quiet interior

Example

Assume the Axiom of Countable Choice. Let $c > 0$ , $x_{0} \in R^{3}$ , $r > 0$ , and let $(u_{0}, u_{1}) \in C_{c}^{3} \times C_{c}^{2}$ be supported in ${\overline{B}}_{r} (x_{0})$ (The support of a function on $R^{n}$ and its compactly supported Riemann integral). Then the Kirchhoff solution $u$ of Kirchhoff's formula in three dimensions satisfies $u (x, t) = 0$ whenever $∣ x - x_{0} ∣ > c t + r$ or (for $c t > r$ ) $∣ x - x_{0} ∣ < c t - r$ ; at time $t$ the pulse is carried by the spherical shell

$c t - r \leq ∣ x - x_{0} ∣ \leq c t + r,$

and the interior $B_{c t - r} (x_{0})$ behind the front is quiet. This is the concrete illustration of the strong Huygens principle in three dimensions (The strong Huygens principle in odd spatial dimensions(b)), and it is the three-dimensional side of the contrast with A two-dimensional pulse has a tail inside the cone.

Facts & Assumptions

Given: ${A C}_{ω}$ ; $c > 0$ , $x_{0} \in R^{3}$ , $r > 0$ , compactly supported data $(u_{0}, u_{1})$ with support in ${\overline{B}}_{r} (x_{0})$ ; the Kirchhoff solution $u$ .

[F1]

Kirchhoff's formula defines the solution and evaluates it from $u_{0}$ , its radial derivative, and $u_{1}$ on the sphere $\partial B_{c t} (x)$ , for $t > 0$ . (Kirchhoff's formula in three dimensions)

[F2]

Shell form of strong Huygens in odd dimensions: if the data are supported in a compact $K$ and $\partial B_{c t} (x) \cap K = \emptyset$ , then $u (x, t) = 0$ . (The strong Huygens principle in odd spatial dimensions, The strong Huygens principle in the homogeneous Cauchy setting)

Verification

1.1givenalgebra

The support ball lies inside the sphere: if $∣ x - x_{0} ∣ < c t - r$ (with $c t > r$ ) and $∣ y - x_{0} ∣ \leq r$ , then $∣ y - x ∣ \leq ∣ y - x_{0} ∣ + ∣ x_{0} - x ∣ < r + (c t - r) = c t$ , so ${\overline{B}}_{r} (x_{0})$ is disjoint from $\partial B_{c t} (x)$ with positive distance; if $∣ x - x_{0} ∣ > c t + r$ and $∣ y - x_{0} ∣ \leq r$ , then $∣ y - x ∣ \geq ∣ x - x_{0} ∣ - ∣ y - x_{0} ∣ > c t + r - r = c t$ , so again the support ball is disjoint from the sphere with positive distance.

2.1givenstep 1.1F1F2algebra∎

Vanishing: in either case of step 1.1 the data are supported in a compact set disjoint from $\partial B_{c t} (x)$ , so [F2] gives $u (x, t) = 0$ ; hence $u (\cdot, t)$ vanishes both outside the outer sphere $∣ x - x_{0} ∣ = c t + r$ and inside the inner sphere $∣ x - x_{0} ∣ = c t - r$ , so its support is contained in the closed shell $c t - r \leq ∣ x - x_{0} ∣ \leq c t + r$ ; the quiet interior behind the front is the case $∣ x - x_{0} ∣ < c t - r$ , and the statement is exactly the three-dimensional instance of the shell form [F2], evaluated from the data on $\partial B_{c t} (x)$ as [F1] prescribes.

Depends on

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Jared Speck, MIT 18.152 Introduction to Partial Differential Equations, Class Meeting #12: Kirchhoff's Formula and Minkowskian Geometry (Fall 2011) (standard reference, not scraped)