Track waves on spheres and inside cones

Try

In three-dimensional space, points a distance ct from an observer form a sphere. A wave traveling at speed c can connect that sphere at initial time to the observer at time t. The geometry suggests averaging data over a sphere, but the normalization matters: averaging a constant must return that constant, regardless of radius.

Define Mᵣf(x) as the average of f(x+rω) over unit directions ω on the sphere, using surface measure divided by 4π. The radius is carried inside f, so the averaging measure itself is fixed.

Predict

For constant initial displacement A and velocity B, predict the solution without any spherical formula. It must be A+Bt. Use this as a test for missing factors of t or missing derivatives in a proposed formula.

Experiment

For sufficiently smooth three-dimensional data, Kirchhoff’s formula is

u(x,t)=∂ₜ[t Mct f(x)]+t Mct g(x).

Initial data	Spherical mean	Contribution
f=A	A	∂ₜ(tA)=A
g=B	B	tB

For a less constant check, let f(x)=|x|² and g=0. Averaging |x+ctω|² eliminates the cross term and gives |x|²+c²t². Differentiating t times this quantity yields u=|x|²+3c²t². Verify uₜₜ=6c² and c²Δu=6c². This example tests the derivative of the changing radius, not just normalization.

Understand

For radial u(r,t), the three-dimensional Laplacian is uᵣᵣ+2uᵣ/r away from the origin. Set v=ru. Then vᵣᵣ=ruᵣᵣ+2uᵣ and vₜₜ=ruₜₜ, so v satisfies the one-dimensional wave equation. Smoothness at the origin requires v(0,t)=0; it cannot be ignored by treating r as an unrestricted coordinate. This reduction supplies one route to spherical-mean formulas.

More generally, spherical means satisfy a radial mean identity, from which the Kirchhoff expression can be differentiated and verified. The full derivation for arbitrary smooth data uses that identity and the small-radius expansion of the mean to check initial displacement and velocity. We state the resulting classical formula here and verify its mechanism on the polynomial example; we do not claim that a radial calculation alone proves the arbitrary-data theorem.

Finite propagation says the initial data outside a ball cannot affect the observation. The sharp Huygens principle for the free wave in three spatial dimensions is stronger: the formula uses only data on the bounding sphere, including the local radial derivative implicit in the displacement term. In one dimension, the velocity contribution integrates over an interval; in two dimensions, the corresponding formula involves the interior of a disk. Therefore finite propagation does not universally imply a sharp trailing edge.

Apply

Use constant velocity and quadratic displacement to check Kirchhoff’s formula with a different speed. Then describe a compactly supported disturbance and determine when its support can intersect an expanding dependence sphere. State why the same timing argument based only on the sphere would be invalid for the one-dimensional velocity formula.

Continue

Elliptic equations next describe spatial balance rather than time propagation. Their mean-value property also uses spheres, but as a rigidity statement: a harmonic interior value equals every surrounding average inside its domain.