d'Alembert, Kirchhoff, and fundamental solutions

For a whole-line wave equation with positive constant speed, prescribe initial displacement f and initial velocity g. The solution is the mean of two shifted displacement values plus a scaled integral of g over the dependence interval. Differentiating the formula checks both initial conditions and the PDE for smooth data. Less regular finite-energy data can be handled by approximation, with derivatives interpreted weakly. A formula for a classical point value should not be applied indiscriminately to arbitrary representatives of rough data.

The two shifted f values travel in opposite directions, but the velocity contribution involves every point between their feet. Thus knowing only the interval endpoints is insufficient for generic one-dimensional wave data. If g vanishes and f is compactly supported, the displacement consists of two departing profiles; if g has nonzero integral, a persistent interior displacement can remain. That interior constant carries no gradient or kinetic energy after the fronts have passed locally, so it is consistent with energy being transported away.

In three spatial dimensions, define the spherical mean by averaging over unit directions after scaling the sphere radius. The Kirchhoff formula differentiates time times the spherical mean of f and adds time times the spherical mean of g, both at radius ct. For smooth compactly supported data these operations are classical and can be justified directly. At small time, the first term approaches f while the derivative of the second supplies g, making the normalization check transparent.

Only values on the expanding sphere occur in this three-dimensional homogeneous constant-coefficient formula. Once that sphere has moved beyond a compact data support, the observation becomes zero. This is the sharp Huygens behavior for the free wave equation in three dimensions, with analogous behavior in odd spatial dimensions at least three. Even dimensions have interior-cone tails, and dimension one has the velocity-integral behavior just described. Boundaries, potentials, curvature, or altered equations can change this conclusion even in a dimension where the free-space formula has it.

A retarded fundamental solution is a distribution supported at nonnegative times whose wave operator gives a space-time point source. Convolution with initial data or a forcing term provides a systematic representation of propagation. Its singular support and its full support are different: a wavefront can be concentrated on a cone boundary while the fundamental solution also has a tail inside. The explicit formulas provide concrete instances of this distinction without requiring a pointwise interpretation of the source distribution.

Finite propagation is the common statement that the retarded influence stays inside or on the forward cone. Huygens behavior says more: in the relevant free-space dimensions the propagation kernel for positive times is supported on the cone surface. Numerical plots of a single sinusoid cannot distinguish these support properties, because a sinusoid is nonzero across the domain from the start. Compactly supported test data and a clearly specified equation are necessary when interpreting a simulation as evidence about tails or propagation fronts.