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For $t>0$, the three-dimensional Kirchhoff formula with zero displacement isThe Strong Huygens principle says that the value depends only on the initial velocity on the sphere of radius $t$, not on points strictly inside it. In particular a compactly supported disturbance leaves no tail at a fixed point after the sphere has passed the support.

For method of descent for the wave equation, extend the two-dimensional initial velocity independently of the third coordinate. Projecting the three-dimensional sphere onto its planar disk gives two sheets, each with area element $tdy/\sqrt{t^2-|y-x{|}^2}$. ThereforeThe local argument can use compact cutoffs in the extra coordinate, so global Schwartz decay there is unnecessary. The weak Huygens principle is finite propagation speed: only data in the closed disk of radius $t$ can influence the point. Unlike the three-dimensional free equation, the full interior contributes and a tail can persist.

For the equation with potential $|x{|}^2$, multiply by $u_t$ and use the product rule:This is the claimed local energy conservation with flux $p=-u_t\nabla u$.

Fix $(t_0,x_0)$ and let $E(t)$ be the integral of $e$ over the shrinking ball $|x-x_0|\le t_0-t$, $0\le t\le t_0-a$ with $a>0$. Its boundary moves inward at speed one. The divergence theorem givesbecause the negative integrand isAll initial data are zero, so $E(0)=0$. Nonnegativity then forces $E(t)=0$. Letting the small terminal ball approach the cone tip proves $u=0$ there; alternatively $u_t=0$ throughout the cone and the initial value fixes $u$. The same argument applied to a difference of solutions whose data agree on the initial ball proves local domain of dependence. Thus the potential equation has propagation speed at most one and satisfies the weak Huygens principle, without requiring finite total energy on all space. It does not assert the strong no-tail property.

The following unheaded requests concern the spherical measure, the wave solution and both uniqueness arguments. The missing part (i) above and this root Solution slot restore their coverage while retaining the existing part (ii) anchor.

For $k=|\xi |$, rotational symmetry of the spherical measure giveswith continuous value $4\pi t^2$ at $k=0$. The spatial Fourier transform of the wave equation gives ${\hat{u}}_{tt}+k^2\hat{u}=0$ with initial data zero and $\hat{g}$. Thus $\hat{u}=\sin (tk)\hat{g}/k$. The convolution theorem gives the Kirchhoff formulaIt is smooth for Schwartz data, solves the transformed equation, and has $u(0,x)=0$, $u_t(0,x)=g(x)$ by the second expression and its small-time limit.

To prove uniqueness for arbitrary $C^2$ solutions, take their difference $v$, with zero initial data, and set $e=(v_t^2+|\nabla v{|}^2)/2$, $p=-v_t\nabla v$. Direct differentiation gives $e_t+\nabla \cdot p=v_t(v_{tt}-\Delta v)=0$. For any centre $x_{\ast }$ and radius $R$, integrate over the shrinking ball $B(x_{\ast },R-t)$. Differentiating the moving-domain energy givessince $v_t{\partial }_{n}v\le (v_t^2+|\nabla v{|}^2)/2$. Initial energy is zero and energy is nonnegative, so it stays zero. In each such cone $v_t$ and $\nabla v$ vanish, and the initial value is zero, hence $v=0$. Choosing $R>T$ and arbitrary centre proves uniqueness at any time $T$, without imposing finite total energy or decay on the competing solution.

For the equation with the smooth positive time-independent potential $V$, use insteadThen ${\partial }_t{e}_V+\nabla \cdot p=v_t(v_{tt}-\Delta v+Vv)=0$. The same shrinking-ball estimate is nonpositive because $V{v}^2\ge 0$. Zero initial data therefore give $v=0$. The potential-wave initial-value problem has at most one $C^2$ solution, with no global energy-growth assumption required. Time independence of $V$ is what makes this particular local energy identity exact.

Rotate the polar axis to $\xi$. Since $W_t$ is a finite sphere measure, its Fourier transform iswith value $t$ at $\xi =0$. Fourier transforming the wave equation gives ${\hat{u}}_{tt}+|\xi {|}^2\hat{u}=0$ with initial displacement zero and velocity $\hat{g}$. Its solution is this multiplier times $\hat{g}$, so inversion gives $u=W_t\ast g$. Thus the Kirchhoff formula isThis extends to arbitrary smooth $g$, without any growth restriction: for each finite spacetime neighborhood, multiply $g$ by a compactly supported smooth cutoff equal to one on every sphere occurring in that neighborhood. The formula is unchanged there, so its verified differential equation remains valid locally. Its initial values follow from $u(t,x)=tg(x)+o(t)$ and differentiation under the sphere integral. Since the neighborhood was arbitrary, it is a smooth solution everywhere for $t\ge 0$.

For the wave equation with d'Alembert operator $\square =-{\partial }_t^2+\Delta$, the wave speed is one. The domain of dependence of $(t,x)$ is the initial ball $\overline{B}(x,t)$. More precisely, two solutions whose Cauchy data agree on a neighbourhood of that ball have the same value at $(t,x)$. Thus, if both Cauchy data have support in $K$, finite propagation speed givesThe Strong Huygens principle is sharper in three spatial dimensions: only the initial data on the sphere $|y-x|=t$ contribute, including the first spatial derivatives of the initial displacement there. The Kirchhoff formula makes this precise:In particular, if the Cauchy data have compact support in $\overline{B}(0,R)$, thenFor $t>R$, the solution therefore vanishes behind the inward edge $|x|=t-R$ as well as outside $|x|=t+R$. The absence of an interior tail is the additional content of the Strong Huygens principle.

In the sense of distribution theory, $H^{\prime }(t)=\delta (t)$. Since $\sin (\alpha t)$ vanishes at zero,and therefore

Let $G_{\mathrm{ret}}(\mathbf{x},t;\mathbf{y})$ solveThe spatial Fourier transform turns this into the preceding oscillator equation with $\alpha =ck$, soInverting with the supplied radial integral gives the three-dimensional retarded Green function

For zero initial displacement and initial velocity $f$, convolution with this kernel gives the Kirchhoff formulaThus the value at $(\mathbf{x},t)$ depends only on the initial data on the sphere of radius $ct$, not on its interior. This is the Huygens principle and describes propagation at the finite wave speed $c$.

For a smooth function $g$ define its spherical meanThe Kirchhoff formuladefines a smooth solution of the three-dimensional wave equation for all positive and negative $t$ and has the prescribed data at $t=0$.

For uniqueness, apply the local energy estimate to the difference of two solutions on a backward light cone. Its energy at the cone tip is bounded by the zero initial energy on the cone base, so the difference and its derivatives vanish. Covering spacetime by such cones proves uniqueness among $C^2$ solutions.

Compact support is unnecessary for existence or uniqueness: the sphere in the Kirchhoff formula is compact for each $(t,x)$, so arbitrary smooth data suffice, and the cone-energy proof is local.