5.1 Spherical means and Kirchhoff’s formula
For a continuous function \(h\) on \(\mathbb R^n\), the spherical mean of \(h\) over the sphere of radius \(r\) about \(x\) is \[ M_h(x, r) = \frac {1}{\omega _n}\int _{|\omega | = 1}h(x + r\omega )\,dS(\omega ) , \] where \(\omega _n\) is the area of the unit sphere (\(4\pi \) in \(\mathbb R^3\)). As \(r \to 0\), \(M_h(x, r) \to h(x)\).
Lemma 5.1.1 (Darboux). If \(h\) is \(C^2\), then \(\partial _r^2M_h + \frac {n - 1}{r}\partial _rM_h = \Delta _xM_h\).
Proof. Differentiating under the integral and applying the divergence theorem on the ball \(B_r(x)\), \[ \partial _rM_h = \frac 1{\omega _n}\int _{|\omega | = 1}\nabla h(x + r\omega )\cdot \omega \,dS = \frac {1}{\omega _nr^{n - 1}}\int _{B_r(x)}\Delta h\,dy . \] So \(r^{n - 1}\partial _rM_h = \frac 1{\omega _n}\int _{B_r(x)}\Delta h\), and differentiating in \(r\) gives \(\partial _r(r^{n - 1}\partial _rM_h) = \frac 1{\omega _n}\int _{\partial B_r(x)}\Delta h\,dS = r^{n - 1}M_{\Delta h}(x, r) = r^{n - 1}\Delta _xM_h\), which is the identity. □
If \(u\) solves the wave equation, apply the lemma to \(h = u(\cdot , t)\): the mean \(U(x; r, t) = M_{u(\cdot , t)}(x, r)\) satisfies the Euler–Poisson–Darboux equation \[ U_{tt} = c^2\Bigl (U_{rr} + \frac {n - 1}{r}U_r\Bigr ) . \] In three dimensions the substitution \(V = rU\) turns this into \(V_{tt} = c^2V_{rr}\), the one-dimensional wave equation, on the half-line \(r > 0\) with \(V(x; 0, t) = 0\). The same calculation shows that the radial solutions in \(\mathbb R^3\) are the spherical waves \[ u = \frac {F(r - ct) + G(r + ct)}{r} , \] outgoing and incoming, whose amplitude decays like \(1/r\) so that the energy crossing each sphere, proportional to \(r^2\cdot r^{-2}\), is the same.
Theorem 5.1.2 (Kirchhoff). Let \(f \in C^3(\mathbb R^3)\) and \(g \in C^2(\mathbb R^3)\). The solution of \(u_{tt} = c^2\Delta u\) in \(\mathbb R^3\) with \(u(x, 0) = f\), \(u_t(x, 0) = g\) is \[ u(x, t) = \frac {\partial }{\partial t}\bigl (tM_f(x, ct)\bigr ) + tM_g(x, ct) , \] that is, \[ u(x, t) = \frac {\partial }{\partial t}\biggl (\frac {1}{4\pi c^2t}\int _{|y - x| = ct}f\,dS\biggr ) + \frac {1}{4\pi c^2t}\int _{|y - x| = ct}g\,dS . \]
Proof. With \(V = rU\) as above, \(V(x; r, 0) = rM_f(x, r) =: \Phi (r)\) and \(V_t(x; r, 0) = rM_g(x, r) =: \Psi (r)\), both odd in \(r\) when extended to \(r < 0\). By the reflection formula of Section 4, for \(0 < r < ct\), \[ V(x; r, t) = \frac 12\bigl (\Phi (ct + r) - \Phi (ct - r)\bigr ) + \frac 1{2c}\int _{ct - r}^{ct + r}\Psi (s)\,ds . \] Since \(u(x, t) = \lim _{r \to 0}U = \lim _{r \to 0}V/r\), dividing by \(r\) and letting \(r \to 0\) gives \(u = \Phi '(ct) + \frac 1c\Psi (ct)\). Now \(\Phi '(ct) = \frac {d}{d\rho }\bigl (\rho M_f(x, \rho )\bigr )\big |_{\rho = ct} = \frac {\partial }{\partial t}\bigl (tM_f(x, ct)\bigr )\) and \(\frac 1c\Psi (ct) = tM_g(x, ct)\). The hypotheses make \(u\) a \(C^2\) solution; the loss of one derivative from \(f\) to \(u\) is a feature of waves in three dimensions, where focusing can amplify singularities. □
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