5.4 Energy and sources
Theorem 5.4.1. For a \(C^2\) solution of \(u_{tt} = c^2\Delta u\) in \(\mathbb R^n\) vanishing for large \(|x|\), the energy \(E = \frac 12\int (u_t^2 + c^2|\nabla u|^2)\,dx\) is constant, and the energy in the shrinking ball \(|x - x_0| \leq R - ct\) is non-increasing. Hence the solution is unique, and \(u(x_0, t_0)\) depends only on the data in \(|y - x_0| \leq ct_0\).
Proof. With \(e = \frac 12(u_t^2 + c^2|\nabla u|^2)\), \(e_t = u_tu_{tt} + c^2\nabla u\cdot \nabla u_t = c^2\operatorname {div}(u_t\nabla u)\). Integrating over the ball \(B(t)\) of radius \(R - ct\), whose boundary moves inward at speed \(c\), \[ \frac {d}{dt}\int _{B(t)}e\,dx = \int _{\partial B(t)}\bigl (c^2u_t\,\partial _\nu u - c\,e\bigr )dS \leq 0 , \] since \(|c^2u_t\partial _\nu u| \leq c\cdot \frac 12(u_t^2 + c^2|\nabla u|^2)\). Uniqueness follows as in one dimension. □
For a source, Duhamel’s principle applied to Kirchhoff’s formula gives the solution of \(u_{tt} = c^2\Delta u + F\) in \(\mathbb R^3\) with zero data. The impulse at time \(s\) contributes \((t - s)M_{F(\cdot , s)}(x, c(t - s))\); integrating in \(s\) and substituting \(r = c(t - s)\) gives the retarded potential \begin {equation} \label {eq:retarded} u(x, t) = \frac {1}{4\pi c^2}\int _{|y - x| \leq ct}\frac {F\bigl (y, t - |x - y|/c\bigr )}{|x - y|}\,dy . \end {equation} The source at \(y\) is felt at \(x\) as it was at the earlier time \(t - |x - y|/c\), the time a signal needs to cover the distance. This is the formula for the potentials of electromagnetism produced by moving charges.
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