The wave equation transports energy rather than dissipating it. Its finite propagation speed sharply contrasts with the instantaneous influence of heat flow and provides the model for hyperbolic equations.
D'Alembert's formula
Theorem. The solution of \[ u_{tt}-c^2u_{xx}=0,\qquad u(x,0)=f(x),\quad u_t(x,0)=g(x), \] on \(\mathbb R\) is \[ u(x,t)=\frac{f(x-ct)+f(x+ct)}2 +\frac1{2c}\int_{x-ct}^{x+ct}g(y)\,\mathrm{d}y. \]
Proof. Introduce characteristic coordinates \(\xi=x+ct\), \(\eta=x-ct\). Since \[ \partial_t^2-c^2\partial_x^2=-4c^2\partial_\xi\partial_\eta, \] every \(C^2\)-solution has the form \(F(\xi)+G(\eta)\). The two initial conditions give \[ F(x)+G(x)=f(x),\qquad cF'(x)-cG'(x)=g(x). \] Integrating the second identity and combining it with the first yields the stated formula.
The value at \((x,t)\) depends only on data in \([x-ct,x+ct]\). Disturbances therefore travel no faster than \(c\).
Energy conservation and uniqueness
Theorem (energy identity). For a smooth solution on \(\mathbb R^n\) that decays sufficiently rapidly, \[ E(t)=\frac12\int_{\mathbb R^n} \bigl(|u_t(x,t)|^2+c^2|\nabla u(x,t)|^2\bigr)\,\mathrm{d}x \] is constant.
Proof. Multiply \(u_{tt}-c^2\Delta u=0\) by \(u_t\), integrate by parts, and obtain \[ \frac{\mathrm{d}}{\mathrm{d}t}E(t) =\int u_tu_{tt}+c^2\nabla u\cdot\nabla u_t\,\mathrm{d}x =\int u_t(u_{tt}-c^2\Delta u)\,\mathrm{d}x=0. \] Applied to the difference of two solutions, conservation of zero initial energy proves uniqueness.
Finite propagation in several dimensions
Proposition (domain of dependence). If the initial displacement and velocity vanish in the ball \(B(x_0,cT)\), then \(u(x_0,T)=0\).
Proof. Integrate the local energy density \[ e=\tfrac12(u_t^2+c^2|\nabla u|^2) \] over the shrinking ball \(B(x_0,c(T-t))\). The local conservation law \[ \partial_t e-\nabla\!\cdot(c^2u_t\nabla u)=0 \] and Reynolds' transport formula show that the cone energy is nonincreasing; the boundary flux is nonpositive by \(2|u_t\partial_\nu u|\le c^{-1}u_t^2+c|\partial_\nu u|^2\). It starts at zero, hence remains zero up to the cone vertex.
Higher-dimensional formulas
In three dimensions, Kirchhoff's formula reads \[ u(x,t)=\partial_t\!\left[ \frac{t}{4\pi}\int_{S^2}f(x+ct\omega)\,\mathrm{d}\omega \right] +\frac{t}{4\pi}\int_{S^2}g(x+ct\omega)\,\mathrm{d}\omega , \] after the customary scaling by \(c\). Only data on the sphere \(|y-x|=ct\) enter; this is Huygens' principle. In two dimensions, the formula averages over the whole disk, so a trailing tail remains inside the light cone.
Weak solutions and forcing
For \(f\in H_0^1(\Omega)\) and \(g\in L^2(\Omega)\), a finite-energy weak solution satisfies \[ u\in C([0,T];H_0^1(\Omega)),\qquad u_t\in C([0,T];L^2(\Omega)) \] and \[ \langle u_{tt},v\rangle+c^2\int_\Omega\nabla u\cdot\nabla v\,\mathrm{d}x =\langle F,v\rangle \] for every \(v\in H_0^1(\Omega)\). Galerkin approximation in Laplacian eigenfunctions, uniform energy estimates, weak compactness, and uniqueness give existence. For an abstract positive self-adjoint operator \(A\), \[ u(t)=\cos(tA^{1/2})f+A^{-1/2}\sin(tA^{1/2})g +\int_0^t A^{-1/2}\sin((t-s)A^{1/2})F(s)\,\mathrm{d}s. \]