Proof of Energy Conservation for the Wave Equation

Energy conservation is a fundamental property of the wave equation, reflecting the Hamiltonian structure of wave propagation. The conserved energy functional provides a priori estimates and uniqueness results.

Statement

Theorem6.3Energy Conservation for the Wave Equation

For a solution $u$ of $u_{tt} = c^2\nabla^2 u$ in $\Omega \times [0,T]$ with $u = 0$ on $\partial\Omega$ , the energy $E(t) = \frac{1}{2}\int_\Omega [u_t^2 + c^2|\nabla u|^2]\,dx$ is constant: $\frac{dE}{dt} = 0$ .

Proof

Differentiate the energy: $\frac{dE}{dt} = \int_\Omega [u_t u_{tt} + c^2 \nabla u \cdot \nabla u_t]\,dx$

Substitute $u_{tt} = c^2\nabla^2 u$ : $= \int_\Omega [u_t c^2\nabla^2 u + c^2 \nabla u \cdot \nabla u_t]\,dx$

Integrate the second term by parts (Green's first identity): $\int_\Omega \nabla u \cdot \nabla u_t\,dx = -\int_\Omega u_t \nabla^2 u\,dx + \oint_{\partial\Omega} u_t \frac{\partial u}{\partial n}\,dS$

Since $u = 0$ on $\partial\Omega$ , we have $u_t = 0$ on $\partial\Omega$ (differentiating the boundary condition in time). Therefore the boundary integral vanishes: $\frac{dE}{dt} = c^2\int_\Omega u_t \nabla^2 u\,dx - c^2\int_\Omega u_t\nabla^2 u\,dx + 0 = 0$

The two volume integrals cancel exactly, proving $dE/dt = 0$ . $\square$

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ExampleUniqueness from Energy Conservation

Uniqueness: If $u_1, u_2$ solve the wave equation with the same initial and boundary data, then $w = u_1 - u_2$ solves the wave equation with zero data. The energy $E_w(t) = \frac{1}{2}\int[w_t^2 + c^2|\nabla w|^2]dx$ satisfies $E_w(0) = 0$ (from zero initial data) and $E_w(t) = E_w(0) = 0$ (by conservation). Since the integrand is non-negative: $w_t = 0$ and $\nabla w = 0$ everywhere, giving $w = \text{const} = 0$ (from zero initial condition). Thus $u_1 = u_2$ .

RemarkEnergy for Damped and Driven Waves

For the damped wave equation $u_{tt} + \gamma u_t = c^2\nabla^2 u$ : $dE/dt = -\gamma\int u_t^2\,dx \leq 0$ . Energy decreases monotonically, with the rate proportional to the kinetic energy. For driven waves $u_{tt} = c^2\nabla^2 u + f$ : $dE/dt = \int fu_t\,dx$ (work done by the forcing). Energy arguments extend to nonlinear wave equations via suitable energy functionals.