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The local wave-energy conservation law

Statement

Let $n \geq 1$ , $c > 0$ , $U \subseteq R^{n}$ open, $I \subseteq R$ an open interval and $u \in C^{2} (U \times I)$ ( $C^{k}$ maps and multi-index derivative notation in Euclidean space). Put $f : = □_{c} u = \partial_{t}^{2} u - c^{2} Δ u$ (Wave equation, Cauchy data and wave speed, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field) and let $e, q$ be the energy density and flux of Wave energy density, energy flux and total energy. Then the pointwise identity

$\partial_{t} e + div q = f u_{t}$

holds on $U \times I$ ; in particular $\partial_{t} e + div q = 0$ for every classical solution of the homogeneous equation. For a homogeneous solution at unit speed the identity reads $\partial_{t} [\frac{1}{2} (u_{t}^{2} + ∣ D u ∣^{2})] = div (u_{t} D u)$ : indeed $q = - c^{2} u_{t} D u$ gives $- div (u_{t} D u) = div q$ at $c = 1$ . No integration, integrability or boundary regularity is used or asserted.

Facts & Assumptions

Given: $n \geq 1$ , $c > 0$ , an open set $U \subseteq R^{n}$ , an open interval $I$ and $u \in C^{2} (U \times I)$ ; the fields $e = \frac{1}{2} (u_{t}^{2} + c^{2} ∣ D u ∣^{2})$ and $q = - c^{2} u_{t} D u$ of Wave energy density, energy flux and total energy; write $D u_{t} : = \partial_{t} (D u)$ and $Δ u = div (D u)$ .

[F1]

Clairaut–Schwarz: on an open set where $u$ is $C^{2}$ , $\partial_{i} \partial_{j} u = \partial_{j} \partial_{i} u$ for every pair of coordinate indices; in particular $\partial_{t} \partial_{i} u = \partial_{i} \partial_{t} u$ , so $D u_{t} = D (u_{t})$ and the mixed derivatives of $u$ commute. (Clairaut--Schwarz theorem for continuous second partial derivatives)

[F2]

Product rule for the divergence: $div (φ F) = ⟨ \nabla φ, F ⟩ + φ div F$ for $C^{1}$ scalar $φ$ and $C^{1}$ field $F$ . (Divergence and curl are linear and satisfy the scalar product rules)

[F3]

The Laplacian is the divergence of the gradient: $Δ f = div \nabla f = \sum_{i < n} \partial_{i} \partial_{i} f$ . (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field)

[F4]

Product rule in one variable: $(g h)^{'} = g^{'} h + g h^{'}$ for differentiable $g, h : R \to R$ . (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ )

[F5]

The Euclidean inner product is symmetric, $⟨ x, y ⟩ = ⟨ y, x ⟩$ , and $∣ z ∣^{2} = ⟨ z, z ⟩$ . (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ )

Proof

1.1givenF1F4F5algebra

Time derivative of the energy density: at every point of $U \times I$ the product rule [F4] gives $\partial_{t} (u_{t}^{2}) = 2 u_{t} u_{t t}$ and, since $∣ D u ∣^{2} = ⟨ D u, D u ⟩$ [F5], $\partial_{t} ∣ D u ∣^{2} = 2 ⟨ D u, \partial_{t} (D u) ⟩ = 2 ⟨ D u, D u_{t} ⟩$ , where $\partial_{t} (D u) = D (u_{t})$ by [F1]; hence $\partial_{t} e = \frac{1}{2} (2 u_{t} u_{t t} + c^{2} \cdot 2 ⟨ D u, D u_{t} ⟩) = u_{t} u_{t t} + c^{2} ⟨ D u, D u_{t} ⟩$ .

1.2givenF1F2F3algebra

Divergence of the flux: the scalar $φ : = - c^{2} u_{t}$ and the field $F : = D u$ are $C^{1}$ on $U \times I$ because $u$ is $C^{2}$ , so the product rule [F2] and [F3] give $div q = div (φ F) = ⟨ \nabla φ, F ⟩ + φ div F = - c^{2} ⟨ \nabla u_{t}, D u ⟩ - c^{2} u_{t} Δ u = - c^{2} ⟨ D u_{t}, D u ⟩ - c^{2} u_{t} Δ u$ , using $\nabla u_{t} = D u_{t}$ from [F1] in the last step.

2.1givenstep 1.1step 1.2F5algebra∎

The balance law: adding the identities of steps 1.1 and 1.2, $\partial_{t} e + div q = u_{t} u_{t t} + c^{2} ⟨ D u, D u_{t} ⟩ - c^{2} ⟨ D u_{t}, D u ⟩ - c^{2} u_{t} Δ u = u_{t} u_{t t} - c^{2} u_{t} Δ u = u_{t} (u_{t t} - c^{2} Δ u) = u_{t} □_{c} u = f u_{t}$ , the inner-product terms cancelling by symmetry [F5]; for $f = 0$ this is $\partial_{t} e + div q = 0$ , and at $c = 1$ it is the stated unit-speed form.

Depends on

Used by

The local conservation law need not integrate to a finite conserved energy Counterexample
Wave energy need not be conserved through an open boundary Counterexample
The energy identity on a truncated wave cone Lemma
Conservation of total wave energy in three admissible settings Theorem
Energy continuous dependence for the forced wave equation Theorem

Dependency tree · two levels

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Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)