CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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Time reversal of the homogeneous wave equation

Statement

Let $c > 0$ , let $I \subseteq R$ be an open interval and let $u \in C^{2} (R^{n} \times I)$ satisfy $□_{c} u = 0$ on $R^{n} \times I$ . For $τ \in I$ define $v (x, t) : = u (x, 2 τ - t)$ for $t \in 2 τ - I$ . Then $□_{c} v = 0$ on $R^{n} \times (2 τ - I)$ , and $v (\cdot, τ) = u (\cdot, τ), \partial_{t} v (\cdot, τ) = - \partial_{t} u (\cdot, τ) .$ Thus the homogeneous wave flow is reversible: the same equation propagates the time-reversed state, and the velocity is negated.

Facts & Assumptions

Given: an open interval $I$ , a parameter $τ \in I$ , a $C^{2}$ function $u$ on $R^{n} \times I$ with $□_{c} u = 0$ , and $v (x, t) = u (x, 2 τ - t)$ on $R^{n} \times (2 τ - I)$ .

[F1]

If $g$ is totally differentiable at $a$ and $f$ at $g (a)$ , then $f \circ g$ is totally differentiable at $a$ with $D (f \circ g) (a) = D f (g (a)) \circ D g (a)$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ). The required total differentiability follows from continuous coordinate partial derivatives (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

[F2]

Sums, products, constant multiples of differentiable functions are differentiable with the usual rules and derivatives; in particular the affine map $t \mapsto 2 τ - t$ has derivative $- 1$ (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 $g (c) \neq 0$ ).

Proof

1.1F1F2algebra

Apply [F1] to the composition $(x, t) \mapsto (x, 2 τ - t) \mapsto u (x, 2 τ - t)$ : the inner map is affine with differential $(h, s) \mapsto (h, - s)$ , so $\partial_{t} v (x, t) = - \partial_{t} u (x, 2 τ - t)$ ; applying the same rule once more, $\partial_{t}^{2} v (x, t) = (- 1)^{2} \partial_{t}^{2} u (x, 2 τ - t) = \partial_{t}^{2} u (x, 2 τ - t)$ . In the spatial directions the inner map is the identity, so $\partial_{x_{j}} v (x, t) = \partial_{x_{j}} u (x, 2 τ - t)$ for each $j$ and hence $Δ v (x, t) = Δ u (x, 2 τ - t)$ .

2.1F2algebra∎

Therefore for every $(x, t)$ with $t \in 2 τ - I$ , $□_{c} v (x, t) = \partial_{t}^{2} u (x, 2 τ - t) - c^{2} Δ u (x, 2 τ - t) = □_{c} u (x, 2 τ - t) = 0$ , while at $t = τ$ the substitutions $2 τ - τ = τ$ give $v (x, τ) = u (x, τ)$ and $\partial_{t} v (x, τ) = - \partial_{t} u (x, τ)$ . This is the reversibility of the homogeneous wave flow.

Depends on

Used by

Time-reversed energy uniqueness from final data Corollary

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (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)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)