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Wave energy need not be conserved through an open boundary

Statement refuted

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the Lebesgue and Riemann integral bridge used below. The claim refuted is that the total energy of a classical wave solution is automatically constant whenever the domain is bounded, without any hypothesis on the boundary flux. Witness: let $c > 0$ , let $Ω = (0, 1) \subseteq R$ and let $F \in C_{c}^{2} (R)$ have $F^{'}$ nonzero somewhere in $(0, 1)$ ; put

$u (x, t) : = F (x - c t),$

the right-moving packet. Then $u$ solves $□_{c} u = 0$ on $R \times R$ , but its energy in the fixed interval,

$E_{Ω} (t) = c^{2} \int_{0}^{1} F^{'} (x - c t)^{2} d x,$

equals $c^{2} \int_{0}^{1} F^{'} (x)^{2} d x > 0$ at $t = 0$ and is $0$ for all sufficiently large $t$ , once the packet has left the interval. The decrease is exactly the boundary flux: with $q = - c^{2} u_{t} u_{x} = c^{3} F^{' 2}$ ,

$\frac{d}{d t} E_{Ω} (t) = q (0, t) - q (1, t),$

so the energy lost through the right endpoint is accounted for, and Conservation of total wave energy in three admissible settings may not be invoked on a domain with an open boundary without the vanishing-flux hypothesis.

Homogeneous-boundary comparison on the half-line. If $v \in C^{2} ([0, \infty) \times I)$ solves $v_{t t} - c^{2} v_{x x} = 0$ on an open time interval $I$ , and for each compact $J \subseteq I$ there is $R_{J} < \infty$ with $v_{x} = v_{t} = 0$ for $x \geq R_{J}$ and $t \in J$ , then $E_{+} (t) : = \frac{1}{2} \int_{0}^{\infty} (v_{t}^{2} + c^{2} v_{x}^{2}) d x$ is constant under either $v (0, t) = 0$ for every $t \in I$ or $v_{x} (0, t) = 0$ for every $t \in I$ (the homogeneous Dirichlet and Neumann comparisons for the linear case of the cited problem). No nonlinear potential term is asserted.

Facts & Assumptions

Given: Countable Choice; $c > 0$ , $Ω = (0, 1)$ , $F \in C_{c}^{2} (R)$ with $F^{'}$ nonzero somewhere in $(0, 1)$ , $u (x, t) = F (x - c t)$ , and the fields $e = \frac{1}{2} (u_{t}^{2} + c^{2} u_{x}^{2})$ , $q = - c^{2} u_{t} u_{x}$ of Wave energy density, energy flux and total energy; for the last part a solution $v$ on $[0, \infty) \times I$ with the stated support hypothesis.

[F1]

The local balance: $\partial_{t} e + \partial_{x} q = f u_{t}$ , hence $\partial_{t} e = - \partial_{x} q$ for a classical solution. (The local wave-energy conservation law)

[F2]

Chain rule, scalar product rule, and equality of mixed second partials for C2 functions. (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , 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 $g (c) \neq 0$ , Clairaut--Schwarz theorem for continuous second partial derivatives)

[F3]

Differentiation under the integral sign: if $x \mapsto f (x, t)$ is integrable for every $t$ , $t \mapsto f (x, t)$ is differentiable for almost every $x$ , and the $t$ -derivative is dominated on the time interval by a fixed integrable function, then $F (t) = \int f (x, t) d x$ is differentiable with $F^{'} (t) = \int \partial_{t} f (x, t) d x$ . (Differentiation under the integral sign)

[F4]

Second fundamental theorem: if $G$ is differentiable on $[a, b]$ with integrable derivative, then $\int_{a}^{b} G^{'} = G (b) - G (a)$ (Darboux integral); on a closed bounded interval continuous functions are integrable, Darboux and Riemann integrals agree, and a bounded Borel Riemann integrable function on a closed interval has the same Lebesgue integral. (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The Darboux and Riemann definitions agree: a bounded $f$ on $[a, b]$ is Darboux integrable with integral $I$ if and only if for every real $ε > 0$ there is a real $δ > 0$ such that $∣ S (f, P, ξ) - I ∣ < ε$ for every tagged partition of mesh below $δ$ , A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral)

[F5]

A continuous function on an interval whose derivative vanishes at every interior point is constant. (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant)

[F6]

The support of $f$ is the closure of ${f \neq 0}$ ; for the translate, $supp F (\cdot - c t) = c t + supp F$ . (The support of a function on $R^{n}$ and its compactly supported Riemann integral)

Proof

1.1givenF1F2algebra

The packet and its flux: by the chain rule [F2], $u_{t} = - c F^{'} (x - c t)$ and $u_{x} = F^{'} (x - c t)$ , so $u_{t t} = c^{2} F^{''} (x - c t)$ , $u_{x x} = F^{''} (x - c t)$ and $□_{c} u = 0$ ; hence [F1] holds and the energy density and flux are $e = c^{2} F^{'} (x - c t)^{2}$ and $q = - c^{2} u_{t} u_{x} = c^{3} F^{'} (x - c t)^{2}$ .

2.1givenstep 1.1F6algebra

Positive initial energy and late vanishing: since $F^{'}$ is continuous and nonzero somewhere in $(0, 1)$ , there are a subinterval of $(0, 1)$ on which $F^{' 2} \geq m > 0$ and hence $E_{Ω} (0) = c^{2} \int_{0}^{1} F^{'} (x)^{2} d x > 0$ ; and by [F6] the support of $x \mapsto u (x, t)$ is $c t + supp F$ , so for every $t > (1 - \min supp F) / c$ the packet is disjoint from $[0, 1]$ and $E_{Ω} (t) = 0$ .

2.2givenstep 1.1F3F4algebra

The flux identity: $e$ and $\partial_{t} e$ are continuous on $[0, 1]$ and bounded on compact time intervals, so [F3] gives $E_{Ω}^{'} (t) = \int_{0}^{1} \partial_{t} e (x, t) d x = - c^{3} \int_{0}^{1} (F^{' 2})^{'} (x - c t) d x$ , and by [F4] the Darboux fundamental theorem applies to the continuous function $x \mapsto F^{' 2} (x - c t)$ on $[0, 1]$ , whose Lebesgue integral equals that Darboux integral, giving $\int_{0}^{1} (F^{' 2})^{'} (x - c t) d x = F^{' 2} (1 - c t) - F^{' 2} (- c t)$ ; therefore $E_{Ω}^{'} (t) = c^{3} (F^{' 2} (- c t) - F^{' 2} (1 - c t)) = q (0, t) - q (1, t)$ .

3.1step 2.1step 2.2algebra

Conclusion for the open boundary: by steps 2.1 and 2.2 the energy $E_{Ω}$ is positive at $t = 0$ , zero for all large $t$ , and its rate of change is exactly the difference of the outward fluxes at the two endpoints; so $E_{Ω}$ is not constant, the drop is accounted for by the flux through the right endpoint, and automatic conservation cannot be inferred without controlling the boundary flux.

4.1givenstep 1.1F2F3F4F5∎

The half-line comparison: fix a compact $J \subseteq I$ and $R \geq R_{J}$ ; for $t \in J$ the integrand of $E_{+}$ vanishes for $x \geq R_{J}$ , so $E_{+} (t) = \frac{1}{2} \int_{0}^{R} (v_{t}^{2} + c^{2} v_{x}^{2}) d x$ , and [F3] with the domination constant $\sup_{[0, R] \times J} ∣ v_{t} v_{t t} + c^{2} v_{x} v_{x t} ∣$ gives $E_{+}^{'} (t) = \int_{0}^{R} (v_{t} v_{t t} + c^{2} v_{x} v_{x t}) d x = c^{2} \int_{0}^{R} (v_{t} v_{x x} + v_{x} v_{x t}) d x = c^{2} \int_{0}^{R} \partial_{x} (v_{t} v_{x}) d x$ , using $v_{t t} = c^{2} v_{x x}$ and the product rule [F2]; by [F4], $\int_{0}^{R} \partial_{x} (v_{t} v_{x}) d x = v_{t} (R, t) v_{x} (R, t) - v_{t} (0, t) v_{x} (0, t)$ ; the upper endpoint term is zero by the support hypothesis, and the lower endpoint term is zero because in the Dirichlet case the trace $t \mapsto v (0, t)$ is identically zero and differentiable with derivative $v_{t} (0, t)$ , while in the Neumann case $v_{x} (0, t) = 0$ directly; hence $E_{+}^{'} (t) = 0$ for every interior $t \in J$ and, $E_{+}$ being continuous on $J$ with vanishing derivative there, [F5] makes $E_{+}$ constant on $J$ ; as $J$ is an arbitrary compact subinterval of $I$ , $E_{+}$ is constant on $I$ .

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Sources

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)