Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Positive-response linear sound in adopted barotropic fluid

Physical scope

First variation about a uniform rest state of the adopted smooth, positive-density barotropic model when its pressure law is the fixed-entropy restriction of the adopted local state relation. The prediction is linear; no finite-amplitude error, lifespan, material measurement or empirical sound-speed claim is asserted.

Statement

Let $I \subset R$ be a nonempty open time interval and $Ω \subset I \times R^{3}$ a nonempty open domain for the model fields. Let $e \in C^{2} (U)$ be the local specific-energy relation on an open $U \subset (0, \infty) \times R$ , with state coordinates $(v, s)$ , $T = e_{s}$ , and $p = - e_{v}$ ; each state partial derivative holds the other coordinate fixed. Fix $(v_{b}, s_{b}) \in U$ with $T_{b} > 0$ and $e_{s s} (v_{b}, s_{b}) > 0$ , set $ρ_{b} = 1 / v_{b}$ , and take a barotropic pressure law $P \in C^{1} ((0, \infty))$ that agrees with $p (1 / ρ, s_{b})$ on an open density interval $J_{ρ} \subset (0, \infty)$ containing $ρ_{b}$ , with $(1 / ρ, s_{b}) \in U$ for every $ρ \in J_{ρ}$ . This is the fixed-entropy restriction of the adopted local state relation. The smooth barotropic Euler model is linearized about the uniform rest state $(ρ, u) = (ρ_{b}, 0)$ . For $C^{2}$ perturbation fields $δ ρ : I \times R^{3} \to R$ and $w : I \times R^{3} \to R^{3}$ , the first variation at $ε = 0$ after $ρ = ρ_{b} + ε δ ρ$ , $u = ε w$ is $\partial_{t} δ ρ + ρ_{b} div w = 0, ρ_{b} \partial_{t} w + P^{'} (ρ_{b}) \nabla δ ρ = 0, \partial_{t t} δ ρ = c_{s}^{2} Δ δ ρ, c_{s}^{2} = P^{'} (ρ_{b}) = v_{b}^{2} e_{v v} (v_{b}, s_{b}) .$ The sign of $c_{s}^{2}$ is not assumed in the base model. If $c_{s}^{2} > 0$ , define $c_{s} = \sqrt{c_{s}^{2}}$ . For every nonzero $k \in R^{3}$ , with $ω = c_{s} ∣ k ∣$ , the fields $δ ρ = A \cos (k \cdot x - ω t), w = \frac{ω A}{ρ_{b} ∣ k ∣^{2}} k \cos (k \cdot x - ω t)$ solve the first-variation system for every real amplitude $A$ . They are exact solutions of the linearized equations, not finite-amplitude approximations with an automatic error or lifespan bound. No empirical value of $c_{s}$ is asserted.

In SI units, $t$ is in seconds, $x$ in metres, $ρ_{b}, δ ρ, A$ in $k g m^{- 3}$ , $u, w, c_{s}$ in $m s^{- 1}$ , $k$ in $m^{- 1}$ , $ω$ in $s^{- 1}$ , and $c_{s}^{2}$ in $m^{2} s^{- 2}$ .

Statement references

David Tong, Fluid Mechanics, Part III lecture notes, §§4.4.2–4.4.4, printed pp. 146–151

An Introduction to Continuum Mechanics, Second Edition, Chapter 6, §§6.5.1–6.5.2, printed pp. 242–244

Facts & Assumptions

Given: A smooth positive-density barotropic model, a local $C^{2}$ state relation, a fixed-entropy pressure law, and the uniform rest state specified in the Statement.

[L1]

Smooth positive-density barotropic Euler model: $ρ > 0$ , the model equations are $\partial_{t} ρ + div (ρ u) = 0$ and $ρ D_{t} u + \nabla P (ρ) = 0$ , and $P \in C^{1} ((0, \infty))$ . Its current batch-11 Definition gives this exact local model and adds no sign condition on $P^{'}$ .

[A1]

Local simple-fluid equilibrium and Gibbs closure: $v = 1 / ρ$ , $e \in C^{2} (U)$ , $T = e_{s} > 0$ , and $p = - e_{v}$ are adopted local state values. Its current batch-11 Postulate states this local equilibrium assumption explicitly.

[L2]

Response derivatives and linear acoustic sign: for $P (ρ) = p (1 / ρ, s_{b})$ , the exact fixed-entropy derivative is $P^{'} (ρ_{b}) = v_{b}^{2} e_{v v} (v_{b}, s_{b})$ , and the first variation of the same barotropic equations is the displayed linear system. The current Statement, hypotheses and complete proof were inspected; step1Decision currently reports ready. This is authoring readiness, not independent review approval.

[A2]

The perturbation fields are differentiated at $ε = 0$ ; this first variation is not asserted to approximate a nonlinear solution for any specified finite $ε$ .

Proof

1.1A1L2givenalgebra

Since $P (ρ) = p (1 / ρ, s_{b})$ on $J_{ρ}$ and $p = - e_{v}$ , the chain rule at fixed $s_{b}$ gives $P^{'} (ρ_{b}) = p_{v} (v_{b}, s_{b}) (- 1 / ρ_{b}^{2}) = (- e_{v v}) (- v_{b}^{2}) = v_{b}^{2} e_{v v}$ . This is $c_{s}^{2}$ by definition.

2.1A2L1step 1.1algebra

Substitute $ρ_{ε} = ρ_{b} + ε δ ρ$ and $u_{ε} = ε w$ into the model equations from [L1], with $ε$ sufficiently close to zero locally at each point that $ρ_{ε} \in J_{ρ}$ . The order- $ε$ continuity term is $\partial_{t} δ ρ + ρ_{b} div w$ . The convective term in momentum is order $ε^{2}$ , while differentiability of $P$ gives $\nabla P (ρ_{ε}) = ε P^{'} (ρ_{b}) \nabla δ ρ + o (ε)$ . Dividing by $ε$ and taking the derivative at zero yields the stated first-variation system.

3.1L2step 2.1algebra

Differentiate the first equation of the first-variation system in time. Take the divergence of the second equation; because $ρ_{b}$ and $P^{'} (ρ_{b}) = c_{s}^{2}$ are constant, substitution gives $\partial_{t t} δ ρ = c_{s}^{2} Δ δ ρ$ .

4.1L2step 2.1givenalgebra∎

Suppose $c_{s}^{2} > 0$ , take $k \neq 0$ , and put $ω = c_{s} ∣ k ∣$ . For the stated fields, $\partial_{t} δ ρ = ω A \sin (k \cdot x - ω t)$ , $\nabla δ ρ = - A k \sin (k \cdot x - ω t)$ , $div w = - ω A \sin (k \cdot x - ω t) / ρ_{b}$ , and $\partial_{t} w = ω^{2} A k \sin (k \cdot x - ω t) / (ρ_{b} ∣ k ∣^{2})$ . The continuity equation follows from the first and third expressions; the momentum equation follows from $ω^{2} = c_{s}^{2} ∣ k ∣^{2}$ .

Positive-response linear sound in adopted barotropic fluid

Statement

Statement references

Facts & Assumptions

Proof

Depends on

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