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Specified small-perturbation material dynamics — Examples

1 · Prerequisites

2 · Summary

The example verifies explicit perfect-branch sound waves and spatially constant neutral modes. It shows by direct substitution why energy contraction does not imply that every perturbation decays. The comparison is conditional on the perfect-branch theorem and its stated ${A C}_{ω}$ hypothesis; it contains no empirical data.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-6-luna)Open item page →

Perfect sound and neutral perturbation modes

Physical scope

Explicit plane-wave and spatially constant solutions of the specified adiabatic perfect-material linear system; no claim that every mode decays, no nonlinear solution, and no empirical data.

Example

Under ${A C}_{ω}$ , use the perfect linear branch of Causal stable perfect-material linear response with $h_{0} > 0$ , $0 < a_{s} \leq 1$ , and reference speed $c > 0$ . For any nonzero wave vector $k \in R^{3}$ in $m^{- 1}$ and real amplitude $A$ in Pa, define the dimensionless phase $ϕ (t, x) = k \cdot x - ω t$ , where $ω = c \sqrt{a_{s}} ∣ k ∣$ has units $s^{- 1}$ , and set

$r (t, x) = A \cos ϕ (t, x), w (t, x) = \frac{A ω}{c h_{0} ∣ k ∣^{2}} k \cos ϕ (t, x) .$

This is an exact sound mode of the adopted linear system. At zero wave vector, the separate constant fields $r = r_{0}$ and $w = w_{0}$ are also exact solutions for arbitrary constants $r_{0} \in R$ in Pa and $w_{0} \in R^{3}$ dimensionless. Every nonzero such choice is a neutral mode, so linear energy contraction does not mean that every perturbation decays. These are calculated model solutions, not observations.

Facts & Assumptions

Given: The perfect-material linear equations and parameter conventions in Causal stable perfect-material linear response.

[A1]

Assume ${A C}_{ω}$ as defined by The Axiom of Countable Choice ( ${A C}_{ω}$ ), because the cited theorem's global existence, contraction, and finite-propagation conclusions use that hypothesis.

[F1]

Causal stable perfect-material linear response: under its stated hypotheses, including ${A C}_{ω}$ , the adopted equations are $r_{t} + c h_{0} div w = 0$ and $w_{t} + (c a_{s} / h_{0}) \nabla r = 0$ , with $h_{0} > 0$ , $0 < a_{s} \leq 1$ , and $c > 0$ . This example uses its exact perfect-system equations and parameter conventions.

Verification

1.1F1givenalgebra

For $k \neq 0$ , $r_{t} = A ω \sin ϕ$ and $div w = - A ω (c h_{0})^{- 1} \sin ϕ$ , so $r_{t} + c h_{0} div w = 0$ . Also $w_{t} = A ω^{2} (c h_{0} ∣ k ∣^{2})^{- 1} k \sin ϕ$ and $\nabla r = - A k \sin ϕ$ . Substitution into the second equation gives a coefficient proportional to $ω^{2} - c^{2} a_{s} ∣ k ∣^{2}$ , which is zero by the definition of $ω$ . Thus both first-order equations hold pointwise.

2.1A1F1givenalgebra∎

At $k = 0$ , define $r = r_{0}$ and $w = w_{0}$ directly rather than using the nonzero-wave formula, whose denominator would vanish. Both spatial and time derivatives are zero, so each side of both equations is zero. Nonzero choices give persistent modes; under [A1] the theorem's contractive energy context therefore does not imply that every perturbation decays.