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Supported-rest effective Cattaneo thermal model — Examples

1 · Prerequisites

2 · Summary

This companion verifies an explicit underdamped periodic Cattaneo mode by direct substitution. Its temperature-departure envelope decays and a sufficiently small nonzero amplitude keeps the forward-time model state inside the selected near-equilibrium domain with positive temperature. The construction is a model calculation, not an experiment or a claim about measured transport coefficients.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Supported thermal finite-speed damped mode

Physical scope

An explicitly verified underdamped periodic solution of the supported-rest Cattaneo linear subsystem; its forward-time positive-temperature interpretation holds for sufficiently small amplitude inside the chosen near-equilibrium domain. No observation or material-specific coefficient is reported.

Example

Use the supported-rest SI conventions inherited through Conditional thermal entropy, stability, and finite propagation: $t$ and $τ$ are in seconds, $x_{1}$ and $L$ in metres, $T_{0}$ and $ϑ$ in K, $C$ in $J m^{- 3} K^{- 1}$ , $κ$ in $W m^{- 1} K^{- 1}$ , and $q$ in $W m^{- 2}$ . The amplitude $A$ and mode function $ϕ$ have temperature units K; the wave number $k$ has units $m^{- 1}$ and the angular frequency $Ω$ has units $s^{- 1}$ . The integer $n$ and Cartesian unit vector $e_{1}$ are dimensionless. Thus $k x_{1}$ , $Ω t$ , and $t / (2 τ)$ are dimensionless, and $(C / k) ϕ^{'}$ has heat-flux units $W m^{- 2}$ .

Choose $C, κ, τ, T_{0}, L > 0$ , an integer $n \geq 1$ , and set $k : = 2 π n / L$ . Assume $k^{2} > C / (4 τ κ)$ and define

$Ω : = \sqrt{\frac{κ k^{2}}{C τ} - \frac{1}{4 τ^{2}}} > 0, ϕ (t) : = A e^{- t / (2 τ)} \cos (Ω t) .$

On $T_{L}^{3}$ , using the postulate's periodic-lift convention so these fields represent smooth spatially $L$ -periodic fields on the Minkowski rest space, let $e_{1} = (1, 0, 0)$ and put

$ϑ (t, x) : = ϕ (t) \cos (k x_{1}), q (t, x) : = - \frac{C}{k} ϕ^{'} (t) \sin (k x_{1}) e_{1} .$

For every real amplitude $A$ , these fields are smooth on $R \times T_{L}^{3}$ and solve the supported-rest Cattaneo linear subsystem. For a nonzero amplitude chosen sufficiently small, $T_{0} + ϑ$ stays positive for all $t \geq 0$ , and the state $(ϑ, q)$ stays in the postulate's declared open near-equilibrium neighborhood. The temperature-departure envelope decays as $e^{- t / (2 τ)}$ . This is a constructed model solution, not a reported observation; its physical interpretation is restricted to forward times inside that neighborhood.

Facts & Assumptions

Given: $C, κ, τ, T_{0}, L > 0$ , integer $n \geq 1$ , $k = 2 π n / L$ , $k^{2} > C / (4 τ κ)$ , and the supported-rest equations of Conditional thermal entropy, stability, and finite propagation. Let $D$ be the open state neighborhood selected in the underlying postulate.

[F1]

Conditional thermal entropy, stability, and finite propagation: the linear subsystem is $C ϑ_{t} + div q = 0$ , $τ q_{t} + q = - κ \nabla ϑ$ .

[F2]

Conditional thermal entropy, stability, and finite propagation: eliminating $q$ gives $τ ϑ_{t t} + ϑ_{t} - (κ / C) Δ ϑ = 0$ .

[F3]

Conditional thermal entropy, stability, and finite propagation: its physical interpretation is restricted to times when $(ϑ, q) \in D$ , the postulate's declared state neighborhood.

Verification

1.1givenF2algebra

The assumed strict inequality gives $Ω > 0$ . Write $ϕ (t) = e^{- t / (2 τ)} y (t)$ with $y (t) = A \cos (Ω t)$ . Differentiation yields $ϕ^{''} + τ^{- 1} ϕ^{'} + (κ k^{2} / (C τ)) ϕ = e^{- t / (2 τ)} [y^{''} + Ω^{2} y] = 0$ , because $y^{''} = - Ω^{2} y$ and $Ω^{2} = κ k^{2} / (C τ) - 1 / (4 τ^{2})$ . Thus $ϕ$ solves the one-mode telegrapher equation from [F2].

1.2givenF1algebra

Differentiating the displayed temperature field in time gives $C ϑ_{t} = C ϕ^{'} (t) \cos (k x_{1})$ , while differentiating the first component of $q$ in space gives $div q = - C ϕ^{'} (t) \cos (k x_{1})$ . Their sum is zero, so the thermal-energy balance in [F1] holds pointwise.

2.1givenF1step 1.1algebra

The flux satisfies $τ q_{t} + q = - (C / k) (τ ϕ^{''} + ϕ^{'}) \sin (k x_{1}) e_{1}$ . The mode equation in step 1.1 is equivalent to $- (C / k) (τ ϕ^{''} + ϕ^{'}) = κ k ϕ$ . Since $\nabla ϑ = - k ϕ \sin (k x_{1}) e_{1}$ , this equals $- κ \nabla ϑ$ , proving the Cattaneo relation in [F1] pointwise.

3.1givenF3algebra∎

For $t \geq 0$ , $∣ ϑ (t, x) ∣ \leq ∣ A ∣ e^{- t / (2 τ)} \leq ∣ A ∣$ , and $∣ ϕ^{'} (t) ∣ \leq ∣ A ∣ e^{- t / (2 τ)} (1 / (2 τ) + Ω)$ , so $∣ q (t, x) ∣ \leq (C / k) ∣ A ∣ (1 / (2 τ) + Ω)$ . Because $D$ in [F3] is open and contains $(0, 0)$ , choose $δ_{T}, δ_{q} > 0$ with $(- δ_{T}, δ_{T}) \times {q : ∣ q ∣ < δ_{q}} \subseteq D$ ; then any nonzero $A$ satisfying $∣ A ∣ < \min (T_{0}, δ_{T})$ and $(C / k) ∣ A ∣ (1 / (2 τ) + Ω) < δ_{q}$ keeps the entire forward solution in $D$ and gives $T_{0} + ϑ \geq T_{0} - ∣ A ∣ > 0$ .