Supported thermal finite-speed damped mode

Use the supported-rest SI conventions inherited through Conditional thermal entropy, stability, and finite propagation: $t$ and $\tau$ are in seconds, $x_1$ and $L$ in metres, $T_0$ and $\vartheta$ in K, $C$ in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$, $\kappa$ in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, and $q$ in $\mathrm{W}{\mathrm{m}}^{-2}$. The amplitude $A$ and mode function $\varphi$ have temperature units K; the wave number $k$ has units ${\mathrm{m}}^{-1}$ and the angular frequency $\mathrm{\Omega }$ has units ${\mathrm{s}}^{-1}$. The integer $n$ and Cartesian unit vector $e_1$ are dimensionless. Thus $k{x}_1$, $\mathrm{\Omega }t$, and $t\mathrm{/}(2\tau )$ are dimensionless, and $(C\mathrm{/}k){\varphi }^{\mathrm{\prime }}$ has heat-flux units $\mathrm{W}{\mathrm{m}}^{-2}$.

Choose $C,\kappa ,\tau ,T_0,L>0$, an integer $n\ge 1$, and set $k:=2\pi n\mathrm{/}L$. Assume $k^2>C\mathrm{/}(4\tau \kappa )$ and define

$\mathrm{\Omega }:=\sqrt{\frac{\kappa k^2}{C\tau }-\frac{1}{4{\tau }^2}}>0,\varphi (t):=A{e}^{-t\mathrm{/}(2\tau )}\cos (\mathrm{\Omega }t)\mathrm{.}$

On ${\mathbb{T}}_L^3$, 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

$\vartheta (t,x):=\varphi (t)\cos (k{x}_1),q(t,x):=-\frac{C}{k}{\varphi }^{\mathrm{\prime }}(t)\sin (k{x}_1)e_1\mathrm{.}$

For every real amplitude $A$, these fields are smooth on $\mathbb{R}\times {\mathbb{T}}_L^3$ and solve the supported-rest Cattaneo linear subsystem. For a nonzero amplitude chosen sufficiently small, $T_0+\vartheta$ stays positive for all $t\ge 0$, and the state $(\vartheta ,q)$ stays in the postulate's declared open near-equilibrium neighborhood. The temperature-departure envelope decays as $e^{-t\mathrm{/}(2\tau )}$. This is a constructed model solution, not a reported observation; its physical interpretation is restricted to forward times inside that neighborhood.