Damped Cattaneo Fourier mode

Choose $C,\kappa ,\tau ,T_0,L>0$ and an integer $n\ge 1$. Set $k:=2\pi n\mathrm{/}L$ and assume $k^2>C\mathrm{/}(4\tau \kappa )$. Let $\mathrm{\Theta }\in \mathbb{R}$ with $0<\mathrm{\mid }\mathrm{\Theta }\mathrm{\mid }<T_0$ and

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

On ${\mathbb{T}}_L^3$, let $e_1=(1,0,0)$ be the first standard basis vector of ${\mathbb{R}}^3$ and define

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

This is a nonzero, smooth, spatially periodic solution for every $t\in \mathbb{R}$ of the linear Cattaneo subsystem

$C{\theta }_t+\mathrm{div}q=0,\tau q_t+q=-\kappa \mathrm{\nabla }\theta \mathrm{.}$

Under the conventional SI interpretation of [F1], $L$ is measured in m, $k$ in m${}^{-1}$, $\omega$ in s${}^{-1}$, and $\mathrm{\Theta },\varphi ,\theta ,T_0$ in K; the units of $C,\kappa ,\tau ,q$ are those stated in [F1]. These unit labels do not assert a material realization.

For $t\ge 0$, the proxy field $T_0+\theta$ remains positive, and its oscillation envelope decays as $e^{-t\mathrm{/}(2\tau )}$. The physical-temperature interpretation is limited to the interval where $T_0+\theta >0$.