= Ångström method worked example
{c}
Consider an original stipulated model, not measured material data: <thermal diffusivity> $\alpha=10^{-4}\,\mathrm{m^2\,s^{-1}}$, period $P=100\,\mathrm s$, driven <temperature> <amplitude> $\Theta_0=4\,\mathrm K$, and <thermometers> at $x_1=0.020\,\mathrm m$ and $x_2=0.050\,\mathrm m$. Without side losses, $a=b=17.7245\,\mathrm{m^{-1}}$ and <thermal penetration depth> $\delta=0.05642\,\mathrm m$. Thus
$$
A_1=2.80612\,\mathrm K,\quad A_2=1.64883\,\mathrm K,\quad
D=\Delta\phi=0.531736,\quad\Delta t=8.46284\,\mathrm s.
$$
The two no-loss estimates and the combined estimate all return $10^{-4}\,\mathrm{m^2\,s^{-1}}$.
Now retain the same <thermal diffusivity> and drive but introduce $\lambda=0.010\,\mathrm{s^{-1}}$. The exact <complex wavenumber> gives $a=19.1863\,\mathrm{m^{-1}}$ and $b=16.3742\,\mathrm{m^{-1}}$, hence
$$
\begin{aligned}
A_1&=2.72527\,\mathrm K,& A_2&=1.53262\,\mathrm K,\\
D&=0.575588,&\Delta\phi&=0.491225\,\mathrm{rad},\\
\Delta t&=7.81809\,\mathrm s,&\alpha_{\rm combined}&=10^{-4}\,\mathrm{m^2\,s^{-1}}.
\end{aligned}
$$
Using attenuation alone with the no-loss formula would give $8.5343\times10^{-5}\,\mathrm{m^2\,s^{-1}}$; using lag alone would give $1.17174\times10^{-4}\,\mathrm{m^2\,s^{-1}}$. Their disagreement is informative. If an independent <volumetric heat capacity> measurement gives $\rho c_p=2.0\times10^6\,\mathrm{J\,m^{-3}\,K^{-1}}$, the combined estimate implies <thermal conductivity> $k=200\,\mathrm{W\,m^{-1}\,K^{-1}}$ and $H=2.0\times10^4\,\mathrm{W\,m^{-3}\,K^{-1}}$.
\Image[/angstrom-periodic-heating-amplitude-phase.png]
{title=Solved <Ångström method> example: spatial <amplitude>, <wave phase> lag, time traces and finite-end bias}
{description=Original analytical predictions using the stipulated parameters above. Dots in the upper panels independently solve the spatial <heat equation> with <finite differences>. The lower-left traces show the linear-loss case at the two sensor positions. The lower-right curve applies the semi-infinite inference to an exact finite rod with an insulated far end: the resulting apparent <thermal diffusivity> can lie above or below the true value. None of these curves are experimental data.}
The exact <complex amplitudes> can be checked independently by discretizing $\alpha\widehat\theta''-(\lambda+i\omega)\widehat\theta=0$ and solving the resulting <linear system> with a <tridiagonal matrix>. Halving the spatial step quarters the discrepancy, as expected for centered <finite differences>. Separately fitting the real time traces to a constant plus <sine> and <cosine> terms recovers both <amplitudes> and their lag; this also checks the sign convention used in the combined estimate.
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