= Thermal diffusion wave
{title2=$\theta(x,t)=\Theta_0e^{-ax}\cos(\omega t-bx)$}
A <thermal diffusion wave> is a periodically forced solution of the <heat equation>, with spatially decreasing <amplitude> and varying <wave phase>. In a semi-infinite rod $x\geq0$, assume constant <thermal diffusivity>, negligible side losses, a nearly uniform <temperature> across each cross-section, and the <boundary conditions> $\theta(0,t)=\Theta_0\cos\omega t$ and $\theta(x,t)\to0$ as $x\to\infty$. A <steady periodic thermal state> then satisfies
$$
\partial_t\theta=\alpha\partial_x^2\theta,\qquad
\widehat\theta(x)=\Theta_0e^{-\gamma x},\qquad
\gamma^2=\frac{i\omega}{\alpha},\qquad
\gamma=a+ib=(1+i)\sqrt{\frac{\omega}{2\alpha}}.
$$
Choose the <square root> in the <complex numbers> with $a>0$ so that the <amplitude> decays into the rod. Substitution of the <phasor> into the <heat equation> proves the formula: $i\omega\widehat\theta=\alpha\widehat\theta''$. Taking the real part gives the displayed <cosine>. In this convention $a$ is the spatial attenuation slope and $b$ is the spatial lag slope, both in $\mathrm{m^{-1}}$; $\gamma$ combines them into one <complex number>. If a <complex wavenumber> $K$ is defined by $e^{i\omega t-iKx}$ instead, $K=b-ia$.
With $\Delta x=x_2-x_1>0$, let $A_j=|\widehat\theta(x_j)|$ and define a positive downstream lag $\Delta\phi$. Then
$$
D=\log\frac{A_1}{A_2}=a\Delta x,\qquad
\Delta\phi=b\Delta x,\qquad
a=b\quad\text{without side losses},
$$
and either measurement determines <thermal diffusivity>:
$$
\alpha=\frac{\omega\Delta x^2}{2D^2}
=\frac{\omega\Delta x^2}{2(\Delta\phi)^2}.
$$
The <logarithm> uses an <amplitude> ratio, not a ratio of squared <amplitudes>. The lag is in <radians> and must be tracked continuously across cycles: a measured <phase difference> modulo $2\pi$ leaves an integer-cycle ambiguity. Use closer sensor spacings or vary <frequency> continuously to resolve that ambiguity. Equal attenuation and lag slopes provide a useful diagnostic of the no-side-loss model.
The apparent <phase velocity> is $\omega/b=\sqrt{2\alpha\omega}$, which depends on <frequency>. A <thermal diffusion wave> therefore does not give a material-independent travel speed for <heat>; it is a forced pattern in a dissipative <diffusion equation>. The ideal <heat equation> still has <infinite propagation speed>, and the observed lag is not the arrival time of a sharp front.
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