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A complex number used as a wavenumber to encode both spatial oscillation and exponential variation of a wave amplitude. With the convention $e^{i\omega t-iKx}$, a disturbance decaying for increasing $x$ has $\mathrm{Im}K<0$. The sign depends on the convention; thermal diffusion waves provide a concrete example.

Write ${\delta }_j=\sqrt{2K/{\omega }_j}$ for each component's thermal penetration depth. Its contribution at depth isThus its amplitude is multiplied by $e^{-x/{\delta }_j}$ and its wave phase lags the surface by $x/{\delta }_j$ radians. The corresponding unwrapped time lag is $x/({\delta }_j{\omega }_j)$; a phase measured from a single periodic signal is observable only modulo $2\pi$.

Higher-frequency variations have smaller penetration depths. In particular,The daily oscillation is therefore strongly attenuated at depths where a substantial annual oscillation survives. This is a thermal diffusion wave, whose phase lag is not the arrival time of a sharp thermal front. The displayed solution is the established periodic response; an additional transient would depend on initial subsurface data not needed for this interpretation.

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 $x\ge 0$, 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 $\theta (x,t)\to 0$ as $x\to \infty$. A steady periodic thermal state then satisfiesChoose 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: $i\omega \hat{\theta }=\alpha {\hat{\theta }}^{\prime \prime }$. 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 $A_j=|\hat{\theta }(x_j)|$ and define a positive downstream lag $\Delta \varphi$. Thenand either measurement determines thermal diffusivity: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.

The thermal penetration depth of a harmonic ground temperature disturbance scales as the inverse square root of its angular frequency. For yearly and daily forcing, a 365-day year gives ${\delta }_Y/{\delta }_D=\sqrt{365}\simeq 19.1$. At ten daily penetration depths the daily component is attenuated by $e^{-10}$, whereas the annual one retains about $59.2\%$ of its surface amplitude. At ten annual penetration depths, even the annual component is attenuated by $e^{-10}$. These are consequences of the linear thermal diffusion wave model, with homogeneous constant thermal diffusivity.

For the no-side-loss thermal diffusion wave, the thermal penetration depth $\delta =1/a=\sqrt{2\alpha /\omega }=\sqrt{\alpha P/\pi }$ is the distance over which the amplitude falls by $e^{-1}$ and the wave phase lags by one radian. It is a periodic counterpart of the thermal diffusion length, with its numerical factor fixed by this definition. A shorter period probes a shallower region; a longer period increases sensitivity to the far-end boundary condition.