Thermal penetration depth ($\delta =\sqrt{2\alpha /\omega }$)

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.