Annual ground-temperature wave ($T-T_0=-T_A{e}^{-z/\delta }\cos (\omega t-z/\delta )$)

A sinusoidal surface temperature applied to a homogeneous semi-infinite conductor produces a periodic depth profile with penetration length $\delta =\sqrt{2\kappa /(\omega C)}$. It follows by substituting a complex exponential into $C{\partial }_{t}T=\kappa {\partial }_z^{2}T$ and taking the root that decays at depth. A depth $z$ attenuates the amplitude by $e^{-z/\delta }$ and delays the phase by $z/\delta$. This is diffusion with phase delay, rather than a nondispersive travelling wave.