  Modeling underground temperatures provides a practical application of the
one-dimensional heat equation. In this work, the one-dimensional heat equation
in surface soil is extended to include heat carried by the vertical flow of
rainwater through the soil. Analytical solutions, with and without water flow,
illustrate the influence of rainwater circulation on the sub-surface
propagation of seasonal temperature variations, an important effect that is
generally neglected in textbooks. The surface temperature variations are damped
by the soil, and this effect was used by the Troglodytae in Egypt or the Petra
in South Jordan to insulate against extreme temperatures. For a realistic case
of horizontally layered geology, a finite difference Python code was developed
for the same purpose. Subsurface temperatures were also measured over a full
year at depths up to 1.8 m and used to estimate the thermal skin depth and
thermal wavelength. This study provides students with a practical example of
how a textbook physics problem can be modified to extract information of
contemporary importance in geophysics and global warming.
