If we assume the ground is made of solid homogenous material with con-
ductivity κ and heat capacity CV, then the temperature at depth z below the
ground and time t responds to the imposed temperature at the surface in
accordance with the diffusion equation


For a sinusoidal imposed temperature with frequency ω and amplitude A at
depth z = 0,

T(0, t) = Tsurface(t) = Taverage + Acos(ωt),


the resulting temperature at depth z and time t is a decaying and oscillating

T(z, t) = Taverage(t) + Ae-z/z0  cos(ωt − z/z0),


where z0 is the characteristic length-scale of both the decay and the oscillation,


The flux of heat (the power per unit area) at depth z is


For example, at the surface, the peak flux is

Box E.19. Working out the natural flux caused by sinusoidal temperature variations.