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

(E.4)

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

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

(E.5)

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

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

(E.6)

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

(E.7)

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

(E.8)

For example, at the surface, the peak flux is

(E.9)
Box E.19. Working out the natural flux caused by sinusoidal temperature variations.