Another thermal boundary condition that frequently arises in natural as well as in engineering systems is the periodic change of the temperature of the surrounding medium. The daily and seasonal changes of solar radiation on the soil or on buildings, the periodic temperature changes in the cylinders of internal combustion engines, the (on-off) temperature control by a thermostat, and the periodic heat transfer in a regenerator are examples for this kind of boundary condition.

The periodic change of temperature of the surroundings may be represented by a simple harmonic oscillation about an average value T̄ with an amplitude (T_{∞, max} – T̄ ):

\[\label{eq1} T_{\infty}(t)= \bar{T}+\left(T_{\infty,\rm max}- \bar{T}\right)\sin(\omega t) \tag{1}\]

If this boundary condition has been maintained for a sufficiently large number of periods (t_P = 2π /ω)t ⪢ t_P, the temperatures inside the body will undergo a periodic change too:

\[\label{eq2} T(x,t)=\bar{T}+\left(T_{\infty,\rm max}- \bar{T}\right)\hat{A}(x)\sin[\omega t+\varphi(x)] \tag{2}\]