One face of a thick uniform layer is subject to sinusoidal temperature variations of angular frequency ω. Show that damped sinusoidal temperature oscillations propagate into the layer and give an expression for the decay length of the oscillation amplitude. A cellar is built underground and is covered by a ceiling, which is 3 m thick and made of limestone. The outside temperature is subject to daily fluctuations of amplitude 10∘C and annual fluctuations of 20∘C. Estimate the magnitude of the daily and annual temperature variations within the cellar. Assuming that January is the coldest month of the year, when will the cellar's temperature be at its lowest?
[The thermal conductivity of limestone is 1.6Wm−1K−1, and the heat capacity of limestone is 2.5×106JK−1m−3.]