6.3 Periodic heating: Fourier’s cellar

If the surface of the ground, \(x = 0\), has temperature \(A\cos \omega t\) about its mean, the temperature at depth \(x\) is the decaying wave \[ u(x, t) = Ae^{-qx}\cos (\omega t - qx), \qquad q = \sqrt {\frac {\omega }{2\kappa }} . \] To check, \(u\) is the real part of \(Ae^{i\omega t - (1 + i)qx}\), and \(\kappa \bigl ((1 + i)q\bigr )^2 = 2i\kappa q^2 = i\omega \). The amplitude falls by the factor \(e^{-qx}\) and the phase lags by \(qx\), so the slower the oscillation, the deeper it reaches. Fourier used this to explain why deep cellars are cool in summer and warm in winter: at the depth \(\pi /q\) the annual cycle is exactly out of phase with the surface, and reduced by \(e^{-\pi } = 0.043\).

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