An integration of the Fourier equation of heat conduction in a semi-infinite medium (air) is presented for the case where the thermal (or eddy) diffusivity varies with height, assuming a harmonic variation of temperature with time. Special attention is paid to the case where the medium consists of an arbitrary number of layers in each of which the diffusivity is represented by a different function of height. Application of these methods resolves some of the difficulties encountered in explaining the observed variation of amplitude and phase of the temperature wave with height. It is concluded that —although theoretically possible —it is not feasible in practice to evaluate the diffusivity from an analysis of observations of this type, and that some results of previous attempts to do so are largely in error. Relations between the waves of temperature and heat flux near the surface are briefly discussed. It is shown that for the diurnal waves these depend, inter alia, on the diffusivity values up to a height of the order of 1000 meters.