The simultaneous diffusion of heat and moisture through soil is described by two coupled partial differential equations in which the diffusion coefficients are highly non-linear functions of the dependent variables, The system has been regarded as analytically intractable for any generality of coupled how. However, for an asymptotically steady state, the equations show a marked periodic stability. Computer simulation indicates that the behaviour quickly becomes entrained to input boundary periodicity for any initial state, regardless of the detailed functional form of the diffusion coefficients. This property allows an harmonic series solution to be assembled. Factors such as amplitude decay, phase shift and wave form evolution may be evaluated. The solution is adapted to boundary conditions pertaining to arid soils and the results validated against the 1968 field data of Rose and the 1973 experiment by Jackson.