The daily and annual variations of temperature at the surface of the earth
may be represented by sine-wave oscillations, with equal amplitudes and periods of 1 day and 365 days respectively. Assume that for (angular) frequency ω the temperature at depth x in the earth is given by u ( x, t ) = A sin( ωt + μx ) exp(- λx ), where λ and μ are constants.
1. Use the diffusion equation to find the values of λ and μ .
2. Find the ratio of the depths below the surface at which the two amplitudes have dropped to 1 / 20 of their surface values.
3. At what time of year is the soil coldest at the greater of these depths, assuming that the smoothed annual variation in temperature at the surface has a minimum on February 1st?