=K 14.3 (a) By considering the heat balance between time t and t + dt of a slab of unit area at depth z and thickness dz, show that its temperature changes at a rate given by at 22T (14.34) at z2 (b) Assume that the temperature at the soil-air interface varies with time as T(0,t) = To +asinot and that heat flow in and out of the soil is only by conduction. Show that the temperature at depth z is T(z,t) = To + a(z)sin(ot - 2 / D (14.35) with D = 2K / 0)12 and a(z) = a(O) exp(-2/D) (Hint: Differentiate the left side of (14.35) with respect to tand the right side twice with respect to z. (c) For a typical soil K = 0.3x10-6ms-! Over a period of one year, calculate D and hence find the peak-peak variation of tem- perature at depths of 1 m, 3 m and 5 m for the case To=15C, a(O) = 20C