b) Assume that the temperature at the soil-air interface varies with time as T(0,t)=TO + asinwt and that heat flow in and out of the solid is only by conduction. Show that the temperature at depth z is: T (2,1)=T, + a(z)sin est D where D=22 (/w) a(z)= a(0)exp(-2/D) 3( -) 1/2 = K (you can start by differentiating left hand side of T(z,t) equation with respect to t and the right hand side twice with respect to z) O'T (start with equation OT OPT You need to find expressions for and at Oz? Ot Oz Differentiate the equation T (2,1)=T, +a(z)sin (ct bz) with respect to time once and double differentiate with respect to z. Once derivatives are complete, plug in the manu starting equation and compare co-efficients of Sin() and Cos (). You might need to solve the equation obtained after comparing the coefficients which will lead you to 1 a = a exp and a = a, exp(-bz); and D 2Kb b = 22 and a = a exp