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(3,1)=T. + a(z)sin( con- D where D= 22 (k/m)" a(z) = a(0)exp(--/D) EK Oz? (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) at aT OT O'T (start with equation You need to find expressions for and at at Oz? Differentiate the equation T (2,1)=T. + a(z)sin ( mt 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 ACLI the equation obtained after comparing the coefficients which will lead you to 1 So to a = 4, exp and a = a, exp(-bz); and D=-= 212 and a = a exp .) 2.b b 1/2 K = = 0