Because the surface temperature is time-periodic, we can assume that the subsurface temperature also varies periodically with time at the same frequency. However, it is not correct to assume that T ′(t) is simply cosωt, as that would imply that the subsurface temperatures are in exact time phase with the surface temperature. In other words, if T ′(t) were cosωt, the maximum and minimum temperatures would be reached at the same times, independent of depth. In fact, we will see that the temperature changes at different depths are not in phase; the maximum temperature at any depth lags behind the maximum temperature at the surface, for example, because of the finite amount of time required for the temperature maximum to diffuse downward. The difference in phase between temperature variations at the surface and those at depth can be accounted for by using both cosωt and sinωt dependences for T ′(t). We generalize Equation (4–75) to T (y, t) = T0 + Y1(y) cosωt+ Y2(y) sinωt. (4.76)
By substituting Equation (4–76) into Equation (4–68), we find −ωY1 = κ d2Y2 dy2 ωY2 = κ d2Y1 dy2 .
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