As we discussed in the text, the origins of Fourier analysis can be found in problems of

Chapter 3, Problem 3.67

(choose chapter or problem)

As we discussed in the text, the origins of Fourier analysis can be found in problems of mathematical physics. In particular, the work of Fourier was motivated by his investigation of heat diffusion. In this problem, we illustrate how the Fourier series enter into the investigation. 13 Consider the problem of determining the temperature at a given depth beneath the surface of the earth as a function of time, where we assume that the temperature at the surface is a given function of time T(t) that is periodic with period I. (The unit of time is one year.) Let T(x, t) denote the temperature at a depth x below the surface at time t. This function obeys the heat diffusion equation with auxiliary condition aT(x, t) at ~k T(x, t) 2 ax2 T(O, t) = T(t). (P3.67-l) (P3.67-2) Here, k is the heat diffusion constant for the earth (k > 0). Suppose that we expand T(t) in a Fourier series: +x T(t) = L anejnZm. (P3.67-3) n= -:o Similarly, let us expand T(x, t) at any given depth x in a Fourier series in t. We obtain +oo T(x, t) = L bn(x)ejnZm, (P3.67- 4) n= -oc where the Fourier coefficients bn(x} depend upon the depth x. 13The problem has been adapted from A. Sommerfeld, Partial Differential Equations in Physics (Nt York: Academic Press, 1949), pp 68-71. (a) Use eqs. (P3.67-1)-(P3.67-4) to show that h11 (x) satisfies the differential equation with auxiliary condition 47T"jn ---yzr-b"(x) (P3.67-5a) (P3.67-5b) Since eq. (P3.67-5a) is a second-order equation, we need a second auxiliary condition. We argue on physical grounds that, far below the earth's surface, the variations in temperature due to surface fluctuations should disappear. That is, lim T(x, t) = a constant. x~x (b) Show that the solution of eqs. (P3.67-5) is bn(x) = [ an exp[- J27Tinl(l + j)xl k], an exp[- J27Tinl(l - j)xlk], n 2:: 0 n s 0 (P3.67-5c) (c) Thus, the temperature oscillations at depth x are damped and phase-shifted versions of the temperature oscillations at the surface. To see this more clearly, let T(t) = ao + a, sin 2m (so that a0 represents the mean yearly temperature). Sketch T(t) and T(x, t) over a one-year period for a0 = 2, and a 1 = 1. Note that at this depth not only are the temperature oscillations significantly damped, but the phase shift is such that it is warmest in winter and coldest in summer. This is exactly the reason why vegetable cellars are constructed!