Ex 100

In a study of frost penetration it was found that the temperature \(T\) at time \(t\) (in days) at a depth \(x\) (in feet) can be modeled by the function
\[T(x, t) = T_0 + T_1 e^{-\lambda x} \sin(\omega t - \lambda x)\]
where \(\omega = 2\pi / 365\) and \(\lambda\) is a positive constant.

  1. Find \(\partial T / \partial x\). What is its physical significance?
  2. Find \(\partial T / \partial t\). What is its physical significance?
  3. Show that \(T\) satisfies the heat equation \(T_t = k T_{xx}\) for a certain constant \(k\).
  4. If \(\lambda = 0.2\), \(T_0 = 0\), and \(T_1 = 10\), use a computer to graph \(T(x, t)\).
  5. What is the physical significance of the term \(-\lambda x\) in the expression \(\sin(\omega t - \lambda x)\)?