Chapter 14.3, Problem 100E

In a study of frost penetration it was found that the temperature T at time t (measured in days) at a depth x (measured in feet) can be modeled by the function

$T\left(x,t\right)=T_0+T_1{e}^{-\lambda x}\sin \left(\omega t-\lambda x\right)$

where $\omega =2\pi /365$ and λ is a positive constant.

(a) Find $\partial T/\partial x$. What is its physical significance?

(b) Find $\partial T/\partial t$. What is its physical significance?

(c) Show that T satisfies the heat equation $T_t=k{T}_{xx}$ for a certain constant k.

(d) If $\lambda =0.2$, $T_0=0$, $T_1=10$, use a computer to graph $T\left(x,t\right)$.

(e) What is the physical significance of the term $-\lambda x$ in the expression $\sin \left(\omega t-\lambda x\right)$?