PHYSICS 312. Midterm. February 10 2003 Allowed aids formula sheet
(2 pages). Calculator. Answer all three questions.

Problem 1:
a: Find the solution to the wave equation

$$\frac{\partial ^2u}{\partial x^2}=\frac{1}{c^2}\frac{\partial ^2u}{\partial t^2},\;-\infty<x<\infty;\; t>0$$

satisfying

$$u(x,0)=e^{-x^2};\;\; \frac{\partial u(x,t)}{\partial t}\vert _{t=0}=-2xce^{-x^2}$$

b: Describe qualitatively the solutions found in a:.
Problem 2:
a: For which values of $\lambda>0$ will

$$\frac{d^2\phi}{dx^2}+\lambda^2\phi=0$$

have solutions (other than $\phi=0$) satisfying

$$\frac{d\phi(0)}{dx}=0;\;\;\phi(a)=0$$

b: Sketch the solutions for the three lowest values of $\lambda$.
Problem 3:
The steady state temperature inside a bar satisfies the differential equation

$$\frac{d}{dx}(\kappa(T)\frac{dT}{dx})=0$$

The ends $x=0$ and $x=L$ are kept at the temperatures $T=0$ and $T=T_0>0$, respectively. The thermal conductivity depends on the temperature according to

$$\kappa(T)=\kappa_0+\alpha T$$

where $\alpha$ is a constant.
a: When integrating the steady state differential equation you will end up with a quadratic equation for $T$. Find this equation.
b: Solve this equation to find the steady state temperature distribution $T(x), \;0<x<L$.
c: Which of the two roots is correct when $\alpha>0$?