EML 5060 Analysis in Mechanical Engineering 4/23/92
Closed book Van Dommelen 12:30-2:30pm

Show all reasoning and intermediate results leading to your answer. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used.

1.

The wave equation w_tt = a² w_xx can be reduced to a first order system by defining u=w_t and v=w_x:

u_t = a² v_x

v_t = u_x

Solve this as a system and verify that the same solution is obtained as we get from a canonical form of the wave equation.

2.

A semi-infinite bar with unit heat conduction coefficient is initially at zero temperature, but at the end heat is being added at a unit rate:

$${\partial T \over \partial t} = {\partial^2 T \over \partial x^2}$$

$$T(x,0)=0 \qquad {\partial T \over \partial x} (0,t)=1$$

Find the temperature in the bar for arbitrary times.

3.

A circular plate has an insulated circumference. The heat conduction coefficient is unity:

$${\partial T \over \partial t} = \nabla^2 T$$

$${\partial T \over \partial r} = 0 \hbox{ at } r=1$$

The initial temperature has an arbitrary variation. Approximately how long will it take until the initial temperature variations decay by a factor 10? Your estimate for this time should be exact under certain conditions. Which ones?