EML 4930/5061 Analysis in M.E. II 12/8/15
Closed book Van Dommelen 8:00-10:00 pm
Show all reasoning and intermediate results leading to your answer, or
credit will be lost. One book of mathematical tables, such as Schaum's
Mathematical Handbook, may be used, as well as a calculator and one
handwritten letter-size single formula sheet.
- (20%) Solve the following PDE and boundary condition for
using the method of characteristics:
Clean up your answer! Very neatly draw a set of characteristics to
fully cover the complete first quadrant. Shade any regions in which
the initial condition does not fix the solution. Indicate any
singular points.
Solution.
- (20%) Use D’Alembert to find the acoustics in
an organ pipe with open ends:
if the initial string deflection and velocity are
In two very neat graphs, show the initial conditions extended to all
that make the boundary conditions automatic. Evaluate the exact
solution . Make sure to explicitly list the value of each
term in the expression for it. Exact value only. Simplify
fully.
Solution.
- (20%) Use the Laplace transform method to solve the following
damped wave propagation problem:
where is a given function. Clean up completely.
Use only the attached Laplace transform tables unless stated
otherwise. Use only one table item in each step you take (except
P2) and list it! Use convolution only where it is unavoidable.
No funny (discontinuous) functions in your answers.
Solution.
- (40%) Use separation of variables to solve the following damped
wave propagation problem:
with the initial conditions
and boundary conditions:
Show all reasoning. Show exactly what problem you are solving using
separation of variables. Fully explore all possible
eigenfunctions.
At the end, write out the fully worked out and fully
simplified solution completely, with all parameters in it clearly
identified. The professor should be able to simply take your final
expressions and put them in a computer program to plot the solution
without having to find stuff elsewhere.
Solution.
Table 1:
Properties of the Laplace Transform.
(, )
|