What is the relationship between the shape of a drum and the sound
the drum makes?
It turns out that this is quite a complicated question. You can
read about the
mathematics of drum design, and the discovery that
you can't always
hear the shape of a drum.
In fact, there is a deep connection between math and music in general. Check
out these links at the Math Archive.
You can read more about partial differential equations on
the Mathematics: the equations of nature page.