Knowing all "sounds" of a drum (all eigenfrequencies), one
can determine its area and its circumference. In 1966, Mark Kac posed
the question if one can hear the shape of a drum, see [1].
It took until 1992 to get this question answered. In [2], Carolyn Gordon, David
Webb and Scott Wolpert explained that it is possible to have a pair of
differnt drums which "sound" exactly the same. Here
is a link to the AMS with a short description and a picture of Carolyn
and David with paper models of the pair of drums which sound the
same. It turned out that there are many more pairs of drum which
sound the same, see [3].
Some more information can be found on an MAA page
and Wikepedia.
Seminar
To investigate the sound of drums, many areas of mathematics can be
used: differential equations, geometry, groups, symmetries,
and, number theory. During the seminar, we will touch upon the
following topics to get a better idea of the problem.
- Sounds of a string: the wave equation, its solutions and
eigenfrequencies. This can be used to find the length of a
string. See e.g. Mathworld and
Wikepedia.
-
Sounds of a drum and the two-dimensional wave equation.
See e.g. Mathworld and again
Wikepedia.
- A rectangular drum and eigenfunctions. See e.g. Mathworld.
- A counting function (Weyl's formula) to find the area and circumference of a drum.
- Shapes of drums that "sound" the same.
References
-
Kac, Mark (1966), "Can one hear the shape of a drum?", American
Mathematical Monthly 73 (4, part 2): 1-23
-
Gordon, Carolyn; Webb, David L.; Wolpert, Scott (1992), "One Cannot
Hear the Shape of a Drum", Bulletin of the American Mathematical
Society 27: 134-138
-
Buser, Peter; Conway, John; Doyle, Peter; Semmler, Klaus-Dieter
(1994), "Some planar isospectral domains", International Mathematics
Research Notices 9: 391ff
