Can You Hear the Shape of a Drum?

If you had perfect pitch and listened to a recording of the sounds a drum made when struck, could you determine the shape of the drum? This question is an example of an inverse problem; inverse problems arise in medical imaging, oil prospecting, spectroscopy, and many other fields. We’ll first discuss the analogous question in the simpler setting of plucking a string. Then we’ll tackle the problem for drums and see that there are some surprises. Finally, I will give a brief indication of how this problem relates to some of my recent research. The emphasis will be on the ideas rather than on the technical details, so there will be pretty pictures instead of equations.