IIRC, the exponential decay is ‘physical’—and you can also Play the sound the drum makes!
But this actually only makes sense if you include the most appropriate “harmonics”—actually, the superposition of various modes, given different weightings, and different decay rates—which have frequencies that are not rational multiples of one another!
I did this a few years ago (before the Community existed), using measured values for the weightings and decay rates—from The Physics of Musical Instruments—and the sound generated was a reasonable approximation to a Timpani or Kettle Drum (but, of course, the body of the drum is actually a key to the sound generated). Here is an example:
Play[1.0 Cos[128 2 Pi t] E^(-((5.25 t)/0.4)) +
2.5 Cos[145 2 Pi t] E^(-((5.25 t)/2.3)) +
1.9 Cos[218 2 Pi t] E^(-((5.25 t)/3.7)) +
1.5 Cos[287 2 Pi t] E^(-((5.25 t)/4.6)) +
0.7 Cos[354 2 Pi t] E^(-((5.25 t)/4.3)), {t, 0, 2}]
And there is also NDEigensystem, which I recently used to try out the “hearing the drum” problem (again I need to find this, and post it!)
