Is there software that, when the input is the shape of a drum, will produce the corresponding audible sound?
1 Answer
The full physics problem is complex, the vibrating membrane displaces the air, which causes a backreaction and signifantly modifies the response. Moreover, the response also depends sensitively on where the membrane is hit. Professional simulators take all of this into account, see for example the COMSOL description.
For a simpler procedure you could just use a numerical Laplace equation solver and pass the frequencies to an audio generator. A worked example using Mathematica is here: NDEigenvalues
gives you the resonant frequencies of a userdefined shape of the membrane, AudioGenerator
then converts these into sound.

1$\begingroup$ From a nonmath standpoint, if you are just trying to make the sound of drums, you can look into socalled "physical modeling" software from the music world. madronalabs.com/products/kaivo : "Kaivo brings some of the latest academic research in physical modeling to a patchable package for the first time. Mathematically speaking, its finite difference time domain (FDTD) models let the player reach inside the instrument and affect the internal vibrations at any point. This allows for a fine degree of realistic detail, like the bridge rattles on a 'gut string' model, for example." $\endgroup$– sfscsCommented Jul 16 at 6:38

$\begingroup$ interesting, but it seems you cannot choose the shape of the drum with that software. $\endgroup$ Commented Jul 16 at 8:33