Is it possible to calculate how a wind instrument will sound just based on its geometry? It seems like it could be done like a cellular automaton, if I enclosed a very-fine 3-d grid within a surface and use cube-cell rules to define how the particles will bounce and propagate the wave based on the surrounding cells. I could feed an audio file converted into a stream of voxels forming longitudinal waves into the instrument's mouthpiece, then measure and record near the output and convert back into a wave file. I searched around and couldn't find acoustic/sound simulators that aren't either extremely expensive or require a business inquiry, and none of them seem to be what I am looking for, which is producing an output sound given a shape and input sound file.
I have a rudimentary understanding of acoustics and waves - I can calculate the speed of sound through air, find a waves harmonics, the doppler shift between moving sources/observer, but these things don't apply as much to an automata-based simulation.. Would modeling sound like this work or would the fidelity be impossible to achieve, or be completely inaccurate because of the complex interactions of air particles bouncing off each other and other materials?
 A: This can be done with low accuracy, but not high accuracy. The approach would be a finite element method (FEM), which indeed works by breaking the geometry into small volumes and solves the appropriate differential equations by coupling adjacent volumes. This works in both the frequency and time domains. One commercial software to do this is COMSOL, and there are others with the general trade off between ease of use and power versus expense.
Probably you would proceed by finding the transfer function of your geometry for each frequency (how each frequency gets amplified or attenuated). Then you would Fourier transform your input sound file, apply the transfer spectrum, and inverse Fourier transform back to the time domain. This would result in a sound waveform modified by the instrument spectrum.
So something along the lines of what you want is possible. However, you could never do this with high accuracy. This is because what gives an instrument it’s timbre is the precise combination of overtones, which are highly dependent on the excitation. Why do brass instrument sound so distinct from wind instruments? Primarily the mouthpiece. And how exactly that mouthpiece is played can be significant. Two people playing the same clarinet can make it sound noticeably different in timbre simply by how they hold their mouth. Of course, it’s possible to simulate all this with ever more complex models. But eventually the assumptions of the models are not going to match reality because reality is simply too complicated (how does one simulate a tongue?). And then it’s just a case of garbage in, garbage out.
