# Does/can physical audio theory exist?

I've been wondering,

Since some sound generating physical phenomena have been physically modelled as physical modelling synthesisers,

Does there exist a "physical audio theory" (or scientific subdomains fit for belonging to "physical audio theory"), which is like a subfield of applied mathematics or physics that describes the theory of sound generators, propagation and such. For example, the Karplus-Strong algorithm and its analysis would belong there and it would be presented as a derivation of string physics. Sure there's the physics of sound under physics, but I have not seen a coherent mathematical theory on "physical modelling synthesis", even if "physical modelling synthesis" could perhaps be laid out as mathematical formalisms, e.g. as an "applied linear algebra" body of theory. It seems like much of audio DSP is just scattered information.

For those interested, I also think there are at least some books concerning "physical audio theory", an example would be this book:

https://mitpress.mit.edu/books/designing-sound

It does contain chapters that predominantly analyze natural phenomena or "sound generating settings" and try to translate those into expressions in mathematics, using concepts from physics. The derivations don't differ that much from e.g. typical engineering analysis. Much like the Karplus-Strong algorithm was developed by analyzing string instruments.

• There are several books related to the topic, including (but not limited to): "Physics and Music" by White, "The Physics of Sound" by Berg, "Music, Physics and Engineering" by Olson, "The Theory of Sound" by Rayleigh, "Musimathics: The Mathematical Foundations of Music" by Loy, and etc. The perceptual end would include many many books on audiology, psychoacoustics and human hearing. They all cover slightly different but greatly overlapping sets of topics, some based around physics. What is an elephant? – hotpaw2 Jan 29 '16 at 19:08
• "It seems like much of audio DSP is just scattered information."  yes, it does seem that way. (somethings things are as they appear.) – robert bristow-johnson Jan 30 '16 at 1:43
• I don't think the question is more opinion-based than a question concerning "foundations" of some theory/thing. If "physical audio theory" is feasible, then there wouldn't be "that many" ways for expressing it, which means that it would be well-defined. – mavavilj Jan 30 '16 at 9:54

Physics-based sound synthesis is a pretty multidisciplinary topic. Quoting from the preface of Physical Audio Signal Processing:

[T]he material of this book is multidisciplinary, building on results from physics, musical acoustics, psychoacoustics, signal processing, control engineering, computer music, and computer science. Such diversity is typical of applied research.

It wouldn't really make sense to have it be some kind of subfield of math or physics; computer graphics is in a similar position. Physics-based sound synthesis is mainly numerically simulating a physical model. The trick, as usual, is making approximations to arrive at something reasonably efficient while not losing too much of what is important. This is how fields like psychoacoustics and computer science come in. There isn't a mathematical theory that will tell you what approximations won't produce unpleasant sounds and will run efficiently on a SHARC DSP. The constraints are too ad-hoc. Of course, you can put the constraints of a particular problem into a general mathematical theory of approximation and optimization to help derive solutions.

• Well computer graphics has some formal mathematical theory. Some Riemannian geometry is pretty accurate "computer graphics theory", even if it doesn't involve coding. Sound to me doesn't have the same kind of thing, probably because it's more difficult to model sound. – mavavilj Jan 29 '16 at 17:28
• There is plenty of formal mathematical theory in physics-based audio, but there is no overarching mathematical theory that essentially encompasses the field. Similarly, in computer graphics, there is not a "computer graphics theory" that simultaneously covers Sobel filters and the rendering equation. Riemannian geometry is used in physics based sound synthesis too. Are you interested in mathematically coherent subsets of physics-based audio? – Derek Elkins Jan 29 '16 at 17:43
• By now, probably yes. – mavavilj Jan 29 '16 at 19:16
• Also, I think aesthetics of sound jumps a bit over "physical audio theory". What I'm looking for is a strictly scientific body of knowledge, not a theory in art. "Physical audio theory" ≈ how is sound we hear generated / generatable. – mavavilj Jan 30 '16 at 9:56
• i upped arrow you, Derek. it might not be accurate to say that there isn't a mathematical theory that will tell you what will run efficiently on a SHArC, since the instruction set (and features like the IIR, FIR, and FFT internal components) are well-defined (sometimes icky, but nevertheless well-defined). – robert bristow-johnson Jan 31 '16 at 1:16