Does/can physical audio theory exist?
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.