# Numerical simulation of sound propagation in air

How does one go about accurately modeling sound propagation in a room (with reflections, absorption, and diffusion characteristics) from the motion of a loud speaker? More specifically what are the governing equations that are needed? Obviously Naiver-Stokes but this is too general? Is there an easier way?

• I doubt the field of acoustics is any "easier" than fluid dynamics. – ja72 Dec 3 '13 at 5:42
• This actually sounds way harder than simple Naiver-Stokes and fluid dynamics. Naviar-Stokes deals with relatively low-velocity fluids whereas sound vibrations cause very high-velocity, high-frequency oscillations. – Brandon Enright Dec 3 '13 at 6:26
• @BrandonEnright: NS equations for compressible media do contain all that is necessary for sound propagation, but they are definitely overkill. – user23660 Dec 3 '13 at 7:47
• I think the hardest part of this would be defining the domain to have multiple reflection points. Randy Leveque's Finite Volume Methods for Hyperbolic Problems covers acoustics in several different places. – Kyle Kanos Dec 3 '13 at 14:37
• If you're looking for practical methods there are many current methods! See: youtube.com/watch?v=FDL39J-i0yQ – user12029 Jan 2 '14 at 10:31

From your description I deduce that approximation of geometrical acoustics should be enough. For its applicability we need to ensure that

1. The sound could be described as small perturbation (so, no nonlinear effects).

2. Wavelengths of sound are much smaller than the dimensions of structures with which the sound interacts.

The main equation for geometrical acoustics would be the eikonal equation.

If for your applications the interference effects are essential, then linear acoustics approximation is needed. Main equation would be the wave equation for pressure and/or velocity potential.

• It's not my field, but eikonal seems a little bit oversimplified. I'm thinking of a 50Hz sound: we then have $\lambda = c/\nu = 340{\rm m\,s^{-1}} / 50{\rm Hz} \approx 7{\rm m}$: rather big compared with chairs, tables, peoples, bookshelves .... so I'd guess pressure / velocity potential wave equation would be the wisest. – WetSavannaAnimal Dec 3 '13 at 9:44

Upon assuming small oscillations and neglecting the viscosity of air, the linear three-dimensionsal wave equation with proper boundary conditions would do in the time domain or alternatively Helmholtz equation in the frequency domain. The boundary value problem can be solved by a numerical method e.g. the finite element method.