Acoustic theory

Acoustic theory is the field relating to mathematical description of sound waves. It is derived from fluid dynamics. See acoustics for the engineering approach.

The propagation of sound waves in a fluid (such as air) can be modeled by an equation of motion (conservation of momentum) and an equation of continuity (conservation of mass). With some simplifications, in particular constant density, they can be given as follows:

\begin{align} \rho_0 \frac{\partial \mathbf{v}}{\partial t} + \nabla p & = 0 \qquad \text{(Momentum balance)} \\ \frac{\partial p}{\partial t} + \kappa~\nabla \cdot \mathbf{v} & = 0 \qquad \text{(Mass balance)} \end{align}

where p(\mathbf{x}, t) is the acoustic pressure and \mathbf{v}(\mathbf{x}, t) is the acoustic fluid velocity vector, \mathbf{x} is the vector of spatial coordinates x,y,z, t is the time, ρ0 is the static mass density of the medium and κ is the bulk modulus of the medium. The bulk modulus can be expressed in terms of the density and the speed of sound in the medium (c0) as

\kappa = \rho_0 c_0^2 ~.