math lessons - 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 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:

$\rho_0 \frac{\partial}{\partial t} \mathbf{v}(\mathbf{x}, t) + \nabla p(\mathbf{x}, t) = 0$

$\frac{\partial}{\partial t} p(\mathbf{x}, t) + \rho_0 c^2 \nabla \cdot \mathbf{v}(\mathbf{x}, t) = 0$

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, $ρ 0$ is the static density of air and $c$ is the speed of sound in air.