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 water) 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 is the acoustic pressure and is the acoustic fluid velocity vector, is the vector of spatial coordinates, is the time, is the static mass density of the medium and 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 as

The acoustic wave equation is a combination of these two sets of balance equations and can be expressed as

\cfrac{\partial^2 \mathbf{v}}{\partial t^2} - c_0^2~\nabla^2\mathbf{v} = 0 \qquad \text{or} \qquad \cfrac{\partial^2 p}{\partial t^2} - c_0^2~\nabla^2 p = 0

The acoustic wave equation (and the mass and momentum balance equations) are often expressed in terms of a scalar potential where . In that case the acoustic wave equation is written as

\cfrac{\partial^2 \varphi}{\partial t^2} - c_0^2~\nabla^2 \varphi = 0

and the momentum balance and mass balance are expressed as

p + \rho_0~\cfrac{\partial\varphi}{\partial t} = 0 ~;~~ \rho + \cfrac{\rho_0}{c_0^2}~\cfrac{\partial\varphi}{\partial t} = 0 ~.

Read more about Acoustic Theory:  Governing Equations in Cylindrical Coordinates

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