The linear wave equation

The equations of continuity and momentum for an inviscid fluid are:

The first thing we do in deriving a wave equation is introduce the assumption that the fluctuations in the fluid dynamical quantities are small. This means that we write quantities as the sum of a mean part and a small fluctuation. These fluctuating parts are so small that their products can be neglected. Decomposing the quantities:

$$\rho = \rho_{0} + \rho'(t),$$

$$\mathbf{v} = \mathbf{v}'(t),$$

$$p = p_{0} + p'(t),$$

where 0 indicates a mean value and a prime symbol a fluctuation.

Applying this assumption to the equations of continuity and momentum and eliminating second order terms (products of small quantities), we find the linearized Euler equations:

To make life easier, we can eliminate the velocity $\mathbf{v}'$ to give us a single equation:

$$\frac{\partial}{\partial t} \left( \frac{\partial\rho'}{\partial ... ...nabla \left( \rho_{0}\frac{\partial\mathbf{v}'}{\partial t} + \nabla p' \right)$$

$$= \frac{\partial^{2}\rho'}{\partial t^{2}} -\nabla^{2}p' = 0.$$ (3)

This is almost the wave equation except that it contains both pressure and density and we would like to deal with only one quantity at a time. To eliminate the density, we need a relationship between it and pressure. This depends on the thermodynamical properties of the fluid, as we will see below. Since we have linearized everything else, we may as well linearize the pressure-density relationship too:

$$p = p_{0} + \left. \frac{\partial p}{\partial\rho} \right\vert _{\r... ... p}{\partial\rho^{2}} \right\vert _{\rho=\rho_{0}}(\rho-\rho_{0})^{2} + \ldots,$$

$$p' = p-p_{0} \approx \left.\frac{\partial p}{\partial\rho} \right\vert _{\rho=\rho_{0}}(\rho-\rho_{0}) = c^{2}\rho',$$

$$c^{2} = \left.\frac{\partial p}{\partial \rho}\right\vert _{\rho=\rho_{0}}.$$

The constant is written $c^{2}$ because it is always positive². Substituting this relationship into equation 3.3, we find a wave equation for the acoustic pressure:

$$\boxed{ \frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}} -\nabla^{2}p = 0 }$$ (4)

This is the most fundamental equation in acoustics. It describes the properties of a sound field in space and time and how those properties evolve. It is quite unlike the incompressible flow equations to which you may be accustomed because it describes very weak processes which happen over large distances. The most fundamental obvious property of the wave equation is that it is linear. This means that the sum of two solutions of the wave equation is also itself a solution.