Acoustic pressure and velocity

When we derived the wave equation, we chose to eliminate velocity and density and concentrated on pressure as our dependent variable. There are two main reasons for doing this: the first is that pressure is a scalar and so is conceptually easier to work with than velocity. In practice, given that we could use a velocity potential, this is not a huge advantage. The second, and more important, reason is that pressure is what we hear and what we measure. Our ears and the microphones we use to measure sound are sensitive to pressure fluctuations, so that is what we choose as our main quantity.

There are times, however, when we will need to use some other quantity. The fundamental theory of aerodynamically generated noise is actually based on density fluctuations (which are usually converted to pressure variations using a linear relationship). A more important relationship is that between pressure and velocity because the acoustic velocity is often used as a boundary condition in calculations involving solid bodies. Remember that acoustics is a branch of fluid dynamics and it is a fluid-dynamical boundary condition that must be satisfied, i.e. usually a velocity.

The linearized momentum equation gives us the relationship we need:

$$\frac{\partial\mathbf{v}'}{\partial t} = -\frac{\nabla p'}{\rho_{0}},$$

in other words, the acoustic velocity is proportional to the pressure gradient. If we write the solution of the wave equation in terms of a velocity potential $\phi=f(t-R/c)$, the pressure and velocity are related via:

$$p = -\rho_{0}\frac{\partial \phi}{\partial t},\quad \mathbf{v} = \nabla\phi,$$

$$v = \frac{p}{\rho c} + \frac{f(t-R/c)}{\rho R^{2}}.$$ (12)

For a wave of constant frequency, the acoustic velocity amplitude $V$ is related to the acoustic pressure by

$$V = -\mathrm{j}\frac{\nabla P}{\rho_{0}\omega}.$$

For a plane wave $\nabla\to\partial/\partial x$ and $V=P/\rho_{0}c$. For large $R$, the pressure-velocity relationship for a spherical wave reduces to this form, as seen in equation 3.12.