particle velocity in the definition of specific acoustic impedance

Question

Specific acoustic impedance is defined as the ratio of acoustic pressure and acoustic particle velocity. My problem is that the particle velocity is a vector, so the above definition doesn't really "compile". What is missing from the above definition?

Answer 1

The specific acoustic impedance, according to Fundamentals of Acoustics by Kinsler et al. (2000), is a

characteristic property of the medium and of the type of wave that is being propagated.

Indeed, the component of the particle velocity vector that you choose depends on the type of wave, but you typically choose it along the direction of propagation. For example, for plane waves propagating in the $x$ direction, you choose the $x$ component of the particle velocity $\vec{v}$, namely $v_x$. For spherical waves, you would do the same, and the direction of propagation here would be along the radial coordinate $r$, so that the impedance is $z = p / (\vec{v} \cdot \hat{r})$.

The pressure $p$ and the particle velocity $\vec{v}$ are generally complex frequency-domain quantities, which means that the impedance may also be complex. The polar angle $\arg(z)$ then represents the phase lag between pressure and velocity. For spherical waves, for example, there is such a phase lag close to the source, which diminishes into the far field. For plane waves in lossless media, however, the pressure and velocity are always in phase.

Things get hairier for more complex wave fields, such as multiple overlapping plane waves propagating in different directions. Here, the particle velocity $\vec{v}$ would not necessarily just go back and forth in one direction like it does for plane and spherical waves, but might swing around in different directions. In practice, however, I don't really think I have seen specific acoustic impedance being used in such cases.

Normal specific acoustic impedance

As a sidenote, you may occasionally come across the normal specific acoustic impedance, especially when looking at reflection and transmission between different media:

$$z_n = \frac{p}{\vec{v}\cdot\hat{n}}$$

In this type of acoustic impedance, the component of $\vec{v}$ is given as the normal component to some interface whose normal vector is $\hat{n}$. Thus, this is a type of impedance where the question of which component to use is given. Therefore, normal impedance can easily be applied to more complex wave fields.