According to this answer (source), the time averaged power $P$ of an acoustic wave is:
$$P = \frac{1}{2} \mu v \omega^2 A^2$$
Where $\mu$ is mass density of the medium, $v$ is speed of sound in the medium, $\omega$ is the angular frequency of the wave, and $A$ is the amplitude of the wave.
Under the assumption that amplitude is in units of pressure (e.g. Pascals), this gives the following units for each variable (letters on right sides indicate units):
$$\begin{aligned} P &= \frac{kg \times m^2}{s^3} \text{ (Watts)} \\ \\ \mu &= \frac{kg}{m^3} \\ \\ v &= \frac{m}{s} \\ \\ \omega &= \frac{rad}{s} \\ \\ A &= \frac{kg}{m \times s^2} \text{ (Pascals)} \end{aligned}$$
But when attempting to balance the units, the units on the right side of that equation ends up as:
$$\begin{aligned} \frac{1}{2} \mu v \omega^2 A^2 &= \frac{kg}{m^3} \times \frac{m}{s} \times \frac{rad^2}{s^2} \times \frac{kg^2}{m^2 \times s^4} \\ \\ &= \frac{kg^3 \times rad^2}{m^4 \times s^7} \end{aligned}$$
These are clearly not units of power.
If I instead work backwards to figure out what amplitude needs to be:
$$\begin{aligned} \frac{kg \times m^2}{s^3} &= \frac{kg}{m^3} \times \frac{m}{s} \times \frac{rad^2}{s^2} \times \text{ ?}^2 \\ \\ \text{ ?}^2 &= \frac{m^4}{rad^2} \\ \\ \text{ ?} &= \frac{m^2}{rad} \end{aligned}$$
Then I get units of amplitude as square meters per radian, which is nonsense.
What's going on here?
Note: I'm concerned with acoustic pressure waves in e.g. air. The source, however, seems to be talking about oscillations on a string, which makes me wonder is this even the correct equation for pressure waves in a medium?
