I work with insects which are sensitive to particle velocity (pressure gradient) via their flagellum ears. I need to monitor the sound level of the playback stimuli they are exposed to, however, I only have a pressure microphone and this is not the physical quantity they are sensitive to.
Is there any relationship between particle velocity and pressure?
Assuming a monopole sound source in free-field, RMS particle velocity $v$ and RMS pressure $p$ are related to each other with (chapter 2 p. 51 from Beranek & Mellow 2012)
$$v(r) = \frac{ p(r) }{Z }\sqrt{1 + (\frac{c}{ 2 \pi f r})^2}$$
with $r$ distance to the source, $Z $ medium impedance, $ c $ sound speed, $ f $ sound frequency.
For distance $r$ far greater than the wave length, particle velocity and pressure are proportional:
$$ v(r) = \frac{p(r)}{Z}$$
As a consequence, as long as your speaker is in the far-field of your insect/microphone ($r >> c/f$), you can monitor the particle-velocity level from the pressure level. If not, it is better to have a pressure-gradient microphone (see SE question How reliable are "particle-velocity" microphones?).
If you have two pressure microphones you can estimate the pressure gradient (and from that, the particle velocity $v$) as
$$v_r(t) \simeq \frac{1}{\rho} \int \frac{p_1(t)-p_2(t)}{\Delta r}dt$$
where $p_1$ and $p_2$ is the sound pressure on the two microphones, $\Delta r $ the spacing of the microphones and $\rho$ the medium density.This measurement estimates only the velocity component oriented along the line connecting the microphones.
Note that the microphones should be phase and amplitude matched.
