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?

2 Answers 2


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) = p(r) / Z * sqrt(1 + (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) = 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 particle velocity formula

where p1 and p2 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.

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