<p>If you have <strong>two</strong> pressure microphones you can estimate the pressure gradient (and from that, the particle velocity <span class="math-container">$v$</span>) as</p>
<p><span class="math-container">$$v_r(t) \simeq \frac{1}{\rho} \int \frac{p_1(t)-p_2(t)}{\Delta r}dt$$</span></p>
<p>where <span class="math-container">$p_1$</span> and <span class="math-container">$p_2$</span> is the sound pressure on the two microphones, <span class="math-container">$\Delta r $</span> the spacing of the microphones and <span class="math-container">$\rho$</span> the medium density.This measurement estimates only the velocity component oriented along the line connecting the microphones.</p>
<p>Note that the microphones should be phase and amplitude matched.</p>

CC BY-SA 4.0
2022-07-05T14:13:04.300Z
2022-10-19T12:36:44.363Z
2022-10-19T12:36:44.363Z
Jakob CD
<p>Assuming a monopole sound source in free-field, RMS particle velocity <span class="math-container">$v$</span> and RMS pressure <span class="math-container">$p$</span> are related to each other with (chapter 2 p. 51 from <a href="https://www.sciencedirect.com/book/9780123914217/acoustics-sound-fields-and-transducers" rel="nofollow noreferrer">Beranek &amp; Mellow 2012</a>)</p>
<p><span class="math-container">$$v(r) = \frac{ p(r) }{Z }\sqrt{1 + (\frac{c}{ 2 \pi f r})^2}$$</span></p>
<p>with <span class="math-container">$r$</span> distance to the source,
<span class="math-container">$Z $</span> medium impedance,
<span class="math-container">$ c $</span> sound speed,
<span class="math-container">$ f $</span> sound frequency.</p>
<p>For distance <span class="math-container">$r$</span> far greater than the wave length, particle velocity and pressure are proportional:</p>
<p><span class="math-container">$$ v(r) = \frac{p(r)}{Z}$$</span></p>
<p>As a consequence, as long as your speaker is in the far-field of your insect/microphone (<span class="math-container">$r &gt;&gt; c/f$</span>), 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 <a href="https://bioacoustics.stackexchange.com/questions/157/how-reliable-are-particle-velocity-microphones/">How reliable are &quot;particle-velocity&quot; microphones?</a>).</p>

CC BY-SA 4.0
2022-07-01T21:39:59.497Z
2022-10-19T12:29:44.600Z
2022-10-19T12:29:44.600Z
Noil
<p>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.</p>
<p>Is there any relationship between particle velocity and pressure?</p>

<particle-velocity><pressure><microphone>
Is it possible to estimate particle velocity from a pressure microphone?
# Is it possible to estimate particle velocity from a pressure microphone?

(Asked by: username_0 on 2022-07-01)

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?

**Tags:** particle-velocity, pressure, microphone


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## 2 answers

### Answer by username_0 on 2022-07-01. Score: 6 | Accepted answer

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](https://www.sciencedirect.com/book/9780123914217/acoustics-sound-fields-and-transducers))

$$v(r) = \frac{ p(r) }{Z }\sqrt{1 + (\frac{c}{ 2 \pi f r})^2}$$

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?](https://bioacoustics.stackexchange.com/questions/157/how-reliable-are-particle-velocity-microphones/)).


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### Answer by username_1 on 2022-07-05. Score: 3

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.


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