Answers.Body: <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>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-07-05T14:13:04.300Z
Answers.LastActivityDate: 2022-10-19T12:36:44.363Z
Answers.LastEditDate: 2022-10-19T12:36:44.363Z
Answers.UserDisplayName: Jakob CD
Answers.Body: <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>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-07-01T21:39:59.497Z
Answers.LastActivityDate: 2022-10-19T12:29:44.600Z
Answers.LastEditDate: 2022-10-19T12:29:44.600Z
Answers.UserDisplayName: Noil
Body: <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>
  
Tags: <particle-velocity><pressure><microphone>
Title: Is it possible to estimate particle velocity from a pressure microphone?
ThreadText: # 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
  
  
  ---
  
  ## 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/)).
  
  
  ---
  
  ### 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.
  
  
  ---
  
  