Working on the pressure equation from the linearized euler equations, I stumble across a very simple problem :

How, from the pressure solution of the specific equation (see http://www.acs.psu.edu/drussell/Publications/MDQSources.pdf EQ. 2) such as :

$$ p(r,\theta,t) = i\frac{Q\rho c k}{4\pi R}e^{i(wt - kr)} $$ Which is the pressure response to a harmonic point source. R : Distance between source and receiver, $\rho$ fluid density, $c$ fluid adiabatic velocity, $w = 2\pi f$ wave pulsation, $k = w/c$ wave number, $Q$ a constant, the complex source strength.

Can you get the pressure amplitude against time at any point $\mathbf{x}$ of the physical domain ?

Thanks a lot


  • $\begingroup$ It might be better if you copied the content from the pdf into the post, as we prefer self-contained posts in case of link-rot. $\endgroup$
    – Kyle Kanos
    May 13 '15 at 13:04
  • $\begingroup$ Done ! I put the equation directly $\endgroup$
    – Amzocks
    May 13 '15 at 13:09
  • 1
    $\begingroup$ I am sorry, it could be my mistake, but I don't get the question. Could you please rephrase? You are looking for the amplitude in space for given time? $\endgroup$ May 14 '15 at 15:08
  • $\begingroup$ Yes indeed the pressure amplitude in Space for given time ! Thanks $\endgroup$
    – Amzocks
    May 15 '15 at 15:57

Well, if I'm not mistaken, it's pretty straightforward. Let $p(r, \theta, t)$ be separated in two functions with variables of time $T$ and spatial variables $\Theta$ (I'm not using $R$, cause it's already defined):

$$ p(r,\theta,t)=\Theta(r,\theta)T(t) $$


$$ T = e^{i\omega t} $$

$$ \Theta = i\frac{Q\rho c k}{4\pi R}e^{-ikr} $$

$T$ is given ("amplitude for given time"), so you only need to calculate $\Theta$.

Now, the only problem is how to distinct $r$ and $R$. If I get you right, you basically want an intensity plot for "test receivers" and in that case $r=R$, therefore:

$$ \Theta = i\frac{Q\rho c k}{4\pi R}e^{-ikR} $$

so you'll get circles of intensity with center in $R=0$.

Is that it?

  • $\begingroup$ Thanks a lot ! Indeed it should be rather simple but what you say implies that at $t=0, p(r,\theta,0) > 0$. Which makes me wonder about the causality of the wave propagation. Isn't it weird ? $\endgroup$
    – Amzocks
    May 18 '15 at 4:56
  • $\begingroup$ Well, it makes sense, because $p(r,\theta,0) = 0$ everywhere except $r=0$ (if you assume a point source located at the origin). That's actually given by initial conditions - you need to state what's going on in $t,r = 0$. The equation then shows a development since then. Nice discussion of these phenomena is in the first parts of Howe's theory of vortex sound $\endgroup$ May 18 '15 at 6:05
  • $\begingroup$ There is no "R" in the original article. The position is determined by the variable "r" both in the amplitude and in the phase terms. The amplitude simply decreases as 1/r and does not depend on time. There is no angular dependence as the source emits equally in all directions (see text of article). $\endgroup$
    – nasu
    Jun 9 '17 at 19:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.