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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

Quentin

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  • $\begingroup$ 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
  • $\begingroup$ 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? $\endgroup$ – Victor Pira 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
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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):

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

then:

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

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

$T$ 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?

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  • $\begingroup$ 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 $\endgroup$ – Victor Pira 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

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