## Derive Acoustic Pressure Relation from Ideal Gass Law

1. The problem statement, all variables and given/known data
Derive $$p(r) = \frac{A}{r}e^{j(\omega t - kr)}$$ from $$pV = nRT$$

2. Relevant equations

3. The attempt at a solution
From ideal gas law I have

$$p(r) = \frac{nRT}{V}$$

R and T are constant, so I can pull them out now and replace them with A. If V is the volume of a spherical shell of thickness dr, I get

$$p(r) = \frac{n A}{4 \pi r^{2} dr}$$

This means that the only thing that changes is the net flow of mass flowing into and out of my spherical shell. Which lead to

$$p(r) = \frac{A cos(\omega t - k r)}{4 \pi r^{2} dr}$$

Putting this in exponential form I get

$$p(r) = \frac{A}{4 \pi r^{2} dr} e^{j(\omega t - k r)}$$

Because the atoms are only moving radially, I can ignore the dr part. This leads to

$$p(r) = \frac{A}{4 \pi r^{2}} e^{j(\omega t - k r)}$$

...I'm still stuck with an $$r^{2}$$ in the denominator instead of just $$r$$. I think I did something wrong somewhere. I'm not sure where. Any help is appreciated.
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 Quote by dimensionless Which lead to $$p(r) = \frac{A cos(\omega t - k r)}{4 \pi r^{2} dr}$$ Putting this in exponential form I get $$p(r) = \frac{A}{4 \pi r^{2} dr} e^{j(\omega t - k r)}$$
Not sure I see how this is done. The Euler relations are

$$\cos[x]=\frac{1}{2}\left(\exp[ix]+\exp[-ix]\right)$$

 Quote by dimensionless Because the atoms are only moving radially, I can ignore the dr part. This leads to $$p(r) = \frac{A}{4 \pi r^{2}} e^{j(\omega t - k r)}$$ ...I'm still stuck with an $$r^{2}$$ in the denominator instead of just $$r$$. I think I did something wrong somewhere. I'm not sure where. Any help is appreciated.
Can you clarify why you ignore $dr$?
 I know that the Euler relation is $$\cos[x]=\frac{1}{2}\left(\exp[ix]+\exp[-ix]\right)$$ but quite I'm sure I've seen the expression $$\exp[ix] = \cos[x] + i \sin[x]$$ used in many places but ignoring the imaginary part so that it just becomes $$\exp[ix] = \cos[x]$$ Actually, it really bothers me, but I've seen it in so many places. I'm ignoring dr, in part because I'm seeking a particular result. As for rational, the particles are moving only as part of a sound wave, so they move parallel to dr. The pressure is force per unit area, and the area is perpendicular to dr...in other words, I need the pressure on a spherical surface rather than in a volume ( p(r) is net air pressure above the equilibrium).

## Derive Acoustic Pressure Relation from Ideal Gass Law

 Quote by dimensionless I know that the Euler relation is $$\cos[x]=\frac{1}{2}\left(\exp[ix]+\exp[-ix]\right)$$ but quite I'm sure I've seen the expression $$\exp[ix] = \cos[x] + i \sin[x]$$ used in many places but ignoring the imaginary part so that it just becomes $$\exp[ix] = \cos[x]$$ Actually, it really bothers me, but I've seen it in so many places.
This is true, and I didn't quite think of this part, mostly because the relation actually is

$$\Re\left[\exp[ix]\right]=\cos[x]$$

That is, the real component of the exponential is the cosine term, while the imaginary, $\Im$, is sine.

 I'm ignoring dr, in part because I'm seeking a particular result. As for rational, the particles are moving only as part of a sound wave, so they move parallel to dr. The pressure is force per unit area, and the area is perpendicular to dr...in other words, I need the pressure on a spherical surface rather than in a volume ( p(r) is net air pressure above the equilibrium).
The pressure does apply itself parallel to the radial component, not perpendicular--draw yourself a picture of a spherical wave, you'll see ;)

I think, rather than using volume, you should consider the density:

$$P=\frac{nkT}{V}=\frac{nkT}{m}\rho$$

Then consider how the density $\rho$ would change with a wave. It is possible that you will actually want to integrate, rather than just ignore the infinitesimal radial projection.
 I think it might be simpler if I defined density $$n/V$$, so that $$P=\frac{n}{V} kT= \rho kT$$ which leads to $$\rho = \frac{n}{4 \pi r^{2} dr}$$ Since the number of mols varies with time, I get $$\rho \cos(\omega t)= \frac{n \cos(\omega t)}{4 \pi r^{2} dr}$$ Conversely, I could hold the volume constant and let the number of mols (or the mass) per unit volume vary so that I get $$\rho = \frac{n}{r^{2} V}$$ At the moment though, I'm still stuck.
 I also have $$\frac{1}{2}m v_{average}^{2} = \frac{3}{2}kT = \frac{3}{2}\frac{R}{n}T$$ this leads to $$v = \sqrt{3 \frac{RT}{nm} }$$ and $$T = \frac{n v^{2}}{3mR}$$ where m is the mass of a particle and v is the velocity. I also have $$F = \frac{dp}{dt}$$ Where p=mv is the momentum. I still can't derive this equation though.

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