(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Derive [tex]p(r) = \frac{A}{r}e^{j(\omega t - kr)}[/tex] from [tex]pV = nRT[/tex]

2. Relevant equations

3. The attempt at a solution

From ideal gas law I have

[tex]p(r) = \frac{nRT}{V}[/tex]

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

[tex]p(r) = \frac{n A}{4 \pi r^{2} dr}[/tex]

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

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

Putting this in exponential form I get

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

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

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

...I'm still stuck with an [tex]r^{2}[/tex] in the denominator instead of just [tex]r[/tex]. I think I did something wrong somewhere. I'm not sure where. Any help is appreciated.

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# Homework Help: Derive Acoustic Pressure Relation from Ideal Gass Law

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