Raina Gomez

2022-09-26

If the distance to a point source of sound is tripled, by what factor does the intensity of the sound change?

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Cremolinoer

Expert

Assuming that the sound is not facing any barriers, we can say that it is propagating through a spherical shape, we can then use the intensity of sound I formula:
$I=\frac{P}{4\pi {r}^{2}}$
Where P is the power of the sound source and $4\pi {r}^{2}$ is the area of a sphere.
Denoting ${I}_{2}$ the intensity of the sound when the distance was tripled, yields:
${I}_{2}=\frac{P}{4\pi {r}_{2}^{2}}$
We now replace ${r}_{2}$ with $3{r}_{1}:$
${I}_{2}=\frac{P}{4\pi \left(3{r}_{1}{\right)}^{2}}=\frac{1}{9}\frac{P}{4\pi {r}_{1}^{2}}=\frac{{I}_{1}}{9}$
The intensity of the sound was reduced to a ninth of the original intensity.
Result:
${I}_{2}=\frac{1}{9}{I}_{1}$

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