Sound Intensity - Spatial Expansion

Spatial Expansion

For a spherical sound source, the intensity in the radial direction as a function of distance r from the centre of the source is:

$I_r = \frac{P_{ac}}{A} = \frac{P_{ac}}{4 \pi r^2} \,$

Here, P_ac (upper case) is the sound power and A the surface area of a sphere of radius r. Thus the sound intensity decreases with 1/r2 the distance from an acoustic point source, while the sound pressure decreases only with 1/r from the distance from an acoustic point source after the 1/r-distance law.

$I \propto {p^2} \propto \dfrac{1}{r^2} \,$

$\dfrac{I_2}{I_1} = \dfrac{{r_1}^2}{{r_2}^2} \,$

$I_2 = I_{1} \dfrac{{r_1}^2}{{r_2}^2} \,$

= sound intensity at close distance
= sound intensity at far distance

Hence

$p \propto \dfrac{1}{r} \,$

where p (lower case) is the RMS sound pressure (acoustic pressure).

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