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Consider a point source in a free field. The source is emitting spherical waves. I know the relationship ${p_1\over r_2}={p_2\over r_1}$ that lets you convert the sound pressure $p_1$ at a point at a distance $r_1$ to the point source the sound pressure $p_2$ at a point at a distance $r_2$ from the point source.

But how can you calculate the sound pressure at a given distance from a point source if you are given the sound pressure at the point source? The corresponding distance would be zero...

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You cannot define the pressure at a point source. A point source does not exist but it is sometimes useful to consider a source as being so small relative to other distances that it is effectively a point. You can then be given the amplitude of the pressure wave at a (small) distance from the point source and use your formula to find the amplitude of the pressure wave at other distances.

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Your model uses a pressure field of the form $C/r$. Hence, at the point source $r = 0$, the sound pressure is infinite, so you cannot be given such a sound pressure, and a fortiori you cannot calculate anything from it.

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