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According to the formula, the intensity of acoustic waves at any given point is: $$I(R) = \frac{P}{4\pi(R)^2}$$ where P is the power of the source and R is the radius of the force (Assuming that the surface of an acostic wave is a sphere)

What, then, is the initial intensity of the wave, as it is just being generated? Is it: $$I_o = \lim_{R\to0}{\frac{P}{4\pi R^2}= \infty}$$

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If the source were truly point-like, yes. But nothing is truly point-like, so no: your formula for $I(r)$ is modified for short distances (where the inner structure of the source becomes relevant). This means that, if your source is a set of speakers, then $I(r)\propto R^{-2}$ is only valid for $R\gg\ell$, where $\ell$ is, say, the radius of the diaphragm of the speaker.

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