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When computing the noise level x amount of meters away from a singer who is singing outdoors, should one rely on the standard formula

$I=\frac{P}{4πr^2}$

(and then convert 'I' into decibels)

or rather on the formula $I=\frac{P}{2πr^2}$, given that the sound waves cannot spread downwards, and are thus forming a "half-sphere" only?

My confusion stems from the fact that I remember calculating the radio waves emitted by an antenna using the formula $I=\frac{P}{2πr^2}$, based on the fact that the antenna was located on the ground, meaning that the waves formed a semi-sphere only. This was confirmed by my textbook.

However, that same textbook uses $I=\frac{P}{4πr^2}$ for the scenario given above, which does not seem to make sense.

Can anybody help out?

Thank you!

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These examples are idealized.

For instance, the example with the antenna appears to be based on the assumption that it has a hemispherical radiation diagram, i.e., has uniform radiation above the ground, which is not possible.

The example with the singer appears to be based on the assumption that the sound is radiated uniformly in all directions (isotropic radiator), which is theoretically possible, but not very feasible (for instance, all sound energy directed toward the ground would have to be fully absorbed).

If we accept these unrealistic assumptions, the formulas look fine.

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  • $\begingroup$ Thank you for the reply! But in the case of the singer: would you say it is more realistic to assume a sphere or a semiphere? And in case the answer is a "sphere": is there really any logical reason why the assumption for the antenna was a "hemisphere"? I.e. is there a physical reason that explains why in one case the book opted for assuming a sphere and in the other a hemisphere? Is it - in relative terms - more realistic to assume one of these two scenarios in one case than in the other? $\endgroup$ – Pregunto Sep 15 '18 at 14:50
  • $\begingroup$ @Pregunto To me, these two examples are similar, once all the simplifications have been accepted. $\endgroup$ – V.F. Sep 15 '18 at 15:05
  • $\begingroup$ And would you yourself recommend the hemisphere or the sphere-model in these two cases? $\endgroup$ – Pregunto Sep 15 '18 at 20:55
  • $\begingroup$ @Pregunto I would recommend nither for the reasons outlined before, but if I had to choose, it would be hemisphere. $\endgroup$ – V.F. Sep 15 '18 at 21:07

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