Problem — Acoustic power of a conversation

Problem

Estimate the acoustic power of a normal conversation at $1\;\text{m}$ distance, knowing that the typical sound level is $\sim 60\;\text{dB}$ and the threshold intensity is $I_0 = 10^{-12}\;\text{W/m}^2$.

Solution

From the level in dB to intensity. $L = 10\,\log_{10}(I/I_0)$, so $I = I_0\,10^{L/10}$:

$I = 10^{-12}\cdot 10^{60/10} = 10^{-12}\cdot 10^{6} = 10^{-6}\;\mathrm{W/m^2}$

From intensity to power. Assuming isotropic emission over a sphere of radius $r = 1\;\text{m}$:

$P = I\cdot 4\pi r^2 = 10^{-6}\cdot 4\pi\cdot 1^2 = 1{,}26\cdot 10^{-5}\;\mathrm{W}$

$\ev{P \sim 10^{-5}\;\mathrm{W}}$

About a dozen microwatts: the vocal cords are very low-power acoustic sources (the ear, however, is extremely sensitive).

Links

Topics: Onde Concepts: Intensità di un’onda Skills: Stime di Fermi