In my physics book, it says that in the human ear, the sensation of loudness is approximately logarithmic. And that the relative sound intensity is directly proportional to a logarithmic ratio expressed as follows:
$ \beta = 10\log(\frac{I}{I_0}) $
So does that mean an average human wont perceive a , for example , sound of intensity $2*10^{-5} \frac{W}{m^2}$ to be a as twice as loud as a sound that has an intensity of $1*10^{-5}\frac{W}{m^2}$ , while they will perceive a sound that is $ 10 dB$ to be twice as loud as one that is $5dB$ ?
Best Answer
From Victor Storm:
This is not correct. Rather, the implied relation from @Alfred Centauri 's source is the following:
loudness units (arbitrary) power 1 10 W 2 100 W 4 1000 W 8 10,000 Wand so on. This is neither a linear relation, nor a logarithmic one. It is a power law, $l_p= \frac{1}{2} P ^{\log_210}\approx\frac{1}{2}P^{0.301}$ , where $l_p$ is the perceived loudness and $P$ is the power in Watts.
Perceived loudness also varies with frequency:
(Courtesy of wikipedia: https://en.wikipedia.org/wiki/Sound_pressure)
An increase of 10 phons corresponds roughly to a doubling of perceived loudness (which has another unit, called the sone), so near the middle of the spectrum the relation given by @Alfred Centauri, that an increase in 10 dB corresponds to a doubling of loudness, is correct.
What about the supposed logarithmic behavior of hearing? Well, it simply appears that this is not necessarily correct. The alternate model proposed by the creator of the sone unit is Stephen's power law, which claims that perception of various senses occurs, well, as a power law. However, it is not clear to me whether a consensus exists on which of these models is better, either in general or for hearing specifically.