# Confusion regarding Relation between Frequency and Loudness

The mathematical relation between intensity and loudness is:

$$I_{L} = 10 \log_{10}\left( \frac{I}{I_{0}} \right)$$

where $$I$$ is the sound intensity and $$I_{0}$$ the reference intensity. The unit of sound intensity is expressed in decibels (dB).

The mathematical relation between intensity and frequency is:

$$I = 2 \pi^{2} \nu^{2} \delta^{2} \rho c$$

with $$I$$ the sound intensity, $$\nu$$ the frequency of sound, $$\delta$$ the amplitude of the sound wave, $$\rho$$ the density of the medium in which sound is travelling and $$c$$ the speed of sound.

This indicates that as frequency increases, loudness must also increase.

Then, why most physics textbooks mention that Loudness does not depend on Frequency?

• . . . . . why most physics textbooks mention that Loudness does not depend on Frequency? Please give an example from a Physics textbook because as it stands it is not a correct statement. Feb 20, 2023 at 14:48
• Your second equation calculates sound intensity, typically in $W/m^2$. "Loudness" is a subjective human experience, as the sensitivity of the human ear to a given sound intensity is highly dependent on frequency. Feb 20, 2023 at 21:00

You may be referring to the Sound Intensity Level (SIL), which is calculated with the first equation you provided. There seems to be a rough rule of thumb being used in the acoustics community relating Sound Intensity Level with loudness, which states that there is "roughly" doubling of loudness for a $$10 ~ dB_{SIL}$$ increase.