# What is the relationship between "loudness" and "sound intensity"?

Mar 21, 2017

Loudness is typically measured in decibels, $\text{dB}$. In these units, the relationship is

${L}_{I} = 10 \log \left(\frac{I}{I} _ 0\right)$

where ${L}_{I}$ is the sound intensity level relative to a reference value, $I$ is the sound's intensity, and ${I}_{0}$ is the intensity of the reference (usually in air).

${I}_{0} = {\text{1 pW/m}}^{2}$ (picowatts per meters squared)

This essentially tells you that we perceive something as being loud in a relative manner.

• If there is a lot of background noise, a song on the car radio will seem quiet, even if the volume is normal.
• In a completely quiet room, someone dropping a pin is noticeably loud, even though it may not be loud on an absolute level.

By the way, notice how this resembles the Beer-Lambert Law of absorption:

$A = - \log \left(\frac{I}{I} _ 0\right)$

So, one can think of loudness then as analogous; the darker the substance, the greater its absorbance. However, there comes a point where it's so dark that the absorbance hardly changes.

The mathematical trend it follows is similar with sound intensity levels in that the relative difference in loudness at higher loudness is smaller than at lower loudness.