Question

What is the sound level in dB for a sound whose intensity is 5.0 x 10-6 watts/m2?

Answer 1

The range of sound intensity that humans can detect is so large ( spans 13 orders of magnitude). The intensity of the faintest sound that is audible is called the Threshold of Hearing. This has an intensity of about `1\times10^{-12}Wm^{-2}`.

Because it is difficult to gain intuition for numbers in such a huge range it is desirable that we come up with a scale to measure sound intensity that falls within a range of 0 and 100. That is the purpose of the decibell scale (dB).

Since logarithm has the property of taking in huge number and returning a small number the dB scale is based on logarithmic scaling. This scale is defined such that the Threshold of Hearing intensity has a sound intensity level of 0.

The intensity level in `dB` of a sound of intensity `I` is defined as:

`(10 dB)\log_{10}(I/I_0); \qquad I_o` - intensity at the threshold of hearing .

This Problem :
`I=5\times10^{-6}Wm^{-2}; \qquad I_o=1\times10^{-12}W.m^{-2}`

The sound intensity level in `dB` is :

`(10 dB)\log_{10}((5\times10^{-6} Wm^{-2})/(1\times10^{-12}Wm^{-2}))=66.99dB`