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

Dec 18, 2014

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 $1 \setminus \times {10}^{- 12} W {m}^{- 2}$.

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 $\mathrm{dB}$ 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 $\mathrm{dB}$ is :

$\left(10 \mathrm{dB}\right) \setminus {\log}_{10} \left(\frac{5 \setminus \times {10}^{- 6} W {m}^{- 2}}{1 \setminus \times {10}^{- 12} W {m}^{- 2}}\right) = 66.99 \mathrm{dB}$