Question

Question: A sound-insulating door reduces the sound level by\({\bf{30}}\,{\bf{dB}}\). What fraction of the sound intensity passes through this door?

Short answer

The \(\frac{1}{{1000}}\) sound intensity passes through the door.

Step-by-step solution

Concept

The equation of sound intensity is,

\(\beta = 10\log \frac{I}{{{I_0}}}\)

The intensity of the emitted sound is\(I\)and the intensity of the threshold audible sound is\({I_0} = {10^{ - 12}}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}\).

Given Data

The sound decreases by \(\beta = - 30\;{\rm{dB}}\).

Calculation

The intensity decreases as it passes through the door. So you can write,

\(\begin{array}{c}\beta = 10\log \frac{I}{{{I_0}}} = - 30\\\log \frac{I}{{{I_0}}} = - 3\\\frac{I}{{{I_0}}} = {10^{ - 3}}\\\frac{I}{{{I_0}}} = \frac{1}{{1000}}\end{array}\)

Hence, the\(\frac{1}{{1000}}\) sound intensity passes through the door.