Suppose a person has a \({\rm{50 - dB}}\)hearing loss at all frequencies. By how many factors of\({\rm{10}}\)will low-intensity sounds need to be amplified to seem normal to this person? Note that smaller amplification is appropriate for more intense sounds to avoid further hearing damage.

The sound intensity level (SIL) or sound intensity level is the level (logarithmic quantity) of sound intensity relative to a reference value.

The sound intensity formula is given by,

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

Here, \(\beta \) is the sound intensity level in \(dB\), \(I\) is the intensity (in \({\rm{W}} \cdot {{\rm{m}}^{{\rm{ - 2}}}}\)) of ultrasound wave and \({{\rm{I}}_{\rm{0}}}{\rm{ = 1}}{{\rm{0}}^{{\rm{ - 12}}}}{\rm{ W}} \cdot {{\rm{m}}^{{\rm{ - 2}}}}\) is the threshold intensity of hearing.

Consider the given data as below.

A person has hearing loss \({\rm{50 dB}}\) that we consider as sound intensity level to calculate the intensity \(I\).

The sound intensity level is given by,

\(\beta = {\rm{ }}50{\rm{ }}dB\)

The threshold intensity of hearing is,

\({I_0} = {10^{ - 12}}{\rm{ }}W \cdot {m^{ - 2}}\)

Determine the intensity \(I\) at \(\beta = {\rm{ }}50{\rm{ }}dB\) by using following formula.

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

\(\begin{aligned}{l}\frac{\beta }{{10}} &= {\log _{10}}\left( {\frac{I}{{{I_0}}}} \right)\\\frac{I}{{{I_0}}} &= {\left( {10} \right)^{\frac{\beta }{{10}}}}\\I %= {I_0}{\left( {10} \right)^{\frac{\beta }{{10}}}}\end{aligned}\)

Now replace \({I_0}\) and \(\beta \) with the given data.

\(\begin{aligned}{c}I &= {10^{ - 12}}{\left( {10} \right)^{\frac{{50}}{{10}}}}\\ &= {10^{ - 12}}{\left( {10} \right)^5}\\ &= {10^{ - 7}}{\rm{ }}W \cdot {m^{ - 2}}\end{aligned}\)

This is the intensity for \({\rm{50 dB}}\) hearing loss then we have to relate this threshold intensity

\(\begin{aligned}{c}\frac{I}{{{I_0}}} &= \frac{{{{10}^{ - 7}}}}{{{{10}^{ - 12}}}}\\ &= {10^5}\end{aligned}\)

A person who has hearing loss \({\rm{50 dB}}\) will have to amplify his intensity \({10^5}\) to get a minimum of hearing.