What is the intensity of a sound at the pain level of 120 dB? Compare it to that of a whisper at 20 dB.

The sound level represents the logarithmic relation between the fractions of total sound intensity to the threshold intensity.

The pain sound level is \({\beta _1} = 120\;{\rm{dB}}\).

The whisper sound level is \({\beta _2} = 20\;{\rm{dB}}\).

The standard value for the threshold intensity of sound is \({I_0} = 1 \times {10^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}\).

The sound intensity for pain sound level is calculated below:

\(\begin{aligned}{c}{\beta _1} = 10\log \left( {\frac{{{I_1}}}{{{I_0}}}} \right)\\\log \left( {\frac{{{I_1}}}{{{I_0}}}} \right) = \frac{{{\beta _1}}}{{10}}\end{aligned}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}\log \left( {\frac{{{I_1}}}{{1 \times {{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}}}} \right) = \frac{{120\;{\rm{dB}}}}{{10}}\\\log \left( {\frac{{{I_1}}}{{1 \times {{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}}}} \right) = 12\;{\rm{dB}}\\\left( {\frac{{{I_1}}}{{1 \times {{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}}}} \right) = {10^{12}}\\{I_1} = 1\;{\rm{W/}}{{\rm{m}}^2}\end{aligned}\)

The sound intensity for whisper sound level is calculated below:

\(\begin{aligned}{c}{\beta _2} = 10\log \left( {\frac{{{I_2}}}{{{I_0}}}} \right)\\\log \left( {\frac{{{I_2}}}{{{I_0}}}} \right) = \frac{{{\beta _2}}}{{10}}\end{aligned}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}\log \left( {\frac{{{I_2}}}{{1 \times {{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}}}} \right) = \frac{{20\;{\rm{dB}}}}{{10}}\\\log \left( {\frac{{{I_2}}}{{1 \times {{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}}}} \right) = 2\;{\rm{dB}}\\\left( {\frac{{{I_2}}}{{1 \times {{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}}}} \right) = {10^2}\\{I_2} = 1 \times {10^{ - 10}}\;{\rm{W/}}{{\rm{m}}^2}\end{aligned}\)

Hence, the sound intensity for pain sound level and whisper sound level are \(1\;{\rm{W/}}{{\rm{m}}^2}\), and \(1 \times {10^{ - 10}}\;{\rm{W/}}{{\rm{m}}^2}\).