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Found in: Page 328

### Physics Principles with Applications

Book edition 7th
Author(s) Douglas C. Giancoli
Pages 978 pages
ISBN 978-0321625922

# What is the resultant sound level when an 81 dB sound and an 87 dB sound are heard simultaneously?

The resultant sound level when both sounds heard simultaneously is $$87.97\;{\rm{dB}}$$.

See the step by step solution

## Determination of sound intensity by sound level

The intensity of the sound is determined by sound level by using its relation with threshold intensity in logarithmic form.

## Given information

Given data:

The sound level for sound 1 is $${\beta _1} = 81\;{\rm{dB}}$$.

The sound level for sound 2 is $${\beta _2} = 87\;{\rm{dB}}$$.

## Evaluation of the resultant sound level

The intensity of sound 1 is calculated below:

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

Substitute the values in the above equation.

\begin{aligned}{c}{I_1} &= {I_0}{10^{\left( {\frac{{81\;{\rm{dB}}}}{{10}}} \right)}}\\ &= 1.259 \times {10^8}{I_0}\end{aligned}

The intensity of sound 2 is calculated below:

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

Substitute the values in the above equation.

\begin{aligned}{c}{I_2} &= {I_0}{10^{\left( {\frac{{87\;{\rm{dB}}}}{{10}}} \right)}}\\ &= 5.012 \times {10^8}{I_0}\end{aligned}

The total intensity of both sounds is calculated below:

$$I = {I_1} + {I_2}$$

Substitute the values in the above equation.

\begin{aligned}{l}I &= \left( {1.259 \times {{10}^8}{I_0}} \right) + \left( {5.012 \times {{10}^8}{I_0}} \right)\\I &= 6.271 \times {10^8}{I_0}\end{aligned}

The resultant sound level when both sounds heard simultaneously is calculated below:

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

Substitute the values in the above equation.

\begin{aligned}{l} &= 10\log \left( {\frac{{6.271 \times {{10}^8}{I_0}}}{{{I_0}}}} \right)\\ &= 87.97\;{\rm{dB}}\end{aligned}

Hence, the resultant sound level when both sounds heard simultaneously is $$87.97\;{\rm{dB}}$$.