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12-73GP

Expert-verifiedFound in: Page 328

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 \(87.97\;{\rm{dB}}\).

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

Given data:

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

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

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 \(87.97\;{\rm{dB}}\).

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