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Two sound waves of slightly different frequency have amplitude ratio 11/9. What is the difference of sound levels in decibels of maximum and minimum intensities heard at a point?

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Solution

$\begin{array}{l} [Step 1] : S L_{2} - S L_{1} = 10 \log \frac{I_{max}}{I_{min}} \\ [Step 2] := 10 \log_{10} [\frac{{(\sqrt{I_{1}} + \sqrt{I_{2}})}^{2}}{{(\sqrt{I_{1}} - \sqrt{I_{2}})}^{2}}] = 10 \log_{10} \frac{{(\frac{a_{1}}{a_{2}} + 1)}^{2}}{{(\frac{a_{1}}{a_{2}} - 1)}^{2}} \end{array}$
$10 \log_{10} \frac{{(\frac{11}{9} + 1)}^{2}}{{(\frac{11}{9} - 1)}^{2}} = 20 \log_{10} \frac{\frac{20}{9}}{\frac{2}{9}} = 20 \log_{10} 10 = 20$ dB

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Travelling wave

Remember concepts with our Masterclasses.

Two sound waves of slightly different frequency have amplitude ratio 11/9. What is the difference of sound levels in decibels of maximum and minimum intensities heard at a point?

[ Step 1]:SL2−SL1=10log⁡ImaxImin[ Step 2]:=10log10I1+I22I1−I22=10log10a1a2+12a1a2−1210log10119+12119−12=20log1020929=20log1010=20dB

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