Question

A sound has an intensity of $3 \times {10^{ - 8}}\,w/{m^2}$. What is the sound level of dB (decibels)? (Given, $\log 3 = 0.477$ )
A. 4.477dB
B. 0.4477dB
C. 44.77dB
D. None of these

Answer 1

Hint: The intensity of sound wave is calculated by using the below formula. The intensity of sound decreases as the distance increases. Decibel is measured on a logarithmic scale with base10.

Complete step by step solution:
Given:
\[{ I = 3 \times {10^8}w{m^{ - 2}} \\
\,B = 10{\log _{10}}\left( {\dfrac{{3 \times {{10}^{ - 8}}}}{{{{10}^{12}}}}} \right) \\
\Rightarrow B = 10{\log _{10}}\left( {3 \times {{10}^{ - 8}} \times {{10}^{12}}} \right) \\
\Rightarrow B = 10{\log _{10}}\left( {3 \times {{10}^4}} \right) \\ } \]
The above log value can be expressed as
\[{ {\log _{10}}{a^n} = n{\log _{10}}a \\
\,\,B = 10{\log _{10}}\left( {3 \times {{10}^4}} \right) \\
\Rightarrow B = 10 \times 4{\log _{10}}3 \\
\Rightarrow B = 10 \times 4 \times 0.477 \\
\therefore B = 44.77dB \\
} \]

Hence option (c) is correct.

Note: : Decibels measures the given intensity to the threshold of hearing intensity so that the threshold takes the value of 0 decibels. To assess sound loudness as distinct from objective intensity measurements, the sensitivity of the ear must be factored in. Before application of the above formula of loudness, the intensity of second should be in the unit of $w/{m^2}$.