Loudness of a Sound Wave

Classical Mechanics Level 2

What is the decibel level of a sound wave that carries a power of 2 W 2 \text{ W} ?

Note: use the reference scale of 2 × 1 0 − 12 W 2 \times 10^{-12} \text{ W} for the threshold of human hearing in computing the decibel level.

240 240 60 60 30 30 120 120

Matt DeCross by Matt DeCross , 27, USA

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1 solution

Matt DeCross
May 10, 2016

Relevant wiki: Amplitude, Frequency, Wave Number, Phase Shift

The decibel level is given by:

L = 10 log ⁡ 10 P P 0 = 10 log ⁡ 10 1 0 12 = 10 ( 12 ) = 120. L = 10 \log_{10} \frac{P}{P_0} = 10 \log_{10} 10^{12} = 10(12) = 120.

Note that in the relevant wiki, the formula is given as a logarithm of ratio of amplitudes. The power is proportional to the intensity, which is proportional to the square of the amplitude, hence the prefactor of 10 instead of 20.

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