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Chapter 14: Problem 65

Show that a doubling of sound intensity corresponds to approximately a 3 -dB increase in the decibel level.

Short Answer

Expert verified
Doubling the sound intensity corresponds to approximately a 3 dB increase in the decibel level.

Step by step solution

01

Recall Decibel Definition

Decibel (dB) is a logarithmic scale used to quantify the intensity of a sound. It is calculated using the formula: \( dB = 10 \cdot log_{10}(\frac{I}{I_0}) \) where \( I \) is the sound intensity and \( I_0 \) is the reference intensity. Note that a decibel is one-tenth of a bel, named in honor of Alexander Graham Bell.
02

Calculate Change in Decibel Level

When the intensity of the sound is doubled i.e. it changes from \( I \) to \( 2I \), the change in decibel level can be calculated using the formula: \[ \Delta dB = 10 \cdot log_{10}(\frac{2I}{I_0}) - 10 \cdot log_{10}(\frac{I}{I_0}) \] This simplifies to \( \Delta dB = 10 \cdot log_{10}(2) \) since the \( I_0 \) reference intensity cancels out.
03

Evaluate

Evaluate the logarithm: since we know \( log_{10}(2) \approx 0.3 \), the change in decibel level is about \( 10 \cdot 0.3 = 3 dB \).

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Most popular questions from this chapter

A uniform cable hangs vertically under its own weight. Show that the speed of waves on the cable is given by \(v=\sqrt{y g},\) where \(y\) is the distance from the bottom of the cable.

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