A high-fidelity microphone is connected to a data-acquisition system that records the…

PHYSICS · Multi-part · ICSE · Physics · Sound · Sound

Problem

A high-fidelity microphone is connected to a data-acquisition system that records the instantaneous pressure variation

\Delta P(t)

of a brief sound pulse. The software reports the following: • The peak-to-peak pressure variation is

\Delta P_{\text{pp}} = 8.0\,\text{Pa}

. • The reference pressure for sound intensity calculations is

P_0 = 20\,\mu\text{Pa}

. Assume: • The root-mean-square (rms) pressure variation is related to the amplitude

P_{\text{max}}

of the pressure oscillation by

P_{\text{rms}} = \dfrac{P_{\text{max}}}{\sqrt{2}}

. • The instantaneous intensity

I

of a plane sound wave in air is proportional to the square of the rms pressure variation. • The sound level in decibels is defined by

L = 10\,\log_{10}\!\left(\dfrac{I}{I_0}\right)

, where

I_0

corresponds to the reference pressure

P_0

. A second recording of an otherwise identical pulse is made after the microphone gain is increased so that the peak-to-peak pressure variation is doubled. (a) For the original pulse, determine the sound level

L_1

in decibels. (b) For the amplified pulse, determine the new sound level

L_2

in decibels. (c) Derive a general expression that relates the change in sound level

\Delta L

(in decibels) to the ratio of peak pressure amplitudes of two sound waves. (d) Although the peak pressure amplitude doubles, listeners do not perceive the second pulse as being twice as loud. Provide a physical explanation for this observation that references both the decibel scale and human auditory response.

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