Question

The relationship between the number of decibels β and the intensity of a sound I

in watts per square meter is
β=10log(I10−12)
(a) Determine the number of decibels of a sound with an intensity of 1 watt per square meter.
(b) Determine the number of decibels of a sound with an intensity of 10−2
watt per square meter.
(c) The intensity of the sound in part (a) is 100 times as great as that in part (b). Is the number of decibels 100 times as great? Explain.

Answer 1

For an intensity of 1 W/m², the decibel level is 120 dB, and for an intensity of 10⁻² W/m², it is 100 dB. Sounds' intensities relate to decibel levels logarithmically; thus, a 100 times increase in intensity raises the decibel level by 20, not 100 times.

Step-by-step explanation:

Decibel Levels of Different Sound Intensities

The relationship between the number of decibels β and the intensity of sound I in watts per square meter is defined by the formula β = 10 log(I / 10-12). Here are the calculated decibel levels for two given intensities:

1. Intensity of 1 watt per square meter (W/m2): Using the formula β = 10 log(1 / 10-12) = 120 dB.
2. Intensity of 10-2 watt per square meter (W/m2): Using the formula β = 10 log(10-2 / 10-12) = 100 dB.

In part (c), while the intensity of the sound in part (a) is indeed 100 times that of part (b), the number of decibels is not 100 times greater. Instead, the decibel level is determined by the logarithmic scale, meaning that an increase in intensity by a factor of 100 results in an increase of 20 decibels.