phy1020.DVI

The threshold of hearingIDI 0 corresponds to a sound level of 0 dB. A sound intensity of 1 W/m^2 cor-
responds to a sound level of 120 dB, and is about where most people start finding the sound to be painfully
loud; 120 dB is called thethreshold of pain.
One useful fact to note about decibels is that each time youdoublethe intensityI(W/m^2 ), youadd3dB
to the sound level. This is because in going from intensityIto2I, the sound level becomes

ˇ^0 D 10 log 10 .2I=I 0 / (13.4)

D 10 log 102 C 10 log 10 .I=I 0 / (13.5)

3:010Cˇ: (13.6)

Similarly, when youhalvethe sound intensityI, yousubtract3 dB from the sound level.
When computing sound levels, you cannot do the computations in decibel units. Instead, you must do the
calculations in intensity units (W/m^2 ), then convert to dB at the end. For example, suppose you are 35 meters
away from a 10-watt isotropic sound source. How loud a sound do you hear? You first find the intensity:
IDP=.4r^2 /D.10W/=Œ4.35m/^2 D6:496 10 ^4 W=m^2. Now convert the result to dB to find the
sound level:ˇD 10 log 10 .I=I 0 /D 10 log 10 .6:496 10 ^4 =10^12 /D 88 dB.

13.3 Nepers

A less common unit for measuring sound level is theneper(Np). Like the decibel, the neper is a logarithmic
scale; but unlike the decibel, it is a measure of the ratio ofamplitudes(not intensities), and uses the natural
logarithm instead of the common logarithm. Since the amplitude is proportional to the square root of the
intensity (A/

p

I), the sound level in nepers is given by Dln

p

I=I 0 ,or

D

1

2

ln

I

I 0

: (13.7)

You may convert between decibels and nepers using the relationship

.Np/Dˇ.dB/

ln 10

20

(13.8)

Every doubling of the intensityI(W/m^2 ) corresponds to adding about^1 / 3 Np to the sound level, and
halvingImeans subtracting about^1 / 3 Np from the sound level.
In terms of nepers, the threshold of hearingD 0 dBD 0 Np; the threshold of painD 120 dBD 14 Np.