Sound: Decibel Levels

The loudness of a sound is commonly expressed in decibels. Loudness is called the 'intensity level' of a sound. This is not to be confused with the intensity of a sound!

The range of 'loudness', from the quietest sound that can be heard to a sound so loud that is produces pain rather than the sensation of hearing, is so large that an exponential scale is used. To measure 'loudness' we compare the sound's intensity to that of the least audible sound.

The quietest sound that can be heard is said to be at the threshold of hearing. The intensity of sound at the threshold of hearing is given the symbol Io Intensity of sound is a mesasure of the ound energy directed onto a metre square each second, so it is measured in watts per square metre (W m-2).

I0 = 1.0 x 10-12 W m-2

To calculate the intensity level of a sound you have to find the ratio of the intensity of that sound to the threshold intensity: I/Io. But as the range is so large we use an exponential scale and you find the log of that ratio. You then measure that intensity level in Bels - the unit was named in recognition of Alexander Graham Bell - but they missed off the last 'l' for some reason!

intensity level (B) = log10 (I/Io)

The bel is a large unit - we usually measure in decibels (dB)

intensity level (dB) = 10 log10 (I/Io)

Example 1

The intensity of sound in an exam room is about 1.0 x 10-7 W m-2. Calculate the decibel level of that sound.

Step 1: Find the ratio of the sound intensity to the threshold intensity

I/Io = (1.0 x 10-7)/(1.0 x 10-12)

= 1.0 x 105 (no unit because it is a ratio)

Step 2: Take the logarithm of the ratio

log10 (1.0 x 105) = 5

Step 3: Multiply the ratio by 10

Intensity level = 50 dB

To find the intensity of a given sound whose intensity level in decibels is known, perform the above steps in reverse order and perform the inverse operation.

Example 2

An underground train produces an intensity level of 100 dB. What is the intensity of the sound which produces an intensity level of 100 dB?

Step 1: Divide the decibel level by 10

100/10 = 10

Step 2: Use that value as the exponent of the ratio

I/Io = 1010

Step 3: Find the intensity in watts per square metre.

I = 1010x Io

= 1010x 1.0 x 10-12

= 1.0 x 10-2 W m-2