**Sound: Decibel Levels**

The loudness of a sound is commonly expressed in decibels. Loudness is called the 'intensity level' 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 I_{o} 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}).

I_{0} = 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) = log_{10} (I/Io)

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

**intensity level (dB) = 10 log**_{10} (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 10^{5}

**Step 2: **Take the logarithm of the ratio

log_{10} (1.0 x 10^{5}) = 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 = 10^{10 }

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

I = 10^{10}x Io

= 10^{10}x 1.0 x 10^{-12}

**= 1.0 x 10**^{-2} W m^{-2}^{}