# Sound intensity level with examples

“Sound energy passing per second through a unit area held perpendicular to the direction of propagation of sound waves is called intensity of sound.”The sound waves transfer energy from the sounding body to the listener. The intensity of sound depends on the amplitude of sound waves. Intensity is a physical quantity and can be measured accurately. The unit of intensity of sound is watt per square meter (Wm^{-2}).

## Sound intensity level

The human ear responds to intensities ranging from 10^{-12 }Wm^{-2} to more than 1Wm^{-2} (which is loud enough to be painful). Because the range is so wide, intensities are scaled by factors of ten. The barely audible and the faintest intensity of sound is 10^{-12 }Wm^{-2 }is taken as reference intensity, called zero bel (a unit named after Alexander Graham Bell).

The loudness of a sound depends not only on the intensity of sound but also on the physical conditions of the ear. The human ear is more sensitive to some frequencies rather than others.

The loudness (L) of a sound is directly proportional to the logarithm of intensity i.e.,

**L ∝ log I**

**L =K log I ……(1)**

Where k is a constant of proportionality. Let L_{0 }be the loudness of the faintest audible sound of intensity I_{0 }and L be the loudness of an unknown sound of intensity I, then by equation (1), we can write

**L _{0} =K log I_{0}……..(2)**

Subtracting equation (2) from (1), we get:

**L – L _{0}=K(log I -log I_{0})**

**=K log I/I _{0}**

This difference,(L-L_{0}), between the loudness L of an unknown sound and the loudness L_{0} is called the intensity level of the unknown sound. Therefore, the intensity level of an unknown sound is given by

**Intensity level = k log I/I _{0 ……..(3)}**

The value of k depends not only on the units of I and I_{0} but also on the unit of intensity level. If intensity I of any unknown sound is 10 times greater than the intensity I_{0 }of the faintest audible sound i.e., I=10 I_{0,}_{ }and the intensity level of such a sound is taken as a unit, called bel, the value of k becomes 1. Therefore, using k=1 in equation (3) becomes:

**Intensity level = log I/I _{0} (bel) …..(4)**

Bel is a very large unit of the intensity level of a sound. Generally, a smaller unit called decibel is used. Decibel is abbreviated as (dB). It must be remembered that 1 bel is equal to 10 dB. If the intensity level is measured in decibels, then equation (4) becomes:

**Intensity level = log I/I _{0 }(dB) ……….(5)**

See Also: Sound Vs Noise

Using equation (5), we can construct a scale for measuring the intensity level of sound. Such scale is known as the “decibel scale”.The intensity level of different sounds in decibel is given in the table:

Sources of sound | Intensity (Wm^{-2}) | Intensity level (dB) |

Nearby jet airplane | 10^{3} | 150 |

Fast train | 10^{1} | 130 |

Siren | 10^{0} | 120 |

Lawn mover | 10^{-2} | 100 |

Vacuum cleaner | 10^{-5} | 70 |

Mosquito buzzing | 10^{-8} | 40 |

whisper | 10^{-9} | 30 |

Rustling of leaves | 10^{-11} | 10 |

Faintest audibel sound | 10^{-12} | 0 |

The decibel scale is a logarithmic measure of the amplitude of sound waves. On a logarithmic scale, equal intervals correspond to multiplying by 10 instead of adding equal amounts.

Logarithmic scale Decibels (dB) | Linear scale Amplitude (m) |

0 | 1 |

20 | 10 |

40 | 100 |

60 | 1,000 |

80 | 10,000 |

100 | 1000,000 |

120 | 1,000,000 |

**Watch also video about intensity of sound:**

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you must correct this formula : Intensity level = log I/I0 (dB) ……….(5)

you have to substitute it by : Intensity level = 10log I/I0 (dB) ……….(5)