# Sound Propagation - the Inverse Square Law

## Doubling of the distance from a noise source reduces the sound pressure level with 6 decibel.

In a free field - a doubling of the distance from a noise source reduces the sound pressure level with 6 decibel.

This - the **Inverse Square Law** - can be expressed in a diagram like

*dL = L _{p2} - L_{p1} *

* = 10 log (R _{2} / R_{1})^{2}*

* = 20 log (R _{2} / R_{1}) (1)*

*where*

*dL = difference in sound pressure level (dB)*

*L _{p1} = sound pressure level at location 1 (dB) *

*L _{p2} = sound pressure level at location 2 (dB) *

*R _{1} = distance from source to location 1 (ft, m) *

*R _{2} = distance from source to location 2 (ft, m)*

A "free field" is defined as a flat surface without obstructions.

#### Example - Rifle Shot and Sound Pressure at Distance

If the sound pressure from a rifle shot is measured to *134 dB* at *1.25 feet* - the reduction in sound pressure level at distance *80 feet* can be calculated as

*dL = 20 log ((80 ft) / (1.25 ft)) *

* = 36 dB *

The sound pressure level at distance *80 ft* can be calculated as

*L _{p2} = (134 dB) - (36 dB)*

* = 98 dB*

Distance(feet) | Sound PressureL_{p}(decibel) |
---|---|

1.25 | 134 |

2.5 | 128 |

5 | 122 |

10 | 116 |

20 | 110 |

40 | 104 |

80 | 98 |

160 | 92 |

320 | 86 |

640 | 78 |

1280 | 74 |

2560 | 68 |

5120 | 62 |

### Inverse Square Law Calculator

Use the calculator below to calculate the sound pressure level at distance.

*L _{p1} - sound pressure level at location 1 (dB)*

* R _{1} - distance from source to location 1 (m, ft)*

* L _{p1} - distance from source to location 2 (m, ft)*

### Example - Noise from a Machine

The noise from a machine in distance *1 m* is measured to *110 dB*. The noise reduction due to the inverse square law to a working area at distance *5 m* can be calculated as

*dL = 20 log ((5 m) / (1 m)) *

* = 14 dB *

The sound pressure level in the working area can be calculated as

*L _{p2} = (110 dB) - (14 dB)*

* = 96 dB*

This noise level is only permitted for a limited amount of time and some action with partial barriers or enclosure of the machine should be done.