# Adding Decibels

## The logarithmic decibel scale is convenient calculating sound power levels for two or more sound sources

The decibel *(dB)* is a logarithmic unit used to express the ratio of two signal values - like power, *sound power* or pressure, voltage, intensity etc. - where one value is a reference value.

### Adding Equal Signal Levels

The total signal level in decibel from equal signal sources can be calculated as

*L _{t} = 10 log (n S / S_{ref}) *

* = 10 log (S / S _{ref}) + 10 log (n) *

* * *= L _{s} + 10 log (n) *

*(1)*

*where*

*L _{t}*

*= total signal level (dB)*

*S** = signal (signal unit) *

*S*

_{ref}*= signal reference (signal unit)*

*n = number of sources*

*L _{s }*

*= signal level from each single source (dB)*

The *signal units* depends on the nature of the signal - *W for power, Pa for pressure* and so on.

**Note!** - adding sound **pressure** levels.

#### Example - Total Sound Power from Two Identical Fans

For sound power it is common to use * 10 ^{-12} W *as the reference sound power. Total

*sound power*from two identical fans each generating

*1 W*in noise power can be calculated as

*L _{t} = 10 log (2 (1 W) / (1 10^{-12} W)) *

* = 123 dB*

Sound power and sound power level are often used to specify the noise or sound emitted from technical equipment like fans, pumps or other machines. The "sound" measured with microphones or sensors (meters) are sound pressure.

### Adding Equal Signals Units Calculator

Adding equal signal sources can be expressed graphically

**Note!** Adding two identical sources (doubling the signal) will increase the total signal level with *3 dB (10 log(2))*.

Number of Sources | Increase in Sound Power Level (dB) |
---|---|

2 | 3 |

3 | 4.8 |

4 | 6 |

5 | 7 |

10 | 10 |

15 | 11.8 |

20 | 13 |

### Adding Signals from Sources with different Strengths

The total signal level from sources with different strengths can be calculated as

L_{t}= 10 log ((S_{1}+ S_{2}... + S_{n}) / S_{ref})(2)

#### Example - Total Sound Power from Two different Fans

The total noise power from two fans - one with sound power *1 W* and the other with sound power * 0.5 W* - can be calculated as

*L _{t} = 10 log (((1 W) + (0.5 W)) / (1 10^{-12} W)) *

* = 122 dB*

Adding two signal sources with different levels can be expressed graphically in decibels as

Signal Level Difference between two Sources (dB) | Decibels to Add to the Highest Signal Level (dB) |
---|---|

0 | 3 |

1 | 2.5 |

2 | 2 |

3 | 2 |

4 | 1.5 |

5 | 1 |

6 | 1 |

7 | 1 |

8 | 0.5 |

9 | 0.5 |

10 | 0.5 |

> 10 | 0 |

#### Example - Adding Sound Power in Decibels

The sound power from one of the fans in the example above can be calculated as

*L _{s1 }= 10 log((1 W) / (1 10^{-12} W))*

* = 120 dB *

The sound power from the other fan can be calculated as

*L _{s2 }= 10 log((0.5 W) / (1 10^{-12} W))*

* = 117 dB *

The difference in decibel is

*L _{s1} - *

*L*

_{s2 }*= (120 dB) - (117 dB) *

*= 3 dB*

From the table or diagram above a difference of *3 dB* requires that *2 dB* must be added to the highest sound pressure source as

*L _{t }= (120 dB) + (2 dB)_{ }*

* = 122 dB*

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