### Definition of decibel

The decibel is a unitless method of expressing the ratio of two quantities. The expression is in terms of the logarithm to base 10 of the ratio instead of the raw ratio. This is done for convenience in expressing the ratio of numbers many magnitudes apart with decibel numbers that are not as large.

If G1 and G2 are two vibration acceleration amplitudes, then their decibel ratio (or decibel difference) is:

 1

For example, if G1 is twice G2, then:

Or G1 is about 6 dB greater than G2. If G1 were half as large as G2 then:

Or G1 is about 6 dB less than G2. It is seen from the dB equation that each factor of 10 in the ratio results in 20 dB difference in amplitudes.

Note from equation 1 that:

 2

That is, for squares of amplitudes (power) each factor of 10 in the ratio results in 10 dB difference in power. Thus, a factor of 1000 in the power ratio is a 30 dB difference in power. Equation 2 is the normal way of defining decibels, and equation 1 results from it.

Also for convenience, decibels are used rather than Bels. From Equation 2, a Bel would be defined as:

Seeing the equation written out in this form makes the definition much clearer.

Thanks to Dr. Bill Case for the definition.

Ryan Simmons
September 1997

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