Level (logarithmic quantity)

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In the International System of Quantities, the level of a quantity is the logarithm of the ratio of the value of that quantity to a reference value of the same quantity.[1][2] Examples are the various types of sound level: sound power level (literally, the level of the sound power, abbreviated SWL), sound exposure level (SEL), sound pressure level (SPL) and particle velocity level (SVL).[3]

Mathematical definitions[edit]

The level and its units are defined in ISO 80000-3.


Level of a quantity Q, denoted LQ, is defined by


  • r is the base of the logarithm;
  • Q is the root-power quantity;
  • Q0 is the reference value of Q.

Level of a field quantity[edit]

Level of a field quantity, denoted LF, is defined by


  • F is the field quantity;
  • F0 is the reference value of F.

For the level of a field quantity, the base of the logarithm is r = e.

Level of a root-power quantity[edit]

A root-power quantity is a field quantity. The Level of a root-power quantity, denoted LF, is therefore


  • F is the root-power quantity;
  • F0 is the reference value of F.

For the level of a root-power quantity, the base of the logarithm is r = e.

Level of a power quantity[edit]

Level of a power quantity, denoted LP, is defined by


  • P is the power quantity;
  • P0 is the reference value of P.

For the level of a power quantity, the base of the logarithm is r = e2.[4]

Units of level[edit]

Power level[edit]

The neper, bel, and decibel (one tenth of a bel) are units of level that are often applied to such quantities as power, intensity, or gain.[5] The neper, bel, and decibel are defined by

  • Np = 1;
  • B = 1/2 ln 10 Np;
  • dB = 0.1 B = 1/20 ln 10 Np.

If F is a root-power quantity:

If P is a power quantity:

If the power quantity P is equal to F2, and if the reference value of the power quantity, P0, is equal to F02, the levels LF and LP are equal.

Frequency level[edit]

The octave is a unit of level (specifically "frequency level", for r = 2) though that concept is seldom seen outside of the standard.[6] A semitone is one twelfth of an octave.

See also[edit]


  1. ^ ISO 80000-3:2006, Quantities and units, Part 2: Space and Time
  2. ^ W. M. Carey, Sound Sources and Levels in the Ocean, IEEE J Oceanic Eng 31:61-75(2006)
  3. ^ ISO 80000-8:2007, Quantities and units, Part 8: Acoustics
  4. ^ Ainslie, M. A. A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound. Acoustics Today, 23 February 2015
  5. ^ Barry Taylor (1995). Guide for the Use of the International System of Units (SI): The Metric System. Diane Publishing Co. p. 28. 
  6. ^ ANSI/ASA S1.1-2013, Acoustical Terminology