Level (logarithmic quantity)

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]

Definitions

Level and its units are defined in ISO 80000-3.

Level

Level of a quantity Q, denoted LQ, is defined by[4]

where

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

Level of a root-power quantity

The level of a root-power quantity (also known as a field quantity), denoted LF, is defined by [5]

where

  • F is the root-power quantity, proportional to the square root of 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

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

where

  • 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.[6]

Units of level

Power level

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.[7] The neper, bel, and decibel are defined by

  • Np = 1;
  • B = 1/2 loge10 Np;
  • dB = 0.1 B = 1/20 loge10 Np.

If F is a root-power quantity:

If P is a power quantity:

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

Frequency level

The octave is a unit of level (specifically "frequency level",[8] for r = 2) though that concept is seldom seen outside of the standard.[9] A semitone is one twelfth of an octave. A cent is one hundredth of a semitone.

See also

References

  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. ↑ ANSI S1.1-2013 Acoustical Terminology, entry 3.01
  5. ↑ e Industriale, F. D. I. C., & D'Amore, F. Effect of moisturizer and lubricant on the finger‒surface sliding contact: tribological and dynamical analysis.
  6. ↑ Ainslie, M. A. A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound. Acoustics Today, 23 February 2015
  7. ↑ Barry Taylor (1995). Guide for the Use of the International System of Units (SI): The Metric System. Diane Publishing Co. p. 28.
  8. ↑ Fletcher, H. (1934). Loudness, pitch and the timbre of musical tones and their relation to the intensity, the frequency and the overtone structure. The Journal of the Acoustical Society of America, 6(2), 59–69.
  9. ↑ ANSI/ASA S1.1-2013, Acoustical Terminology
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