Sound pressure

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This article is about the measurement of audible sound. For the music album, see Sound Pressure Level.
Not to be confused with Sound energy density.
Sound measurements
 Sound pressure  p, SPL
 Particle velocity  v, SVL
 Particle displacement  ξ
 Sound intensity  I, SIL
 Sound power  P, SWL, Q
 Sound energy  W
 Sound energy density  w
 Sound exposure  E, SEL
 Acoustic impedance  Z
 Speed of sound  c
 Audio frequency  AF
 Transmission loss  TL

Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average, or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone.

Mathematical definition[edit]

Sound pressure diagram:
1. silence;
2. audible sound;
3. atmospheric pressure;
4. sound pressure

A sound wave in a transmission medium causes a deviation (sound pressure, a dynamic pressure) in the local ambient pressure, a static pressure.
Sound pressure, denoted p and measured in Pa, is given by:

p_\mathrm{total} = p_\mathrm{stat} + p,


  • ptotal is the total pressure, measured in Pa;
  • pstat is the static pressure, measured in Pa.

Sound measurements[edit]

Sound intensity[edit]

In a sound wave, the complementary variable to sound pressure is the particle velocity. Together they determine the sound intensity of the wave.
Sound intensity, denoted I and measured in W·m−2, is given by:

\mathbf I = p \mathbf v,


  • p is the sound pressure, measured in Pa;
  • v is the particle velocity, measured in m·s−1.

Acoustic impedance[edit]

Acoustic impedance, denoted Z and measured in Pa·m−3·s, is given by:[1]

Z(s) = \frac{\hat{p}(s)}{\hat{Q}(s)},


  • \hat{p}(s) is the Laplace transform of sound pressure, measured in Pa;
  • \hat{Q}(s) is the Laplace transform of sound volume flow rate, measured in m3·s−1.

Specific acoustic impedance, denoted z and measured in Pa·m−1·s, is given by:[1]

z(s) = \frac{\hat{p}(s)}{\hat{v}(s)},


  • \hat{p}(s) is the Laplace transform of sound pressure, measured in Pa;
  • \hat{v}(s) is the Laplace transform of particle velocity, measured in m·s−1.

Particle displacement[edit]

A progressive sine wave for particle displacement is expressed by:

\xi(\mathbf{r},\, t) = \xi_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\xi, 0}),

so particle velocity and sound pressure are equal to:

v(\mathbf{r},\, t) = \frac{\partial \xi}{\partial t} (\mathbf{r},\, t) = \omega \xi_\mathrm{m} \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\xi, 0} + \frac{\pi}{2}\right) = v_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{v, 0}),
p(\mathbf{r},\, t) = -\rho c^2 \frac{\partial \xi}{\partial x} (\mathbf{r},\, t) = \rho c^2 k_x \xi_\mathrm{m} \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\xi, 0} + \frac{\pi}{2}\right) = p_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{p, 0}),

where x is the space variable along the direction of propagation of the sound wave.

Taking the Laplace transforms of v and p with respect to time yields:

\hat{v}(\mathbf{r},\, s) = v_\mathrm{m} \frac{s \cos \varphi_{v,0} - \omega \sin \varphi_{v,0}}{s^2 + \omega^2},
\hat{p}(\mathbf{r},\, s) = p_\mathrm{m} \frac{s \cos \varphi_{p,0} - \omega \sin \varphi_{p,0}}{s^2 + \omega^2},

Since \varphi_{v,0} = \varphi_{p,0}, the amplitude of the specific acoustic impedance is equal to:

z_\mathrm{m}(\mathbf{r},\, s) = |z(\mathbf{r},\, s)| = \left|\frac{\hat{p}(\mathbf{r},\, s)}{\hat{v}(\mathbf{r},\, s)}\right| = \frac{p_\mathrm{m}}{v_\mathrm{m}} = \frac{\rho c^2 k_x}{\omega}.

Consequently, the amplitude of the sound pressure is connected to the amplitude of the particle displacement:

\xi_\mathrm{m} = \frac{v_\mathrm{m}}{\omega} = \frac{p_\mathrm{m}}{\omega z_\mathrm{m}(\mathbf{r},\, s)}.

Distance law[edit]

When measuring the sound created by an object, it is important to measure the distance from the object as well, since the sound pressure decreases with distance from a point source with a 1/r relationship (and not 1/r2, like sound intensity).[2]

The distance law of sound pressure p for a spherical sound wave at a distance r from a punctual sound source is given by[citation needed]:

p \propto \frac{1}{r}.

If the sound pressure p1 is measured at a distance r1, the sound pressure p2 at another position r2 can be calculated:

\frac{p_2} {p_1} = \frac{r_1}{r_2},
p_2 = p_{1} \frac{r_1}{r_2}.

The sound pressure may vary in direction from the source, as well, so measurements at different angles may be necessary, depending on the situation.[citation needed] An obvious example of a source that varies in level in different directions is a bullhorn.

Sound pressure level[edit]

For other uses, see Sound level.

Sound pressure level (SPL) or acoustic pressure level is a logarithmic measure of the effective sound pressure of a sound relative to a reference value.
Sound pressure level, denoted Lp and measured in dB, above a standard reference level, is given by:

L_p = 10 \log_{10}\left(\frac{{p_\mathrm{rms}}^2}{{p_0}^2}\right) = 20 \log_{10}\left(\frac{p_\mathrm{rms}}{p_0}\right)~\mathrm{dB (SPL)},


  • prms is the root mean square sound pressure, measured in Pa;[3]
  • p0 is the reference sound pressure, measured in Pa.

Sometimes variants are used such as dB (SPL), dBSPL, or dBSPL. These variants are not recognized as units in the SI.[4] The unit dB (SPL) is sometimes abbreviated to just "dB", which can give the erroneous impression that a dB is an absolute unit by itself.

The commonly used reference sound pressure in air is p0 = 20 μPa (RMS) or 0.0002 dynes/cm2,[5] which is usually considered the threshold of human hearing (roughly the sound of a mosquito flying 3 m away). Most sound level measurements will be made relative to this level, meaning 1 Pa will equal an SPL of 94 dB. In other media, such as underwater, a reference level of 1 μPa is used.[6] These references are defined in ANSI S1.1-1994.[7]

The lower limit of audibility is defined as SPL of 0 dB, but the upper limit is not as clearly defined. While 1 atm (194 dB Peak or 191 dB SPL) is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere, larger sound waves can be present in other atmospheres or other media such as under water, or through the Earth.

Ears detect changes in sound pressure. Human hearing does not have a flat spectral sensitivity (frequency response) relative to frequency versus amplitude. Humans do not perceive low- and high-frequency sounds as well as they perceive sounds near 2,000 Hz, as shown in the equal-loudness contour. Because the frequency response of human hearing changes with amplitude, three weightings have been established for measuring sound pressure: A, B and C. A-weighting applies to sound pressures levels up to 55 dB, B-weighting applies to sound pressures levels between 55 and 85 dB, and C-weighting is for measuring sound pressure levels above 85 dB.[citation needed]

In order to distinguish the different sound measures a suffix is used: A-weighted sound pressure level is written either as dBA or LA. B-weighted sound pressure level is written either as dBB or LB, and C-weighted sound pressure level is written either as dBC or LC. Unweighted sound pressure level is called "linear sound pressure level" and is often written as dBL or just L. Some sound measuring instruments use the letter "Z" as an indication of linear SPL.


The distance of the measuring microphone from a sound source is often omitted when SPL measurements are quoted, making the data useless. In the case of ambient environmental measurements of "background" noise, distance need not be quoted as no single source is present, but when measuring the noise level of a specific piece of equipment the distance should always be stated. A distance of one metre (1 m) from the source is a frequently used standard distance. Because of the effects of reflected noise within a closed room, the use of an anechoic chamber allows for sound to be comparable to measurements made in a free field environment.

When sound level Lp1 is measured at a distance r1, the sound level Lp2 at the distance r2 is

L_{p_2} = L_{p_1} + 20 \log_{10} \left( \frac{r_1}{r_2} \right) dB.

Multiple sources[edit]

The formula for the sum of the sound pressure levels of n incoherent radiating sources is

L_\Sigma = 10 \log_{10} \left(\frac{{p_1}^2 + {p_2}^2 + \ldots + {p_n}^2}{{p_0}^2}\right) = 10 \log_{10} \left(\left(\frac{p_1}{p_0}\right)^2 + \left(\frac{p_2}{p_0}\right)^2 + \ldots + \left(\frac{p_n}{p_0}\right)^2\right).

From the formula of the sound pressure level we find

\left(\frac{p_i}{p_0}\right)^2 = 10^{\frac{L_i}{10}},\quad i = 1,\, 2,\, \ldots,\, n.

This inserted in the formula for the sound pressure level to calculate the sum level shows

L_\Sigma = 10 \log_{10} \left(10^{\frac{L_1}{10}} + 10^{\frac{L_2}{10}} + \ldots + 10^{\frac{L_n}{10}} \right)~\mathrm{dB}.

Examples of sound pressure[edit]

Examples of sound pressure in air at standard atmospheric pressure
Source of sound Sound pressure*
Sound pressure level
Shockwave (distorted sound waves > 1 atm; waveform valleys are clipped at zero pressure) >101,325 >194
Theoretical limit for undistorted sound at 1 atmosphere environmental pressure 101,325 194
Stun grenades 6,000–20,000 170–180
Simple open-ended thermoacoustic device[8] 12,619 176
.30-06 rifle being fired 1 m to shooter's side 7,265 171
Rocket launch equipment acoustic tests 4000 165
LRAD 1000Xi Long Range Acoustic Device at 1 m[9] 893 153
Jet engine at 1 m 632 150
Threshold of pain[10][11][12] 63-200 130-140
Loudest human voice at 1 inch[13] 110 135
Trumpet at 0.5 m[14] 63.2 130
Vuvuzela horn at 1 m[15] 20 120
Risk of instantaneous noise-induced hearing loss 20 120
Jet engine at 100 m 6.32–200 110–140
Non-electric chainsaw at 1 m[16] 6.32 110
Jack hammer at 1 m 2 100
Traffic on a busy roadway at 10 m 0.2–0.632 80–90
Hearing damage (over long-term exposure, need not be continuous)[17] 0.356 85
Passenger car at 10 m (2–20)×10−2 60–80
EPA-identified maximum to protect against hearing loss and other disruptive effects from noise, such as sleep disturbance, stress, learning detriment, etc.[18] 6.32×10−2 70
Handheld electric mixer 65
TV (set at home level) at 1 m 2×10−2 60
Washing machine, dishwasher[19] 42–53
Normal conversation at 1 m (2–20)×10−3 40–60
Very calm room (2–6.32)×10−4 20–30
Light leaf rustling, calm breathing 6.32×10−5 10
Auditory threshold at 1 kHz[17] 2×10−5 0

*All values listed are the effective sound pressure unless otherwise stated.

See also[edit]



  1. ^ a b Wolfe, J. "What is acoustic impedance and why is it important?". University of New South Wales, Dept. of Physics, Music Acoustics. Retrieved 1 January 2014. 
  2. ^ Longhurst, R.S. (1967). Geometrical and Physical Optics. Norwich: Longmans. 
  3. ^ Bies, David A., and Hansen, Colin. (2003). Engineering Noise Control.
  4. ^ Thompson and Taylor 2008, Guide for the Use of the International System of Units (SI), NIST Special Publication SP811
  5. ^ Ross Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.
  6. ^ Morfey, Christopher L. (2001). Dictionary of Acoustics. San Diego: Academic Press. ISBN 978-0125069403. 
  7. ^ "Noise Terms Glossary". Retrieved 2012-10-14. 
  8. ^ Hatazawa, M., Sugita, H., Ogawa, T. & Seo, Y. (Jan. 2004), ‘Performance of a thermoacoustic sound wave generator driven with waste heat of automobile gasoline engine,’ Transactions of the Japan Society of Mechanical Engineers (Part B) Vol. 16, No. 1, 292–299. [1]
  9. ^ "LRAD Corporation Product Overview for LRAD 1000Xi". Retrieved 29 May 2014. 
  10. ^ Nave, Carl R. (2006). "Threshold of Pain". HyperPhysics. SciLinks. Retrieved 2009-06-16. 
  11. ^ Franks, John R.; Stephenson, Mark R.; Merry, Carol J., eds. (June 1996). Preventing Occupational Hearing Loss - A Practical Guide. National Institute for Occupational Safety and Health. p. 88. Retrieved 2009-07-15. 
  12. ^ Realistic Maximum Sound Pressure Levels for Dynamic Microphones - Shure
  13. ^ Realistic Maximum Sound Pressure Levels for Dynamic Microphones - Shure
  14. ^ Recording Brass & Reeds
  15. ^ Swanepoel, De Wet; Hall III, James W; Koekemoer, Dirk (February 2010). "Vuvuzela – good for your team, bad for your ears" (PDF). South African Medical Journal 100 (4): 99–100. PMID 20459912. 
  16. ^ "Decibel Table - SPL - Loudness Comparison Chart". "sengpielaudio". Retrieved 5 Mar 2012. 
  17. ^ a b William Hamby. "Ultimate Sound Pressure Level Decibel Table". Archived from the original on 2010-07-27. 
  18. ^ "EPA Identifies Noise Levels Affecting Health and Welfare" (Press release). Environmental Protection Agency. April 2, 1974. Retrieved October 17, 2014. 
  19. ^ "Active Water". Bosch. p. 17. Retrieved 4 March 2012. 
  • Beranek, Leo L, "Acoustics" (1993) Acoustical Society of America. ISBN 0-88318-494-X
  • Daniel R. Raichel, "The Science and Applications of Acoustics" (2006), Springer New York, ISBN 1441920803

External links[edit]