Sound pressure
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2008-07-01T11:28:00Z
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{{Sound measurements}}
'''Sound pressure''' is the local [[pressure]] deviation from the ambient (average, or equilibrium) pressure caused by a [[sound]] [[wave]]. Sound pressure can be measured using a [[microphone]] in air and a [[hydrophone]] in water. The SI unit for sound pressure is the [[pascal (unit)|pascal]] (symbol: Pa). The instantaneous sound pressure is the deviation from the local ambient pressure ''p''<sub>0</sub> caused by a [[sound]] wave at a given location and given instant in time. The effective sound pressure is the [[root mean square]] of the instantaneous sound pressure over a given interval of time (or space). In a sound wave, the complementary variable to sound pressure is the [[particle velocity|acoustic particle velocity]]. For small amplitudes, sound pressure and particle velocity are linearly related and their ratio is the [[acoustic impedance]]. The acoustic impedance depends on both the characteristics of the wave and the [[Transmission medium|medium]]. The local instantaneous [[sound intensity]] is the product of the sound pressure and the acoustic particle velocity and is, therefore, a vector quantity.
The sound pressure deviation ''p'' is
:<math>
p = \frac{F}{A} \,
</math>
where
:''F'' = force,
:''A'' = area.
The entire pressure ''p''<sub>total</sub> is
:<math>
p_\mathrm{total} = p_0 + p \,
</math>
where
:''p''<sub>0</sub> = local ambient pressure,
:''p'' = sound pressure deviation.
==Sound pressure level==
'''Sound pressure level''' (SPL) or [[sound]] [[level]] ''L''<sub>p</sub> is a [[logarithmic scale|logarithmic measure]] of the [[root mean square|rms]] sound [[pressure]] of a sound relative to a reference value. It is measured in [[decibel]] (dB).
:<math>
L_p=10 \log_{10}\left(\frac{p^2_{\mathrm{{rms}}}}{p^2_{\mathrm{ref}}}\right) =20 \log_{10}\left(\frac{p_{\mathrm{rms}}}{p_{\mathrm{ref}}}\right)\mbox{ dB} \,
</math>
where <math>p_{\mathrm{ref}}</math> is the reference sound pressure and <math>p_{\mathrm{rms}}</math> is the rms sound pressure being measured.<ref>Sometimes reference sound pressure is denoted ''p''<sub>0</sub>, not to be confused with the (much higher) ambient pressure.</ref>
Sometimes variants are used such as dB (SPL), dBSPL, or dB<sub>SPL</sub>. These variants are not permitted by [[SI]].<ref>[http://physics.nist.gov/Pubs/pdf.html Taylor 1995, Guide for the Use of the International System of Units (SI), NIST Special Publication SP811]</ref>
The commonly used reference sound pressure in air is <math>p_{\mathrm{ref}}</math> = 20 [[micropascal|µPa]] (rms), which is usually considered the [[threshold of human hearing]] (roughly the sound of a [[mosquito]] flying 3 m away). When dealing with [[hearing (sense)|hearing]], the perceived loudness of a sound correlates roughly logarithmically to its sound pressure. ''See also [[Weber-Fechner law]].'' Most measurements of audio equipment will be made relative to this level, meaning 1 pascal will equal 94 dB of sound pressure.
In other media, such as [[underwater acoustics|underwater]], a reference level of 1 µPa is more often used.<ref name="Morfey">C. L. Morfey, Dictionary of Acoustics (Academic Press, San Diego, 2001).</Ref>
These references are defined in [[American National Standards Institute|ANSI]] S1.1-1994.<ref>[http://www.quietnoise.com/glossary.htm Glossary of Noise Terms] — ''Sound pressure level'' definition</ref>
The unit dB (SPL) is often abbreviated to just "dB", which gives some the erroneous notion that a dB is an absolute unit by itself.
The human [[ear]] is a sound pressure sensitive detector. It does not have a flat [[spectral response]], so the sound pressure is often [[frequency]] weighted such that the measured level will match the perceived level. When weighted in this way the measurement is referred to as a [[sound level]]. The [[International Electrotechnical Commission]] (IEC) has defined several weighting schemes. [[A-weighting]] attempts to match the response of the human ear to pure tones, while C-weighting is used to measure peak sound levels.<ref>[http://www.cirrusresearch.co.uk/glossary.html Glossary of Terms] — Cirrus Research plc.</ref> If the (unweighted) SPL is desired, many instruments allow a "flat" or unweighted measurement to be made. ''See also [[Weighting filter]].''
When measuring the sound created by an object, it is important to measure the distance from the object as well, since the SPL decreases in distance from a [[point source]] with 1/''r'' (and not with [[Inverse-square law|1/''r''<sup>2</sub>]], like [[sound intensity]]). It often varies in direction from the source, as well, so many measurements may be necessary, depending on the situation. An obvious example of a source that varies in level in different directions is a [[bullhorn]].
Sound pressure ''p'' in N/m² or Pa is
:<math>
p = Zv = \frac{J}{v} = \sqrt{JZ} \,
</math>
where
: ''Z'' is [[acoustic impedance]], [[sound impedance]], or [[characteristic impedance]], in Pa·s/m
: ''v'' is [[particle velocity]] in m/s
: ''J'' is [[acoustic intensity]] or [[sound intensity]], in W/m<sup>2</sup>
Sound pressure ''p'' is connected to '''[[particle displacement]]''' (or particle amplitude) ξ, in m, by
:<math>
\xi = \frac{v}{2 \pi f} = \frac{v}{\omega} = \frac{p}{Z \omega} = \frac{p}{ 2 \pi f Z} \,
</math>.
Sound pressure ''p'' is
:<math>
p = \rho c \omega \xi = Z \omega \xi = { 2 \pi f \xi Z} = \frac{a Z}{\omega} = c \sqrt{\rho E} = \sqrt{\frac{P_{ac} Z}{A}} \,
</math>,
normally in units of N/m² = Pa.
where:
{| class="wikitable"
! Symbol !! [[SI Unit]] !! Meaning
|-
! ''p''
| [[pascal (unit)|pascal]]s || sound pressure
|-
! ''f''
| [[hertz]] || [[frequency]]
|-
! ''ρ''
| [[kilogram|kg]]/[[Metre|m]]³ || [[density of air]]
|-
! ''c''
| [[Metre|m]]/[[second|s]] || [[speed of sound]]
|-
! ''v''
| [[Meters per second|m/s]] || [[particle velocity]]
|-
! <math>\omega</math> = 2 · <math>\pi</math> · ''f''
| [[radians]]/[[second|s]] || [[angular frequency]]
|-
! ''ξ''
| [[meter]]s || [[particle displacement]]
|-
! ''Z = c • ρ''
| [[Newton|N]]·[[second|s]]/[[Metre|m]]³ || [[acoustic impedance]]
|-
! ''a''
| [[Metre|m]]/[[second|s]]² || [[particle acceleration]]
|-
! ''J''
| [[Watt|W]]/[[Metre|m]]² || [[sound intensity]]
|-
! ''E''
| [[Watt|W]]·[[second|s]]/[[Metre|m]]³ || [[sound energy density]]
|-
! ''P''<sub>ac</sub>
| [[watt]]s || [[sound power]] or [[acoustic power]]
|-
! ''A''
| [[Metre|m]]² || [[Area]]
|}
The '''distance law''' for the sound pressure ''p'' is inverse-proportional to the distance ''r'' of a punctual sound source.
:<math>
p \propto \frac{1}{r} \,
</math> (proportional)
:<math>
\frac{p_1} {p_2} = \frac{r_2}{r_1} \,
</math>
:<math>
p_1 = p_{2} \cdot r_{2} \cdot \frac{1}{r_1} \,
</math>
The assumption of 1/''r''² with the square is here wrong. That is only correct for [[sound intensity]].
''Note: The often used term "intensity of sound pressure" is not correct. Use "[[Magnitude (mathematics)|magnitude]]", "[[wikt:strength|strength]]", "[[amplitude]]", or "[[level]]" instead. "[[Sound intensity]]" is [[sound power]] per unit area, while "pressure" is a measure of force per unit area. Intensity is not equivalent to pressure.''
:<math>
I \sim {p^2} \sim \dfrac{1}{r^2} \,
</math>
Hence
<math>
p \sim \dfrac{1}{r} \,
</math>
=== Examples of sound pressure and sound pressure levels ===
Sound pressure in air:
<!-- This section is linked from [[Sound]] -->
{| class="wikitable"
! Source of sound !! Sound pressure !! Sound pressure level
|-
!   !! [[pascal (unit)|pascal]] !! [[Decibel|dB]] re 20 μPa
|-
|Theoretical limit for undistorted sound at<br>1 [[atmosphere (unit)|atmosphere]] environmental [[pressure]] || align="right" | 101,325 Pa || align="right"| 191 dB
|-
|[[Krakatoa]] explosion at 100 [[statute mile|mile]]s (160 km) in air || align="right" | 20,000 Pa || align="right" | 180 dB
|-
|Simple open-ended [[thermoacoustics|thermoacoustic device]] <ref>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.
[http://md1.csa.com/partners/viewrecord.php?requester=gs&collection=TRD&recid=200407211336MT&q=Performance+of+a+thermoacoustic+sound+wave+generator+driven+with+waste+heat+of+automobile+gasoline+engine&uid=790404233&setcookie=yes]
</ref> || align="right" | 12,000 Pa || align="right" | 176 dB
|-
|[[M1 Garand]] being fired at 1 m || align="right" | 5,000 Pa || align="right" | 168 dB
|-
|[[Jet engine]] at 30 [[metre|m]] || align="right" | 630 Pa || align="right" | 150 dB
|-
|[[Rifle]] being fired at 1 m || align="right" | 200 Pa || align="right" | 140 dB
|-
|[[Threshold of pain]] || align="right" | 100 Pa || align="right" | 130 dB
|-
|The Who live in concert 31st of May 1976 at 32 metres <ref>Wikipedia Article The [[Loudest_band_in_the_world]]</ref>|| align="right" | || align="right" | 126 dB
|-
|[[Hearing damage]] (due to short-term exposure) || align="right" | 20 Pa || align="right" | approx. 120 dB
|-
|[[Jet]] at 100 m || align="right" | 6 – 200 Pa || align="right" | 110 – 140 dB
|-
|[[Jack hammer]] at 1 m || align="right" | 2 Pa || align="right" | approx. 100 dB
|-
|[[Hearing damage]] (due to long-term exposure) || align="right" | 6×10<sup>−1</sup> Pa || align="right" | approx. 85 dB
|-
|Major road at 10 m || align="right" | 2×10<sup>−1</sup> – 6×10<sup>−1</sup> Pa || align="right" | 80 – 90 dB
|-
|[[automobile|Passenger car]] at 10 m || align="right" | 2×10<sup>−2</sup> – 2×10<sup>−1</sup> Pa || align="right" | 60 – 80 dB
|-
|TV (set at home level) at 1 m || align="right" | 2×10<sup>−2</sup> Pa || align="right" | approx. 60 dB
|-
|Normal talking at 1 m || align="right" | 2×10<sup>−3</sup> – 2×10<sup>−2</sup> Pa || align="right" | 40 – 60 dB
|-
|Very calm room || align="right" | 2×10<sup>−4</sup> – 6×10<sup>−4</sup> Pa || align="right" | 20 – 30 dB
|-
|Leaves rustling, calm breathing || align="right" | 6×10<sup>−5</sup> Pa || align="right" | 10 dB
|-
|[[Auditory threshold]] at 2 kHz || align="right" | 2×10<sup>−5</sup> Pa || align="right" | 0 dB
|}
Sound pressure in water:
{| class="wikitable"
! Source of sound !! Sound pressure !! Sound pressure level
|-
!   !! [[pascal (unit)|pascal]] !! [[Decibel|dB]] re 1 μPa
|-
|[[Auditory threshold]] of a diver at 1 kHz ||align="right" | '''2.2 · 10<sup>-3</sup> Pa''' ||align="right" | 67 dB<ref>{{cite journal |author=Parvin S.J., Searle S.L. and Gilbert M.J. |title=Exposure of divers to underwater sound in the frequency range from to 2250 Hz |journal=Undersea Hyperb Med. Abstract |volume=28 |issue=Supl |date=[[2001]] |issn=1066-2936 |oclc=26915585 |url=http://archive.rubicon-foundation.org/984 |accessdate=2008-05-05 }}</ref>
|}
The formula for the sum of the sound pressure levels of ''n'' incoherent radiating sources is
:<math>
L_\Sigma = 10\,\cdot\,{\rm log}_{10} \left(\frac{p^2_1 + p^2_2 + \cdots + p^2_n}{p^2_{\mathrm{ref}}}\right)
= 10\,\cdot\,{\rm log}_{10} \left(\left({\frac{p_1}{p_{\mathrm{ref}}}}\right)^2 + \left({\frac{p_2}{p_{\mathrm{ref}}}}\right)^2 + \cdots + \left({\frac{p_n}{p_{\mathrm{ref}}}}\right)^2\right)
</math>
From the formula of the sound pressure level we find
:<math>
\left({\frac{p_i}{p_{\mathrm{ref}}}}\right)^2 = 10^{\frac{L_i}{10}},\qquad i=1,2,\cdots,n
</math>
This inserted in the formula for the sound pressure level to calculate the sum level shows
:<math>
L_\Sigma = 10\,\cdot\,{\rm log}_{10} \left(10^{\frac{L_1}{10}} + 10^{\frac{L_2}{10}} + \cdots + 10^{\frac{L_n}{10}} \right)\,{\rm dB}
</math>
===Beyond 191 dB===
As sound pressure levels approach 191 dB in air at sea level, their waveforms become [[Distortion|distorted]]; the exact level at which this happens varies with the [[barometric pressure]]. Sound waves are made up of [[rarefaction]] and [[Physical compression|compression]] cycles, but when the compression half of the wave cycle is double atmospheric pressure, the rarefaction half of the cycle approaches a perfect vacuum (no further air molecules to remove). At this point, the only possible increase in sound level that could be achieved is on the compression side of the waveform. (The rarefaction half of a sine wave would be [[clipping (audio)|clipped]] at any level above about 191 dB.) Any wave approaching these intensities is no longer considered sound, but a [[shock wave]]. Examples of such an occurrence are large-scale manned [[rocket launch]]es, [[sonic boom]]s, munitions [[explosion]]s, [[thunder]], [[earthquake]]s and volcanic explosions.
== See also ==
*[[Decibel]], especially [[Decibel#Acoustics|the ''Acoustics'' section]]
*[[Sone]]
*[[Loudness]]
*[[Weber-Fechner law#The case of sound|Weber-Fechner law (The case of Sound)]]
*[[Stevens' power law]]
*[[Sound power level]]
*[[Sound level meter]]
*[[Amplitude]]
*[[Acoustics]]
==Notes and References==
{{Reflist}}
*Beranek, Leo L, "Acoustics" (1993) Acoustical Society of America. ISBN 0-88318-494-X
*Morfey, Christopher L, "Dictionary of Acoustics" (2001) Academic Press, San Diego.
== External links ==
*[http://www.sengpielaudio.com/calculator-soundlevel.htm Conversion of sound pressure to sound pressure level and vice versa]
*[http://www.sengpielaudio.com/TableOfSoundPressureLevels.htm Table of Sound Levels - Corresponding Sound Pressure and Sound Intensity]
*[http://www.makeitlouder.com/Decibel%20Level%20Chart.txt William Hamby (2004) ''Ultimate Sound Pressure Level Decibel Table'']
*[http://www.sengpielaudio.com/calculator-ak-ohm.htm Ohm's law as acoustic equivalent - calculations]
*[http://www.rane.com/par-s.html#SPL Definition of sound pressure level]
*[http://www-ccrma.stanford.edu/~jos/mdft/DB_SPL.html A table of SPL values]
*[http://www.sengpielaudio.com/RelationshipsOfAcousticQuantities.pdf Relationships of acoustic quantities associated with a plane progressive acoustic sound wave - pdf]
*[http://www.usmotors.com/products/ProFacts/sound_power_and_sound_pressure.htm Sound pressure and sound power - two commonly confused characteristics of sound]
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