Sound pressure

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. The SI unit of sound pressure is the pascal (Pa).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Mathematical definition

File:Sound pressure diagram.svg

Sound pressure diagram: Template:Ordered list

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, is defined by <math display="block">p_\text{total} = p_\text{stat} + p,</math> where

• p_total is the total pressure,
• p_stat is the static pressure.

Sound measurements

Sound intensity

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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 in SI units, is defined by <math display="block">\mathbf I = p \mathbf v,</math> where

• p is the sound pressure,
• v is the particle velocity.

Acoustic impedance

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Acoustic impedance, denoted Z and measured in Pa·m⁻³·s in SI units, is defined by<ref name="Wolfe">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">Z(s) = \frac{\hat{p}(s)}{\hat{Q}(s)},</math> where

• <math>\hat{p}(s)</math> is the Laplace transform of sound pressure,Template:Citation needed
• <math>\hat{Q}(s)</math> is the Laplace transform of sound volume flow rate.

Specific acoustic impedance, denoted z and measured in Pa·m⁻¹·s in SI units, is defined by<ref name="Wolfe" /> <math display="block">z(s) = \frac{\hat{p}(s)}{\hat{v}(s)},</math> where

• <math>\hat{p}(s)</math> is the Laplace transform of sound pressure,
• <math>\hat{v}(s)</math> is the Laplace transform of particle velocity.

Particle displacement

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The particle displacement of a progressive sine wave is given by <math display="block">\delta(\mathbf{r}, t) = \delta_\text{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0}),</math> where

• <math>\delta_\text{m}</math> is the amplitude of the particle displacement,
• <math>\varphi_{\delta, 0}</math> is the phase shift of the particle displacement,
• k is the angular wavevector,
• ω is the angular frequency.

It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave x are given by <math display="block">v(\mathbf{r}, t) = \frac{\partial \delta}{\partial t} (\mathbf{r}, t) = \omega \delta_\text{m} \cos\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0} + \frac{\pi}{2}\right) = v_\text{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{v, 0}),</math> <math display="block">p(\mathbf{r}, t) = -\rho c^2 \frac{\partial \delta}{\partial x} (\mathbf{r}, t) = \rho c^2 k_x \delta_\text{m} \cos\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0} + \frac{\pi}{2}\right) = p_\text{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{p, 0}),</math> where

• v_m is the amplitude of the particle velocity,
• <math>\varphi_{v, 0}</math> is the phase shift of the particle velocity,
• p_m is the amplitude of the acoustic pressure,
• <math>\varphi_{p, 0}</math> is the phase shift of the acoustic pressure.

Taking the Laplace transforms of v and p with respect to time yields <math display="block">\hat{v}(\mathbf{r}, s) = v_\text{m} \frac{s \cos \varphi_{v,0} - \omega \sin \varphi_{v,0}}{s^2 + \omega^2},</math> <math display="block">\hat{p}(\mathbf{r}, s) = p_\text{m} \frac{s \cos \varphi_{p,0} - \omega \sin \varphi_{p,0}}{s^2 + \omega^2}.</math>

Since <math>\varphi_{v,0} = \varphi_{p,0}</math>, the amplitude of the specific acoustic impedance is given by <math display="block">z_\text{m}(\mathbf{r}, s) = |z(\mathbf{r}, s)| = \left|\frac{\hat{p}(\mathbf{r}, s)}{\hat{v}(\mathbf{r}, s)}\right| = \frac{p_\text{m}}{v_\text{m}} = \frac{\rho c^2 k_x}{\omega}.</math>

Consequently, the amplitude of the particle displacement is related to that of the acoustic velocity and the sound pressure by <math display="block">\delta_\text{m} = \frac{v_\text{m}}{\omega},</math> <math display="block">\delta_\text{m} = \frac{p_\text{m}}{\omega z_\text{m}(\mathbf{r}, s)}.</math>

Inverse-proportional law

When measuring the sound pressure created by a sound source, it is important to measure the distance from the object as well, since the sound pressure of a spherical sound wave decreases as 1/r from the centre of the sphere (and not as 1/r², like the sound intensity):<ref>Template:Cite book</ref> <math display="block">p(r) \propto \frac{1}{r}.</math>

This relationship is an inverse-proportional law.

If the sound pressure p₁ is measured at a distance r₁ from the centre of the sphere, the sound pressure p₂ at another position r₂ can be calculated: <math display="block">p_2 = \frac{r_1}{r_2}\,p_1.</math>

The inverse-proportional law for sound pressure comes from the inverse-square law for sound intensity: <math display="block">I(r) \propto \frac{1}{r^2}.</math> Indeed, <math display="block">I(r) = p(r) v(r) = p(r)\left[p * z^{-1}\right](r) \propto p^2(r),</math> where

• <math>v</math> is the particle velocity,
• <math>*</math> is the convolution operator,
• z⁻¹ is the convolution inverse of the specific acoustic impedance,

hence the inverse-proportional law: <math display="block">p(r) \propto \frac{1}{r}.</math>

Sound pressure level

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Sound pressure level (SPL) or acoustic pressure level (APL) is a logarithmic measure of the effective pressure of a sound relative to a reference value.

Sound pressure level, denoted L_p and measured in dB,<ref name="IEC60027-3">"Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units", IEC 60027-3 Ed. 3.0, International Electrotechnical Commission, 19 July 2002.</ref> is defined by:<ref>Template:Cite book</ref> <math display="block">L_p = \ln\left(\frac{p}{p_0}\right) ~ \text{Np} = 2 \log_{10}\left(\frac{p}{p_0}\right)~\text{B} = 20 \log_{10}\left(\frac{p}{p_0}\right)~\text{dB},</math> where

• p is the root mean square sound pressure,<ref>Template:Cite book</ref>
• p₀ is a reference sound pressure,
• Template:Nowrap is the neper,
• Template:Nowrap is the bel,
• Template:Nowrap is the decibel.

Template:AnchorThe commonly used reference sound pressure in air is<ref>Ross Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.</ref> Template:Block indent which is often considered as the threshold of human hearing (roughly the sound of a mosquito flying 3 m away). The proper notations for sound pressure level using this reference are Template:Nobreak or Template:Nobreak, but the suffix notations Template:Nobreak, Template:Nobreak, dBSPL, and dB_SPL are very common, even if they are not accepted by the SI.<ref name="NIST2008">Thompson, A. and Taylor, B. N. Sec. 8.7: "Logarithmic quantities and units: level, neper, bel", Guide for the Use of the International System of Units (SI) 2008 Edition, NIST Special Publication 811, 2nd printing (November 2008), SP811 PDF.</ref>

Most sound-level measurements will be made relative to this reference, meaning Template:Nobreak will equal an SPL of <math>20 \log_{10}\left(\frac{1}{2\times10^{-5}}\right)~\text{dB}\approx 94~\text{dB}</math>. In other media, such as underwater, a reference level of Template:Nobreak is used.<ref name="Morfey">Template:Cite book</ref> These references are defined in ANSI S1.1-2013.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The main instrument for measuring sound levels in the environment is the sound level meter. Most sound level meters provide readings in A, C, and Z-weighted decibels and must meet international standards such as IEC 61672-2013.

Examples

The lower limit of audibility is defined as SPL of Template:Nobreak, but the upper limit is not as clearly defined. While Template:Nobreak (Template:Nobreak or Template:Nobreak)<ref name=":0">Template:Cite book</ref><ref name=":1">Template:Cite book</ref> is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere (i. e., if the thermodynamic properties of the air are disregarded; in reality, the sound waves become progressively non-linear starting over 150 dB), larger sound waves can be present in other atmospheres or other media, such as underwater or through the Earth.<ref name="Audio1">Template:Cite book</ref>

File:Lindos1.svg

Equal-loudness contour, showing sound-pressure-vs-frequency at different perceived loudness levels

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 between 3,000 and 4,000 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.

In order to distinguish the different sound measures, a suffix is used: A-weighted sound pressure level is written either as dB_A or L_A. B-weighted sound pressure level is written either as dB_B or L_B, and C-weighted sound pressure level is written either as dB_C or L_C. Unweighted sound pressure level is called "linear sound pressure level" and is often written as dB_L or just L. Some sound measuring instruments use the letter "Z" as an indication of linear SPL.<ref name="Audio1" />

Distance

The distance of the measuring microphone from a sound source is often omitted when SPL measurements are quoted, making the data useless, due to the inherent effect of the inverse proportional law. 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 sound to be comparable to measurements made in a free field environment.<ref name="Audio1" />

According to the inverse proportional law, when sound level L_p1 is measured at a distance r₁, the sound level L_p2 at the distance r₂ is <math display="block">L_{p_2} = L_{p_1} + 20 \log_{10}\left( \frac{r_1}{r_2} \right)~\text{dB}.</math>

Multiple sources

The formula for the sum of the sound pressure levels of n incoherent radiating sources is <math display="block">L_\Sigma = 10 \log_{10}\left(\frac{p_1^2 + p_2^2 + \dots + p_n^2}{p_0^2}\right)~\text{dB} = 10 \log_{10}\left[\left(\frac{p_1}{p_0}\right)^2 + \left(\frac{p_2}{p_0}\right)^2 + \dots + \left(\frac{p_n}{p_0}\right)^2\right]~\text{dB}.</math>

Inserting the formulas <math display="block">\left(\frac{p_i}{p_0}\right)^2 = 10^{\frac{L_i}{10~\text{dB}}},\quad i = 1, 2, \ldots, n</math> in the formula for the sum of the sound pressure levels yields <math display="block">L_\Sigma = 10 \log_{10} \left(10^{\frac{L_1}{10~\text{dB}}} + 10^{\frac{L_2}{10~\text{dB}}} + \dots + 10^{\frac{L_n}{10~\text{dB}}} \right)~\text{dB}.</math>

Examples of sound pressure

Examples of sound pressure in air at standard atmospheric pressure

Source of sound	Distance	Sound pressure levelTemplate:Efn
		(Pa)	(dBSPL)
Shock wave (distorted sound waves > 1 atm; waveform valleys are clipped at zero pressure)<ref name=":0" /><ref name=":1" />		>1.01×105	>191
Simple open-ended thermoacoustic device<ref>Template:Cite journal</ref>	Template:Clarify	1.26×104	176
1883 eruption of Krakatoa<ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref><ref name="winchester">Template:Cite book</ref>	165 km		172
.30-06 rifle being fired	1 m to shooter's side	7.09×103	171
Firecracker<ref>Template:Cite journal</ref>	0.5 m	7.09×103	171
Stun grenade<ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>	Ambient	1.60×103 ...8.00×103	158–172
Template:Convert party balloon inflated to rupture<ref name="balloonpoploudness">Template:Cite journal</ref>	At ear	4.92×103	168
Template:Convert diameter balloon crushed to rupture<ref name="balloonpoploudness" />	At ear	1.79×103	159
Template:Convert party balloon inflated to rupture<ref name="balloonpoploudness" />	0.5 m	1.42×103	157
Template:Convert diameter balloon popped with a pin<ref name="balloonpoploudness" />	At ear	1.13×103	155
LRAD 1000Xi Long Range Acoustic Device<ref name="lradproductsoverview">{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>	1 m	8.93×102	153
Template:Convert party balloon inflated to rupture<ref name="balloonpoploudness" />	1 m	731	151
Jet engine<ref name="Audio1" />	1 m	632	150
Template:Convert diameter balloon crushed to rupture<ref name="balloonpoploudness" />	0.95 m	448	147
Template:Convert diameter balloon popped with a pin<ref name="balloonpoploudness" />	1 m	282.5	143
Loudest human voice<ref name="Shure" />	1 inch	110	135
Trumpet<ref>Recording Brass & Reeds.</ref>	0.5 m	63.2	130
Vuvuzela horn<ref>Template:Cite journal</ref>	1 m	20.0	120
Threshold of pain<ref name="painthres">{{#invoke:citation/CS1|citation	CitationClass=web }}</ref><ref name="dictionary">Template:Cite book</ref><ref name="Shure">Realistic Maximum Sound Pressure Levels for Dynamic Microphones – Shure.</ref>	At ear	20–200	120–140
Risk of instantaneous noise-induced hearing loss	At ear	20.0	120
Jet engine	100–30 m	6.32–200	110–140
Two-stroke chainsaw<ref name="sengpielaudio">{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>	1 m	6.32	110
Jackhammer	1 m	2.00	100
Traffic on a busy roadway (combustion engines)	10 m	0.20–0.63	80–90
Hearing damage (over long-term exposure, need not be continuous)<ref name="Hamby" />	At ear	0.36	85
Passenger car (combustion engine)	10 m	0.02–0.20	60–80
Traffic on a busy roadway (electric vehicles) <ref>Template:Citation</ref>	10 m	0.20–0.63	65-75
EPA-identified maximum to protect against hearing loss and other disruptive effects from noise, such as sleep disturbance, stress, learning detriment, etc.<ref>Template:Cite press release</ref>	Ambient	0.06	70
TV (set at home level)	1 m	0.02	60
Normal conversation	1 m	2×10−3–0.02	40–60
citation	CitationClass=web }}</ref>	10 m	0.02–0.20	38-48
Very calm room	Ambient	2.00×10−4 ...6.32×10−4	20–30
Light leaf rustling, calm breathing<ref name="Audio1" />	Ambient	6.32×10−5	10
Auditory threshold at 1 kHz<ref name="Hamby">{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>	At ear	2.00×10−5	0
Anechoic chamber, Orfield Labs, A-weighted<ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>	Ambient	6.80×10−6	−9.4
Anechoic chamber, University of Salford, A-weighted<ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>	Ambient	4.80×10−6	−12.4
Anechoic chamber, Microsoft, A-weighted<ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>	Ambient	1.90×10−6	−20.35

See also

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References

General

• Beranek, Leo L., Acoustics (1993), Acoustical Society of America, Template:ISBN.
• Daniel R. Raichel, The Science and Applications of Acoustics (2006), Springer New York, Template:ISBN.

External links

• Template:Commons category-inline
• Sound Pressure and Sound Power, Two Commonly Confused Characteristics of Sound
• Decibel (Loudness) Comparison Chart