# Sound pressure

Sound measurements
Sound pressure p, SPL
Particle velocity v, SVL
Particle displacement ξ
Sound intensity I, SIL
Sound power Pac
Sound power level SWL
Sound energy
Sound exposure E
Sound exposure level SEL
Sound energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound
Audio frequency AF

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 for sound pressure p is the pascal (symbol: Pa).

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

Sound pressure level (SPL) or sound level is a logarithmic measure of the effective sound pressure of a sound relative to a reference value. It is measured in decibels (dB) above a standard reference level. The standard reference sound pressure in air or other gases is 20 µPa, which is usually considered the threshold of human hearing (at 1 kHz).

## Instantaneous sound pressure

The instantaneous sound pressure is the deviation from the local ambient pressure $p_{0}$ 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).

Total pressure $p_{\rm total}$ is given by:

$p_{\rm total} = p_{0} + p_{\rm osc} \,$

where:

$p_{0}$ = local ambient atmospheric (air) pressure,
$p_{\rm osc}$ = sound pressure deviation.

### Intensity

In a sound wave, the complementary variable to sound pressure is the acoustic particle velocity. Together they determine the acoustic intensity of the wave. The local instantaneous sound intensity is the product of the sound pressure and the acoustic particle velocity.

$\vec{I} = p \vec{v}$

### Acoustic impedance

For small amplitudes, sound pressure and volume 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.

The acoustic impedance is given by[1]

$Z = \frac{p}{U}$

where

Z is acoustic impedance or sound impedance
p is sound pressure
U is volume velocity

### Particle displacement

Sound pressure p is connected to particle displacement (or particle amplitude) ξ by

$\xi = \frac{v}{2 \pi f} = \frac{v}{\omega} = \frac{p}{Z \omega} = \frac{p}{ 2 \pi f Z} \,$.

Sound pressure p is

$p = \rho c 2 \pi f \xi = \rho c \omega \xi = Z \omega \xi = { 2 \pi f \xi Z} = \frac{a Z}{\omega} = Z v = c \sqrt{\rho E} = \sqrt{\frac{P_{ac} Z}{A}} \,$,

normally in units of N/m² = Pa.

where:

Symbol SI Unit Meaning
p pascals sound pressure
f hertz frequency
ρ kg/m³ density of medium
c m/s speed of sound
v m/s particle velocity
ω = 2 π f radians/s angular frequency
ξ meters particle displacement
Z = c ρ N·s/m³ acoustic impedance
a m/s² particle acceleration
I W/m² sound intensity
E W·s/m³ sound energy density
Pac watts sound power or acoustic power
A m² Area

### Distance law

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 for the sound pressure p in 3D is inversely proportional to the distance r of a punctual sound source.[citation needed]

$p \propto \dfrac{1}{r} \,$

If sound pressure $p_1\,$, is measured at a distance $r_1\,$, one can calculate the sound pressure $p_2\,$ at another position $r_2\,$,

$\frac{p_2} {p_1} = \frac{r_1}{r_2} \,$
$p_2 = p_{1} \cdot \dfrac{r_1}{r_2} \,$

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

Sound pressure level (SPL) or sound level $L_p$ is a logarithmic measure of the effective sound pressure of a sound relative to a reference value. It is measured in decibels (dB) above a standard reference level.

$L_p=10 \log_{10}\left(\frac{{p_{\mathrm{{rms}}}}^2}{{p_{\mathrm{ref}}}^2}\right) =20 \log_{10}\left(\frac{p_{\mathrm{rms}}}{p_{\mathrm{ref}}}\right)\mbox{ dB} ,$

where $p_{\mathrm{ref}}$ is the reference sound pressure and $p_{\mathrm{rms}}$ is the rms sound pressure being measured.[3][note 1]

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 $p_{\mathrm{ref}}$ = 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 pascal 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.

### Distance

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 $L_{p1}$ is measured at a distance $r_1$, the sound level $L_{p2}$ at the distance $r_2$ is

$L_{p2} = L_{p1} + 20 \cdot \log_{10} \left( \frac{r_1}{r_2} \right)$

### Multiple sources

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

$L_\Sigma = 10\,\cdot\,{\rm log}_{10} \left(\frac{{p_1}^2 + {p_2}^2 + \cdots + {p_n}^2}{{p_{\mathrm{ref}}}^2}\right) = 10\,\cdot\,{\rm log}_{10} \left(\left({\frac{p_1}{p_{\mathrm{ref}}}}\right)^2 + \left({\frac{p_2}{p_{\mathrm{ref}}}}\right)^2 + \cdots + \left({\frac{p_n}{p_{\mathrm{ref}}}}\right)^2\right)$

From the formula of the sound pressure level we find

$\left({\frac{p_i}{p_{\mathrm{ref}}}}\right)^2 = 10^{\frac{L_i}{10}},\qquad i=1,2,\cdots,n$

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

$L_\Sigma = 10\,\cdot\,{\rm log}_{10} \left(10^{\frac{L_1}{10}} + 10^{\frac{L_2}{10}} + \cdots + 10^{\frac{L_n}{10}} \right)\,{\rm dB}$

### Examples of sound pressure and sound pressure levels

Sound pressure in air:

Source of sound Sound pressure Sound pressure level
Sound in air pascal* dB ref 20 μPa
Shockwave (distorted sound waves > 1 atm; waveform valleys are clipped at zero pressure) >101,325 Pa (peak) >194 dB(peak)
Theoretical limit for undistorted sound at 1 atmosphere environmental pressure 101,325 Pa (peak) ~194.094 dB (peak)
Stun grenades 6,000–20,000 Pa (peak) 170–180 dB (peak)
Simple open-ended thermoacoustic device[8] 12,619 Pa 176 dB
.30-06 rifle being fired 1 m to shooter's side 7,265 Pa (peak) 171 dB (peak)
M1 Garand rifle being fired at 1 m 5,023 Pa (peak) 168 dB (peak)
Rocket launch equipment acoustic tests ~4000 Pa ~165 dB
Jet engine at 30 m 632 Pa 150 dB
Threshold of pain 63.2 Pa 130 dB
Vuvuzela horn at 1 m 20 Pa 120 dB(A)[9]
Risk of instantaneous noise-induced hearing loss 20 Pa approx. 120 dB
Jet engine at 100 m 6.32 – 200 Pa 110 – 140 dB
Non-electric chainsaw at 1 m 6.3 Pa 110 dB[10]
Jack hammer at 1 m 2 Pa approx. 100 dB
Traffic on a busy roadway at 10 m 2×10−1 – 6.32×10−1 Pa 80 – 90 dB
Hearing damage (over long-term exposure, need not be continuous) 0.356 Pa 85 dB[11]
Passenger car at 10 m 2×10−2 – 2×10−1 Pa 60 – 80 dB
EPA-identified maximum to protect against hearing loss and other disruptive effects from noise, such as sleep disturbance, stress, learning detriment, etc. 70 dB[12]
Handheld electric mixer 65 dB
TV (set at home level) at 1 m 2×10−2 Pa approx. 60 dB
Washing machine, dishwasher 42-53 dB[13]
Normal conversation at 1 m 2×10−3 – 2×10−2 Pa 40 – 60 dB
Very calm room 2×10−4 – 6.32×10−4 Pa 20 – 30 dB
Light leaf rustling, calm breathing 6.32×10−5 Pa 10 dB
Auditory threshold at 1 kHz 2×10−5 Pa 0 dB[11]

*All Values are RMS unless otherwise stated.

## Notes

1. ^ Sometimes reference sound pressure is denoted p0, not to be confused with the (much higher) ambient pressure.

## References

1. ^ 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. ^ 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.
10. ^ "Decibel Table - SPL - Loudness Comparison Chart". "sengpielaudio". Retrieved 5 Mar 2012.
11. ^ a b William Hamby. "Ultimate Sound Pressure Level Decibel Table". Archived from the original on 2010-07-27.
12. ^ EPA Identifies Noise Levels Affecting Health and Welfare, 1974-04-02, retrieved 2010-11-01
13. ^ "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