# Sound pressure

﻿
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 energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound c
Audio frequency AF
v · d · e

Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average, or equilibrium) atmospheric 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 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 commonly used "zero" reference sound pressure in air is 20 µPa RMS, 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 p0 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 ptotal is given by:

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

where:

p0 = local ambient atmospheric (air) pressure,
posc = 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 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.

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 particle 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

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

normally in units of N/m² = Pa.

where:

Symbol SI Unit Meaning
p pascals sound pressure
f hertz frequency
ρ kg/m³ density of air
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).

The distance law for the sound pressure p in 3D is inverse-proportional to the distance r of a punctual sound source.

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

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. 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 Lp 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 pref is the reference sound pressure and prms is the rms sound pressure being measured.[2][note 1]

Sometimes variants are used such as dB (SPL), dBSPL, or dBSPL. These variants are not recognized as units in the SI.[3] 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 pref = 20 µPa (rms), 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 SPL of 94 dB. In other media, such as underwater, a reference level of 1 µPa is more often used.[4] These references are defined in ANSI S1.1-1994.[5]

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.

The lower limit of audibility is therefore defined as SPL of 0 dB, but the upper limit is not as clearly defined. While 1 atm (SPL of 191 dB) 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 on Earth in the form of shock waves.

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 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.

### 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 RMS dB re 20 μPa
Shockwave (distorted sound waves > 1 atm; waveform valleys are clipped at zero pressure) >101,325 Pa >194 dB
Theoretical limit for undistorted sound at 1 atmosphere environmental pressure 101,325 Pa ~194.094 dB
Stun grenades 6,000–20,000 Pa 170–180 dB
Rocket launch equipment acoustic tests ~4000 Pa ~165 dB
Simple open-ended thermoacoustic device[6] 12,619 Pa 176 dB
.30-06 rifle being fired 1 m to shooter's side 7,265 Pa 171 dB (peak)
M1 Garand rifle being fired at 1 m 5,023 Pa 168 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)[7]
Hearing damage (possible) 20 Pa approx. 120 dB
Jet engine at 100 m 6.32 – 200 Pa 110 – 140 dB
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[8]
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[9]
TV (set at home level) at 1 m 2×10−2 Pa approx. 60 dB
Handheld electric mixer 65 dB
Washing machine, dish washer 50-53 dB
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[8]

## Notes

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

## References

1. ^ "What is acoustic impedance and why is it important?". Retrieved 2011-08-11.
2. ^ Bies, David A., and Hansen, Colin. (2003). Engineering Noise Control.
3. ^ Taylor 1995, Guide for the Use of the International System of Units (SI), NIST Special Publication SP811
4. ^ C. L. Morfey, Dictionary of Acoustics (Academic Press, San Diego, 2001).
5. ^ Glossary of Noise TermsSound pressure level definition
6. ^ 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]
7. ^ 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.
8. ^ a b William Hamby. "Ultimate Sound Pressure Level Decibel Table". Archived from the original on 2010-07-27.
9. ^ EPA Identifies Noise Levels Affecting Health and Welfare, 1974-04-02, retrieved 2010-11-01
• 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.

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