Sound pressure

Sound measurements
Sound pressure p, SPL
Particle velocity v, SVL
Particle displacement ξ
Sound intensity I, SIL
Sound power P_ac
Sound power level SWL
Sound energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound c
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. 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).

1 Instantaneous sound pressure
2 Sound pressure level
- 2.1 Multiple sources
- 2.2 Examples of sound pressure and sound pressure levels
3 See also
4 Notes
5 References
6 External links

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 $p_{total}$ is given by:

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

where:

$p_{0}$ = local ambient atmospheric (air) pressure,

$p_{osc}$ = 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 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

$p = v A Z$

where

Z is acoustic impedance, sound impedance, or characteristic impedance, in Pa·s/m

v is particle velocity in m/s

I is acoustic intensity or sound intensity, in W/m²

A is surface area in m²

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

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

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
$\omega$ = 2 · $\pi$ · 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
P_ac	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.

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

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 assumption of 1/r² with the square is here wrong. That is only correct for sound intensity.

$I \sim {p^2} \sim \dfrac{1}{r^2} \,$

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.

Note: The often used term "intensity of sound pressure" is not correct. Use "magnitude", "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.

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.^[1]^{[note 1]}

Sometimes variants are used such as dB (SPL), dBSPL, or dB_SPL. These variants are not recognized as units in the SI.^[2]

The unit dB (SPL) is often 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), which is usually considered the threshold of human hearing (roughly the sound of a mosquito flying 3 m away).

The lower limit of audibility is therefore defined as 0 dB, but the upper limit is not as clearly defined. While 1 atm (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.

When dealing with hearing, the perceived loudness of a sound correlates roughly logarithmically to its sound pressure (see 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, a reference level of 1 µPa is more often used.^[3]

These references are defined in ANSI S1.1-1994.^[4]

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 up to 55 dB SPL, B-weighting applies to sound pressures between 55 and 85 dB SPL, and C-weighting is for measuring sound pressure above 85 dB SPL.

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 the 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
Theoretical limit based on Atmospheric pressure records, Min & Max western Pacific Ocean 1979 (Typhoon Tip) & Agata, U.S.S.R. 1968	87,000 - 108,400 Pa	192.8–194.7 dB
Krakatoa explosion - unmeasured^[5]
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 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
TV (set at home level) at 1 m	2×10⁻² Pa	approx. 60 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]

Sound pressure in water, at the source:

Source of sound	Sound pressure	Sound pressure level
Sound under water	pascal	dB re 1 μPa
Sperm Whale	631,000 Pa	236 dB^[9]
Pistol shrimp	79,500 Pa	218 dB^[10]
Fin Whale	100-2000 Pa	160-186 dB^[11]
Humpback Whale	16-500 Pa	144-174 dB^[11]
Bowhead Whale	2-2,818 Pa	128-189 dB^[11]
Blue Whale	56-2,500 Pa	155-188 dB^[11]
Southern Right Whale	398-2240 Pa	172-187 dB^[11]
Gray Whale	12-1,780 Pa	142-185 dB^[11]
Auditory threshold of a diver at 1 kHz	2.2 × 10⁻³ Pa	67 dB^[12]

Notes

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

References

↑ Bies, David A., and Hansen, Colin. (2003). Engineering Noise Control.
↑ Taylor 1995, Guide for the Use of the International System of Units (SI), NIST Special Publication SP811
↑ C. L. Morfey, Dictionary of Acoustics (Academic Press, San Diego, 2001).
↑ Glossary of Noise Terms — Sound pressure level definition
↑ Simon Winchester, Krakatoa: The Day the World Exploded: August 27, 1883. Page 219. (Penguin/Viking, 2003, ISBN0-670-91430-4)
↑ 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]
↑ 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. http://www.scielo.org.za/pdf/samj/v100n2/v100n2a15.pdf.
↑ ^8.0 ^8.1 William Hamby. "Ultimate Sound Pressure Level Decibel Table". Archived from the original on 2010-07-27. http://www.webcitation.org/5rXlLRYsP.
↑ "Møhl et al. (2003). The monopulsed nature of sperm whale clicks. J.Acoust.Soc.Am. 114, 1143-1154". ASA. http://dx.doi.org/10.1121/1.1586258. Retrieved 2010-08-19.
↑ Derbyshire, David (2008-11-13). "Deadly pistol shrimp that stuns prey with sound as loud as Concorde found in UK waters". London: Daily Mail. http://www.dailymail.co.uk/sciencetech/article-1085398/Deadly-pistol-shrimp-stuns-prey-sound-loud-Concorde-UK-waters.html. Retrieved 2010-05-16.
↑ ^11.0 ^11.1 ^11.2 ^11.3 ^11.4 ^11.5 "SURTASS-LFA". US Navy. http://www.surtass-lfa-eis.com/Terms/. Retrieved 2010-05-16. /
↑ Parvin S.J., Searle S.L. and Gilbert M.J. (2001). "Exposure of divers to underwater sound in the frequency range from 800 to 2250 Hz". Undersea Hyperb Med. Abstract 28 (Supl). ISSN 1066-2936. OCLC 26915585. http://archive.rubicon-foundation.org/984. Retrieved 2008-05-05.

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.