Sound pressure

# Sound pressure

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Sound pressure or acoustic pressure is the local pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

deviation from the ambient (average, or equilibrium) atmospheric pressure
Atmospheric pressure
Atmospheric pressure is the force per unit area exerted into a surface by the weight of air above that surface in the atmosphere of Earth . In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point...

caused by a sound wave. Sound pressure can be measured using a microphone
Microphone
A microphone is an acoustic-to-electric transducer or sensor that converts sound into an electrical signal. In 1877, Emile Berliner invented the first microphone used as a telephone voice transmitter...

in air and a hydrophone
Hydrophone
A hydrophone is a microphone designed to be used underwater for recording or listening to underwater sound. Most hydrophones are based on a piezoelectric transducer that generates electricity when subjected to a pressure change...

in water. The SI unit for sound pressure p is the pascal
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

(symbol: Pa).
Sound pressure level (SPL) or sound level is a logarithmic measure
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...

of the effective sound pressure of a sound relative to a reference value. It is measured in decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

s (dB) above a standard reference level. The commonly used "zero" reference sound pressure in air is 20 µPa RMS
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...

, which is usually considered the threshold of human hearing (at 1 kHz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

).

## Instantaneous sound pressure

The instantaneous sound pressure is the deviation from the local ambient pressure caused by a sound wave at a given location and given instant in time.

The effective sound pressure is the root mean square
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...

of the instantaneous sound pressure over a given interval of time (or space).

Total pressure is given by:

where: = local ambient atmospheric (air) pressure, = sound pressure deviation.

### Intensity

In a sound wave, the complementary variable to sound pressure is the acoustic particle velocity
Particle velocity
Particle velocity is the velocity v of a particle in a medium as it transmits a wave. In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string....

.
Together they determine the acoustic intensity of the wave. The local instantaneous sound intensity
Sound intensity
Sound intensity or acoustic intensity is defined as the sound power Pac per unit area A. The usual context is the noise measurement of sound intensity in the air at a listener's location.-Acoustic intensity:...

is the product of the sound pressure and the acoustic particle velocity.

### Acoustic impedance

For small amplitudes, sound pressure and particle velocity are linearly related and their ratio is the acoustic impedance
Acoustic impedance
The acoustic impedance at a particular frequency indicates how much sound pressure is generated by a given air vibration at that frequency. The acoustic impedance Z is frequency dependent and is very useful, for example, for describing the behaviour of musical wind instruments...

. The acoustic impedance depends on both the characteristics of the wave and the transmission medium
Transmission medium
A transmission medium is a material substance that can propagate energy waves...

.

The acoustic impedance is given by

where
Z is acoustic impedance
Acoustic impedance
The acoustic impedance at a particular frequency indicates how much sound pressure is generated by a given air vibration at that frequency. The acoustic impedance Z is frequency dependent and is very useful, for example, for describing the behaviour of musical wind instruments...

or sound impedance
p is sound pressure
U is particle velocity
Particle velocity
Particle velocity is the velocity v of a particle in a medium as it transmits a wave. In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string....

### Particle displacement

Sound pressure p is connected to particle displacement
Particle displacement
Particle displacement or particle amplitude is a measurement of distance of the movement of a particle from its equilibrium position in a medium as it transmits a wave....

(or particle amplitude) ξ by.

Sound pressure p is,

normally in units of N/m² = Pa.

where:
Symbol SI Unit  Meaning
p pascal
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

s
sound pressure
f hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

ρ kg
Kilogram
The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...

/m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

³
density of air
Density of air
The density of air, ρ , is the mass per unit volume of Earth's atmosphere, and is a useful value in aeronautics and other sciences. Air density decreases with increasing altitude, as does air pressure. It also changes with variances in temperature or humidity...

c m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

/s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

v m/s  particle velocity
Particle velocity
Particle velocity is the velocity v of a particle in a medium as it transmits a wave. In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string....

= 2 · · f radians/s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

ξ meters particle displacement
Particle displacement
Particle displacement or particle amplitude is a measurement of distance of the movement of a particle from its equilibrium position in a medium as it transmits a wave....

Z = c • ρ s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

/m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

³
acoustic impedance
Acoustic impedance
The acoustic impedance at a particular frequency indicates how much sound pressure is generated by a given air vibration at that frequency. The acoustic impedance Z is frequency dependent and is very useful, for example, for describing the behaviour of musical wind instruments...

a m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

/s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

²
particle acceleration
Particle acceleration
In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the particle acceleration or sound acceleration with the symbol a in metre/second². In acoustics or physics, acceleration is defined as the rate of change of velocity. It is thus a vector...

I W
Watt
The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...

/m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

²
sound intensity
Sound intensity
Sound intensity or acoustic intensity is defined as the sound power Pac per unit area A. The usual context is the noise measurement of sound intensity in the air at a listener's location.-Acoustic intensity:...

E W
Watt
The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...

·s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....

/m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

³
sound energy density
Sound energy density
The sound energy density or sound density is an adequate measure to describe the sound field at a given point as a sound energy value...

Pac watt
Watt
The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...

s
sound power or acoustic power
A m
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

²
Area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

### 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
Inverse-square law
In physics, an inverse-square law is any physical law stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity....

, 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.

If sound pressure , is measured at a distance , one can calculate the sound pressure at another position ,

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 is a logarithmic measure of the effective sound pressure of a sound relative to a reference value. It is measured in decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

s (dB) above a standard reference level.
where is the reference sound pressure and is the rms sound pressure being measured.Sometimes reference sound pressure is denoted p0, not to be confused with the (much higher) ambient pressure.

Sometimes variants are used such as dB (SPL), dBSPL, or dBSPL. These variants are not recognized as units in the SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

. 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 = 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
Underwater acoustics
Underwater acoustics is the study of the propagation of sound in water and the interaction of the mechanical waves that constitute sound with the water and its boundaries. The water may be in the ocean, a lake or a tank. Typical frequencies associated with underwater acoustics are between 10 Hz and...

, a reference level of 1 µPa is more often used. These references are defined in ANSI
American National Standards Institute
The American National Standards Institute is a private non-profit organization that oversees the development of voluntary consensus standards for products, services, processes, systems, and personnel in the United States. The organization also coordinates U.S. standards with international...

S1.1-1994.

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
Metre
The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

(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
Anechoic chamber
An anechoic chamber is a room designed to stop reflections of either sound or electromagnetic waves.They are also insulated from exterior sources of noise...

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
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

, but the upper limit is not as clearly defined. While 1 atm
Atmosphere (unit)
The standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of pressure. For practical purposes it has been replaced by the bar which is 105 Pa...

(SPL of 194 dB) is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

. Larger sound waves can be present in other atmosphere
Atmosphere
An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...

s or other media such as under water, or within the Earth in the form of shock waves
Shock Waves
Shock Waves, , is a 1977 horror movie written and directed by Ken Wiederhorn...

.

Ears detect changes in sound pressure. Human hearing does not have a flat spectral sensitivity
Spectral sensitivity
Spectral sensitivity is the relative efficiency of detection, of light or other signal, as a function of the frequency or wavelength of the signal....

(frequency response
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...

) 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
Equal-loudness contour
An equal-loudness contour is a measure of sound pressure , over the frequency spectrum, for which a listener perceives a constant loudness when presented with pure steady tones. The unit of measurement for loudness levels is the phon, and is arrived at by reference to equal-loudness contours...

. 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.

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

From the formula of the sound pressure level we find

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

### Examples of sound pressure and sound pressure levels

Sound pressure in air:
Source of sound Sound pressure Sound pressure level
Sound in air pascal
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

RMS
dB
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

re 20 μPa
Shockwave
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

(distorted sound waves > 1 atm
Atmosphere (unit)
The standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of pressure. For practical purposes it has been replaced by the bar which is 105 Pa...

; waveform valleys are clipped at zero pressure)
>101,325 Pa >194 dB
Theoretical limit for undistorted sound at 1 atmosphere
Atmosphere (unit)
The standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of pressure. For practical purposes it has been replaced by the bar which is 105 Pa...

environmental pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

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
Thermoacoustics
Thermoacoustics is about the interaction between thermodynamic and acoustic phenomena. Thermoacoustics is a relatively new field of science and engineering. Few devices based on this principle have been made thus far...

device
12,619 Pa 176 dB
.30-06 rifle
.30-06 Springfield
The .30-06 Springfield cartridge or 7.62×63mm in metric notation, was introduced to the United States Army in 1906 and standardized, and was in use until the 1960s and early 1970s. It replaced the .30-03, 6 mm Lee Navy, and .30 US Army...

being fired 1 m to shooter's side
7,265 Pa 171 dB (peak)
M1 Garand
M1 Garand
The M1 Garand , was the first semi-automatic rifle to be generally issued to the infantry of any nation. Called "the greatest battle implement ever devised" by General George S...

rifle being fired at 1 m
5,023 Pa 168 dB
Jet engine
Jet engine
A jet engine is a reaction engine that discharges a fast moving jet to generate thrust by jet propulsion and in accordance with Newton's laws of motion. This broad definition of jet engines includes turbojets, turbofans, rockets, ramjets, pulse jets...

at 30 m
632 Pa 150 dB
Threshold of pain
Threshold of pain
The threshold of pain is the point at which pain begins to be felt. It is an entirely subjective phenomenon. The intensity at which a stimulus begins to evoke pain is the threshold intensity. So, if a hotplate on a person's skin begins to hurt at 42°C , then that is the pain threshold temperature...

63.2 Pa 130 dB
Vuvuzela
Vuvuzela
The vuvuzela , also known as lepatata Mambu , colloquially known in South Africa as "Moerstripper", is a plastic horn, about long, which produces a loud monotone note, typically around B3 . Some models are made in two parts to facilitate storage, and this design also allows pitch variation...

horn at 1 m
20 Pa 120 dB(A)
Hearing damage (possible) 20 Pa approx. 120 dB
Jet engine
Jet engine
A jet engine is a reaction engine that discharges a fast moving jet to generate thrust by jet propulsion and in accordance with Newton's laws of motion. This broad definition of jet engines includes turbojets, turbofans, rockets, ramjets, pulse jets...

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
Passenger car
Automobile
An automobile, autocar, motor car or car is a wheeled motor vehicle used for transporting passengers, which also carries its own engine or motor...

at 10 m
2×10−2 – 2×10−1 Pa 60 – 80 dB
EPA
United States Environmental Protection Agency
The U.S. Environmental Protection Agency is an agency of the federal government of the United States charged with protecting human health and the environment, by writing and enforcing regulations based on laws passed by Congress...

-identified maximum to protect against hearing loss and other disruptive effects from noise, such as sleep disturbance, stress, learning detriment, etc.
70 dB
TV (set at home level) at 1 m 2×10−2 Pa approx. 60 dB
Handheld electric mixer
Mixer (cooking)
A mixer is a kitchen appliance intended for mixing, folding, beating, and whipping food ingredients. Mixers come in two major variations, hand mixers and stand mixers....

65 dB
Washing machine
Washing machine
A washing machine is a machine designed to wash laundry, such as clothing, towels and sheets...

, 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

• Acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

• Amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

• Decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

, especially the Acoustics section
• Phon
Phon
The phon was proposed in DIN 45631 and ISO 532 B as a unit of perceived loudness level LN for pure tones by S. S. Stevens.-Definition:The purpose of the phon scale is to compensate for the effect of frequency on the perceived loudness of tones...

(unit)
• Loudness
Loudness
Loudness is the quality of a sound that is primarily a psychological correlate of physical strength . More formally, it is defined as "that attribute of auditory sensation in terms of which sounds can be ordered on a scale extending from quiet to loud."Loudness, a subjective measure, is often...

• Sone
Sone
The sone was proposed as a unit of perceived loudness by Stanley Smith Stevens in 1936. In acoustics, loudness is the subjective perception of sound intensity...

(unit)
• Sound level meter
Sound level meter
Sound level meters measure sound pressure level and are commonly used in noise pollution studies for the quantification of almost any noise, but especially for industrial, environmental and aircraft noise. However, the reading given by a sound level meter does not correlate well to...

• Sound power level
• Stevens' power law
Stevens' power law
Stevens' power law is a proposed relationship between the magnitude of a physical stimulus and its perceived intensity or strength. It is often considered to supersede the Weber–Fechner law on the basis that it describes a wider range of sensations, although critics argue that the validity of the...

• Weber–Fechner law
Weber–Fechner law
The Weber–Fechner law is a confusing term, because it combines two different laws. Some authors use the term to mean Weber's law, and others Fechner's law. Fechner himself added confusion to the literature by calling his own law Weber's law...

, especially The case of sound