Sound Pressure Level
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Sound pressure or acoustic pressure is the local
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
deviation from the ambient (average or equilibrium)
atmospheric pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, 7 ...
, caused by a
sound wave In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
. In air, sound pressure can be measured using a
microphone A microphone, colloquially called a mic or mike (), is a transducer that converts sound into an electrical signal. Microphones are used in many applications such as telephones, hearing aids, public address systems for concert halls and public ...
, and in water with a
hydrophone A hydrophone ( grc, ὕδωρ + φωνή, , water + sound) 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 an electric potenti ...
. The
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
of sound pressure is the
pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, Fren ...
(Pa).


Mathematical definition

A sound wave in a
transmission medium A transmission medium is a system or substance that can mediate the propagation of signals for the purposes of telecommunication. Signals are typically imposed on a wave of some kind suitable for the chosen medium. For example, data can modulate ...
causes a deviation (sound pressure, a ''dynamic'' pressure) in the local ambient pressure, a ''static'' pressure. Sound pressure, denoted ''p'', is defined by p_\text = p_\text + p, where * ''p''total is the total pressure, * ''p''stat is the static pressure.


Sound measurements


Sound intensity

In a sound wave, the complementary variable to sound pressure is the
particle velocity Particle velocity is the velocity of a particle (real or imagined) in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can ...
. 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 \mathbf I = p \mathbf v, where * ''p'' is the sound pressure, * v is the particle velocity.


Acoustic impedance

''Acoustic impedance'', denoted ''Z'' and measured in Pa·m−3·s in SI units, is defined by Z(s) = \frac, where * \hat(s) is the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
of sound pressure, * \hat(s) 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 z(s) = \frac, where * \hat(s) is the Laplace transform of sound pressure, * \hat(s) is the Laplace transform of particle velocity.


Particle displacement

The ''particle displacement'' of a ''progressive
sine wave A sine wave, sinusoidal wave, or just sinusoid is a curve, mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph of a function, graph. It is a type of continuous wave and also a Smoothness, smooth p ...
'' is given by \delta(\mathbf, t) = \delta_\text \cos(\mathbf \cdot \mathbf - \omega t + \varphi_), where * \delta_\text is the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
of the particle displacement, * \varphi_ is the
phase shift In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...
of the particle displacement, * k is the
angular wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
, * ''ω'' is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
. It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave ''x'' are given by v(\mathbf, t) = \frac (\mathbf, t) = \omega \delta_\text \cos\left(\mathbf \cdot \mathbf - \omega t + \varphi_ + \frac\right) = v_\text \cos(\mathbf \cdot \mathbf - \omega t + \varphi_), p(\mathbf, t) = -\rho c^2 \frac (\mathbf, t) = \rho c^2 k_x \delta_\text \cos\left(\mathbf \cdot \mathbf - \omega t + \varphi_ + \frac\right) = p_\text \cos(\mathbf \cdot \mathbf - \omega t + \varphi_), where * ''v''m is the amplitude of the particle velocity, * \varphi_ is the phase shift of the particle velocity, * ''p''m is the amplitude of the acoustic pressure, * \varphi_ is the phase shift of the acoustic pressure. Taking the Laplace transforms of ''v'' and ''p'' with respect to time yields \hat(\mathbf, s) = v_\text \frac, \hat(\mathbf, s) = p_\text \frac. Since \varphi_ = \varphi_, the amplitude of the specific acoustic impedance is given by z_\text(\mathbf, s) = , z(\mathbf, s), = \left, \frac\ = \frac = \frac. Consequently, the amplitude of the particle displacement is related to that of the acoustic velocity and the sound pressure by \delta_\text = \frac, \delta_\text = \frac.


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/''r''2, like the sound intensity): p(r) \propto \frac. This relationship is an ''inverse-proportional law''. If the sound pressure ''p''1 is measured at a distance ''r''1 from the centre of the sphere, the sound pressure ''p''2 at another position ''r''2 can be calculated: p_2 = \frac\,p_1. The inverse-proportional law for sound pressure comes from the inverse-square law for sound intensity: I(r) \propto \frac. Indeed, I(r) = p(r) v(r) = p(r)\left * z^\rightr) \propto p^2(r), where * * is the
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
operator, * ''z''−1 is the convolution inverse of the
specific acoustic impedance Acoustic impedance and specific acoustic impedance are measures of the opposition that a system presents to the acoustic flow resulting from an acoustic pressure applied to the system. The SI unit of acoustic impedance is the pascal-second per cub ...
, hence the inverse-proportional law: p(r) \propto \frac. The sound pressure may vary in direction from the centre of the sphere as well, so measurements at different angles may be necessary, depending on the situation. An obvious example of a sound source whose spherical sound wave varies in level in different directions is a bullhorn.


Sound pressure level

Sound pressure level (SPL) or acoustic pressure level is a
logarithmic measure In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used f ...
of the effective pressure of a sound relative to a reference value. Sound pressure level, denoted ''L''''p'' and measured in dB,"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.
is defined by: L_p = \ln\left(\frac\right) ~ \text = 2 \log_\left(\frac\right)~\text = 20 \log_\left(\frac\right)~\text, where * ''p'' is the
root mean square In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
sound pressure, * ''p''0 is a reference sound pressure, * is the
neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As ...
, * is the bel, * is the
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
. The commonly used reference sound pressure in air is which is often considered as the
threshold of human hearing The absolute threshold of hearing (ATH) is the minimum sound level of a pure tone that an average human ear with normal hearing can hear with no other sound present. The absolute threshold relates to the sound that can just be heard by the organ ...
(roughly the sound of a mosquito flying 3 m away). The proper notations for sound pressure level using this reference are or , but the suffix notations , , dBSPL, or dBSPL are very common, even if they are not accepted by the SI.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), SP81
PDF
Most sound-level measurements will be made relative to this reference, meaning will equal an SPL of . In other media, such as underwater, a reference level of is used. These references are defined in
ANSI The American National Standards Institute (ANSI ) 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 organi ...
S1.1-2013. 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 , but the upper limit is not as clearly defined. While ( or ) is the largest pressure variation an undistorted sound wave can have in
Earth's atmosphere The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing for ...
(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
atmosphere An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A s ...
s or other media, such as underwater or through the Earth. Ears detect changes in sound pressure. Human hearing does not have a flat
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. In visual neuroscience, spectral sensitivity is used to describe the different characteristics o ...
(
frequency response In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
) relative to frequency versus
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
. 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 An equal-loudness contour is a measure of sound pressure level, 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 ...
. 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 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, 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 The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
(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 An anechoic chamber (''an-echoic'' meaning "non-reflective") is a room designed to stop reflections of either sound or electromagnetic waves. They are also often isolated from energy entering from their surroundings. This combination means ...
allows sound to be comparable to measurements made in a free field environment. According to the inverse proportional law, when sound level ''L''''p''1 is measured at a distance ''r''1, the sound level ''L''''p''2 at the distance ''r''2 is L_ = L_ + 20 \log_\left( \frac \right)~\text.


Multiple sources

The formula for the sum of the sound pressure levels of ''n'' incoherent radiating sources is L_\Sigma = 10 \log_\left(\frac\right)~\text = 10 \log_\left left(\frac\right)^2 + \left(\frac\right)^2 + \dots + \left(\frac\right)^2\right\text. Inserting the formulas \left(\frac\right)^2 = 10^,\quad i = 1, 2, \ldots, n in the formula for the sum of the sound pressure levels yields L_\Sigma = 10 \log_ \left(10^ + 10^ + \dots + 10^ \right)~\text.


Examples of sound pressure


See also

*
Acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
*
Phon The phon is a logarithmic physical unit, unit of loudness level for tones and complex sounds. Loudness is measured in sone which is a linear unit. Human sensitivity to sound is variable across different frequencies; therefore, although two diff ...
(unit) *
Loudness In acoustics, loudness is the subjectivity, subjective perception of sound pressure. 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". The rel ...
*
Sone The sone () is a unit of loudness, the subjective perception of sound pressure. The study of perceived loudness is included in the topic of psychoacoustics and employs methods of psychophysics. Doubling the perceived loudness doubles the son ...
(unit) * Sound level meter *
Stevens's power law Stevens' power law is an empirical relationship in psychophysics between an increased intensity or strength in a physical stimulus and the perceived magnitude increase in the sensation created by the stimulus. It is often considered to supersede ...
*
Weber–Fechner law The Weber–Fechner laws are two related hypotheses in the field of psychophysics, known as Weber's law and Fechner's law. Both laws relate to human perception, more specifically the relation between the actual change in a physical stimulus and ...
, especially The case of sound


References

;General *Beranek, Leo L., ''Acoustics'' (1993), Acoustical Society of America, . *Daniel R. Raichel, ''The Science and Applications of Acoustics'' (2006), Springer New York, .


External links

*
Sound Pressure and Sound Power, Effect and Cause
* ttp://www.sengpielaudio.com/TableOfSoundPressureLevels.htm Table of Sound Levels, Corresponding Sound Pressure and Sound Intensitybr>Ohm's Law as Acoustic Equivalent, CalculationsRelationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave
* ttp://www.sengpielaudio.com/calculator-levelchange.htm How Many Decibels Is Twice as Loud? Sound Level Change and the Respective Factor of Sound Pressure or Sound Intensitybr>Decibel (Loudness) Comparison Chart
{{Authority control Acoustics Sound Sound measurements Physical quantities Acoustic equations