Sound pressure

File:Sound pressure diagram.svg

A sound wave in a transmission medium causes a deviation (sound pressure, a dynamic pressure) in the local ambient pressure, a static pressure.

Sound pressure, denoted p, is defined by

$$p\_\backslash text\{total\}\; =\; p\_\backslash text\{stat\}\; +\; p,$$

where

• p_total is the total pressure,
• p_stat is the static pressure.

{{main|Sound intensity}}

In a sound wave, the complementary variable to sound pressure is the particle velocity. 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

$$\backslash mathbf\; I\; =\; p\; \backslash mathbf\; v,$$

where

• p is the sound pressure,
• v is the particle velocity.

{{main|Acoustic impedance}}

Acoustic impedance, denoted Z and measured in Pa·m⁻³·s in SI units, is defined by{{cite web |last=Wolfe |first=J. |title=What is acoustic impedance and why is it important? |url=http://www.phys.unsw.edu.au/jw/z.html |publisher=University of New South Wales, Dept. of Physics, Music Acoustics |access-date=1 January 2014}}

$$Z(s)\; =\; \backslash frac\{\backslash hat\{p\}(s)\}\{\backslash hat\{Q\}(s)\},$$

where

• $\backslash hat\{p\}(s)$ is the Laplace transform of sound pressure,{{citation needed|date=April 2015}}
• $\backslash hat\{Q\}(s)$ 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

$$z(s)\; =\; \backslash frac\{\backslash hat\{p\}(s)\}\{\backslash hat\{v\}(s)\},$$

where

• $\backslash hat\{p\}(s)$ is the Laplace transform of sound pressure,
• $\backslash hat\{v\}(s)$ is the Laplace transform of particle velocity.

{{main|Particle displacement}}

The particle displacement of a progressive sine wave is given by

$$\backslash delta(\backslash mathbf\{r\},\; t)\; =\; \backslash delta\_\backslash text\{m\}\; \backslash cos(\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\; -\; \backslash omega\; t\; +\; \backslash varphi\_\{\backslash delta,\; 0\}),$$

where

• $\backslash delta\_\backslash text\{m\}$ is the amplitude of the particle displacement,
• $\backslash varphi\_\{\backslash delta,\; 0\}$ is the phase shift of the particle displacement,
• k is the angular wavevector,
• ω is the angular frequency.

It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave x are given by

$$v(\backslash mathbf\{r\},\; t)\; =\; \backslash frac\{\backslash partial\; \backslash delta\}\{\backslash partial\; t\}\; (\backslash mathbf\{r\},\; t)\; =\; \backslash omega\; \backslash delta\_\backslash text\{m\}\; \backslash cos\backslash left(\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\; -\; \backslash omega\; t\; +\; \backslash varphi\_\{\backslash delta,\; 0\}\; +\; \backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; v\_\backslash text\{m\}\; \backslash cos(\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\; -\; \backslash omega\; t\; +\; \backslash varphi\_\{v,\; 0\}),$$

$$p(\backslash mathbf\{r\},\; t)\; =\; -\backslash rho\; c^2\; \backslash frac\{\backslash partial\; \backslash delta\}\{\backslash partial\; x\}\; (\backslash mathbf\{r\},\; t)\; =\; \backslash rho\; c^2\; k\_x\; \backslash delta\_\backslash text\{m\}\; \backslash cos\backslash left(\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\; -\; \backslash omega\; t\; +\; \backslash varphi\_\{\backslash delta,\; 0\}\; +\; \backslash frac\{\backslash pi\}\{2\}\backslash right)\; =\; p\_\backslash text\{m\}\; \backslash cos(\backslash mathbf\{k\}\; \backslash cdot\; \backslash mathbf\{r\}\; -\; \backslash omega\; t\; +\; \backslash varphi\_\{p,\; 0\}),$$

where

• v_m is the amplitude of the particle velocity,
• $\backslash varphi\_\{v,\; 0\}$ is the phase shift of the particle velocity,
• p_m is the amplitude of the acoustic pressure,
• $\backslash varphi\_\{p,\; 0\}$ is the phase shift of the acoustic pressure.

Taking the Laplace transforms of v and p with respect to time yields

$$\backslash hat\{v\}(\backslash mathbf\{r\},\; s)\; =\; v\_\backslash text\{m\}\; \backslash frac\{s\; \backslash cos\; \backslash varphi\_\{v,0\}\; -\; \backslash omega\; \backslash sin\; \backslash varphi\_\{v,0\}\}\{s^2\; +\; \backslash omega^2\},$$

$$\backslash hat\{p\}(\backslash mathbf\{r\},\; s)\; =\; p\_\backslash text\{m\}\; \backslash frac\{s\; \backslash cos\; \backslash varphi\_\{p,0\}\; -\; \backslash omega\; \backslash sin\; \backslash varphi\_\{p,0\}\}\{s^2\; +\; \backslash omega^2\}.$$

Since $\backslash varphi\_\{v,0\}\; =\; \backslash varphi\_\{p,0\}$, the amplitude of the specific acoustic impedance is given by

$$z\_\backslash text\{m\}(\backslash mathbf\{r\},\; s)\; =\; |z(\backslash mathbf\{r\},\; s)|\; =\; \backslash left|\backslash frac\{\backslash hat\{p\}(\backslash mathbf\{r\},\; s)\}\{\backslash hat\{v\}(\backslash mathbf\{r\},\; s)\}\backslash right|\; =\; \backslash frac\{p\_\backslash text\{m\}\}\{v\_\backslash text\{m\}\}\; =\; \backslash frac\{\backslash rho\; c^2\; k\_x\}\{\backslash omega\}.$$

Consequently, the amplitude of the particle displacement is related to that of the acoustic velocity and the sound pressure by

$$\backslash delta\_\backslash text\{m\}\; =\; \backslash frac\{v\_\backslash text\{m\}\}\{\backslash omega\},$$

$$\backslash delta\_\backslash text\{m\}\; =\; \backslash frac\{p\_\backslash text\{m\}\}\{\backslash omega\; z\_\backslash text\{m\}(\backslash mathbf\{r\},\; s)\}.$$

{{Further|Inverse-square 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/r², like the sound intensity):{{cite book |last=Longhurst |first=R. S. |title=Geometrical and Physical Optics |url=https://archive.org/details/geometricalphysi0000long |url-access=registration |year=1967 |publisher=Longmans |location=Norwich}}

$$p(r)\; \backslash propto\; \backslash frac\{1\}\{r\}.$$

This relationship is an inverse-proportional law.

If the sound pressure p₁ is measured at a distance r₁ from the centre of the sphere, the sound pressure p₂ at another position r₂ can be calculated:

$$p\_2\; =\; \backslash frac\{r\_1\}\{r\_2\}\backslash ,p\_1.$$

The inverse-proportional law for sound pressure comes from the inverse-square law for sound intensity:

$$I(r)\; \backslash propto\; \backslash frac\{1\}\{r^2\}.$$

Indeed,

$$I(r)\; =\; p(r)\; v(r)\; =\; p(r)\backslash left[p\; *\; z^\{-1\}\backslash right](r)\; \backslash propto\; p^2(r),$$

where

• $v$ is the particle velocity,
• $*$ is the convolution operator,
• z⁻¹ is the convolution inverse of the specific acoustic impedance,

hence the inverse-proportional law:

$$p(r)\; \backslash propto\; \backslash frac\{1\}\{r\}.$$

{{Other uses|Sound level (disambiguation){{!}}Sound level}}

Sound pressure level (SPL) or acoustic pressure level (APL) is a logarithmic measure of the effective pressure of a sound relative to a reference value.

Sound pressure level, denoted L_p and measured in dB,[http://webstore.iec.ch/webstore/webstore.nsf/artnum/028981 "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:{{cite book |vauthors=Attenborough K, Postema M |title=A Pocket-Sized Introduction to Acoustics |date=2008 |publisher=The University of Hull |location=Kingston upon Hull |url=https://hal.archives-ouvertes.fr/hal-03188302/document |isbn=978-90-812588-2-1 |doi=10.5281/zenodo.7504060 }}

$$L\_p\; =\; \backslash ln\backslash left(\backslash frac\{p\}\{p\_0\}\backslash right)\; ~\; \backslash text\{Np\}\; =\; 2\; \backslash log\_\{10\}\backslash left(\backslash frac\{p\}\{p\_0\}\backslash right)~\backslash text\{B\}\; =\; 20\; \backslash log\_\{10\}\backslash left(\backslash frac\{p\}\{p\_0\}\backslash right)~\backslash text\{dB\},$$

where

• p is the root mean square sound pressure,{{cite book |last1=Bies |first1=David A. |last2=Hansen |first2=Colin |date=2003 |title=Engineering Noise Control }}
• p₀ is a reference sound pressure,
• {{nowrap|1=1 Np}} is the neper,
• {{nowrap|1=1 B = ({{sfrac|2}} ln 10) Np}} is the bel,
• {{nowrap|1=1 dB = ({{sfrac|20}} ln 10) Np}} is the decibel.

{{Anchor|Reference}}The commonly used reference sound pressure in air isRoss Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.

{{Block indent | em = 1.5 | text = p₀ = 20 μPa,}}

which is often considered as the threshold of human hearing (roughly the sound of a mosquito flying 3 m away). The proper notations for sound pressure level using this reference are {{nobreak|L_{p/(20 μPa)}}} or {{nobreak|L_p (re 20 μPa)}}, but the suffix notations {{nobreak|dB SPL}}, {{nobreak|dB(SPL)}}, dBSPL, and dB_SPL 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), SP811 [http://physics.nist.gov/cuu/pdf/sp811.pdf PDF].

Most sound-level measurements will be made relative to this reference, meaning {{nobreak|1 Pa}} will equal an SPL of $20\; \backslash log\_\{10\}\backslash left(\backslash frac\{1\}\{2\backslash times10^\{-5\}\}\backslash right)~\backslash text\{dB\}\backslash approx\; 94~\backslash text\{dB\}$. In other media, such as underwater, a reference level of {{nobreak|1 μPa}} is used.{{cite book |last=Morfey |first=Christopher L. |title=Dictionary of Acoustics |year=2001 |publisher=Academic Press |location=San Diego |isbn=978-0125069403}} These references are defined in ANSI S1.1-2013.{{cite web |url=http://www.memtechacoustical.com/noise-terms-glossary |title=Noise Terms Glossary |access-date=2012-10-14 }}

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.

The lower limit of audibility is defined as SPL of {{nobreak|0 dB}}, but the upper limit is not as clearly defined. While {{nobreak|1 atm}} ({{nobreak|194 dB peak}} or {{nobreak|191 dB SPL}}){{Cite book |last=Self |first=Douglas |url=https://books.google.com/books?id=L2PdDwAAQBAJ&dq=194+dB+SPL&pg=PP36 |title=Small Signal Audio Design |date=2020-04-17 |publisher=CRC Press |isbn=978-1-000-05044-8 |quote=this limit is reached when the rarefaction creates a vacuum, because you can't have a lower pressure than that. This corresponds to about +194 dB SPL. }}{{Cite book |last1=Guignard |first1=J. C. |last2=King |first2=P.F. |author3=North Atlantic Treaty Organization Advisory Group for Aerospace Research and Development Aerospace Medical Panel |date=1972 |url=https://books.google.com/books?id=LvRJAQAAIAAJ&q=191+dB+SPL |title=Aeromedical Aspects of Vibration and Noise |publisher=North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development |quote=In air at an assumed atmospheric pressure of 1 bar (100,000 N/m²) this occurs theoretically at approximately 191 dB SPL (working with rms values }} is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere (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 atmospheres or other media, such as underwater or through the Earth.{{cite book |last=Winer |first=Ethan |date=2013 |title=The Audio Expert |location=New York and London |publisher=Focal Press |chapter=1 |isbn=978-0-240-82100-9 }}

File:Lindos1.svg, showing sound-pressure-vs-frequency at different perceived loudness levels]]

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 between 3,000 and 4,000 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.

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 use the letter "Z" as an indication of linear SPL.

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 (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 sound to be comparable to measurements made in a free field environment.

According to the inverse proportional law, when sound level L_p1 is measured at a distance r₁, the sound level L_p2 at the distance r₂ is

$$L\_\{p\_2\}\; =\; L\_\{p\_1\}\; +\; 20\; \backslash log\_\{10\}\backslash left(\; \backslash frac\{r\_1\}\{r\_2\}\; \backslash right)~\backslash text\{dB\}.$$

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

$$L\_\backslash Sigma\; =\; 10\; \backslash log\_\{10\}\backslash left(\backslash frac\{p\_1^2\; +\; p\_2^2\; +\; \backslash dots\; +\; p\_n^2\}\{p\_0^2\}\backslash right)~\backslash text\{dB\}\; =$$

10 \log_{10}\left[\left(\frac{p_1}{p_0}\right)^2 + \left(\frac{p_2}{p_0}\right)^2 + \dots + \left(\frac{p_n}{p_0}\right)^2\right]~\text{dB}.

Inserting the formulas

$$\backslash left(\backslash frac\{p\_i\}\{p\_0\}\backslash right)^2\; =\; 10^\{\backslash frac\{L\_i\}\{10~\backslash text\{dB\}\}\},\backslash quad\; i\; =\; 1,\; 2,\; \backslash ldots,\; n$$

in the formula for the sum of the sound pressure levels yields

$$L\_\backslash Sigma\; =\; 10\; \backslash log\_\{10\}\; \backslash left(10^\{\backslash frac\{L\_1\}\{10~\backslash text\{dB\}\}\}\; +\; 10^\{\backslash frac\{L\_2\}\{10~\backslash text\{dB\}\}\}\; +\; \backslash dots\; +\; 10^\{\backslash frac\{L\_n\}\{10~\backslash text\{dB\}\}\}\; \backslash right)~\backslash text\{dB\}.$$

{{Notelist}}

• {{Annotated link|Acoustics}}
• {{Annotated link|Phon}}
• {{Annotated link|Loudness}}
• {{Annotated link|Sone}}
• {{Annotated link|Sound level meter}}
• {{Annotated link|Stevens's power law}}
• {{Annotated link|Weber–Fechner law}}

{{Reflist}}

;General

• 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}}.

• {{Commons category-inline}}
• [http://www.usmotors.com/products/ProFacts/sound_power_and_sound_pressure.htm Sound Pressure and Sound Power, Two Commonly Confused Characteristics of Sound]
• [http://www.gcaudio.com/resources/howtos/loudness.html Decibel (Loudness) Comparison Chart]

{{Orders of magnitude}}

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Category:Acoustic equations

Category:Acoustics

Category:Physical quantities

Category:Sound

Category:Sound measurements