Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone. The SI unit of sound pressure is the pascal (Pa).

Mathematical definition

thumb|upright=1.1|Sound pressure diagram:

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

<math display="block">p_\text{total} = p_\text{stat} + p,</math>

where

  • p<sub>total</sub> is the total pressure,
  • p<sub>stat</sub> is the static pressure.

Sound measurements

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<sup>−2</sup> in SI units, is defined by

<math display="block">\mathbf I = p \mathbf v,</math>

where

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

Acoustic impedance

Acoustic impedance, denoted Z and measured in Pa·m<sup>−3</sup>·s in SI units, is defined by

<math display="block">Z(s) = \frac{\hat{p}(s)}{\hat{Q}(s)},</math>

where

  • <math>\hat{p}(s)</math> is the Laplace transform of sound pressure,
  • <math>\hat{Q}(s)</math> is the Laplace transform of sound volume flow rate.

Specific acoustic impedance, denoted z and measured in Pa·m<sup>−1</sup>·s in SI units, is defined by

<math display="block">p(r) \propto \frac{1}{r}.</math>

This relationship is an inverse-proportional law.

If the sound pressure p<sub>1</sub> is measured at a distance r<sub>1</sub> from the centre of the sphere, the sound pressure p<sub>2</sub> at another position r<sub>2</sub> can be calculated:

<math display="block">p_2 = \frac{r_1}{r_2}\,p_1.</math>

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

<math display="block">I(r) \propto \frac{1}{r^2}.</math>

Indeed,

<math display="block">I(r) = p(r) v(r) = p(r)\left[p * z^{-1}\right](r) \propto p^2(r),</math>

where

  • <math>v</math> is the particle velocity,
  • <math>*</math> is the convolution operator,
  • z<sup>−1</sup> is the convolution inverse of the specific acoustic impedance,

hence the inverse-proportional law:

<math display="block">p(r) \propto \frac{1}{r}.</math>

Sound pressure level

<!--"Sound pressure level" redirects here.-->

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<sub>p</sub> and measured in dB, is defined by:

<math display="block">L_p = \ln\left(\frac{p^2}{p_0^2}\right) ~ \text{Np} = 2 \log_{10}\left(\frac{p}{p_0}\right)~\text{B} = 20 \log_{10}\left(\frac{p}{p_0}\right)~\text{dB},</math>

where

  • p is the root mean square sound pressure,
  • p<sub>0</sub> is a reference sound pressure,
  • is the neper,
  • is the bel,
  • is the decibel.

The commonly used reference sound pressure in air is

which is often considered as the threshold of human hearing (roughly the sound of a mosquito flying 3&nbsp;m away). The proper notations for sound pressure level using this reference are or , but the suffix notations , , dBSPL, and dB<sub>SPL</sub> are very common, even if they are not accepted by the SI.

Most sound-level measurements will be made relative to this reference, meaning will equal an SPL of <math>20 \log_{10}\left(\frac{1}{2\times10^{-5\right)~\text{dB}\approx 94~\text{dB}</math>. In other media, such as underwater, a reference level of is used. These references are defined in ANSI 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 (i. e., if the thermodynamic properties of the air are disregarded; in reality, the sound waves become progressively non-linear starting over 150&nbsp;dB), larger sound waves can be present in other atmospheres or other media, such as underwater or through the Earth.

thumb|upright=1.1|[[Equal-loudness contour, 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&nbsp;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<sub>A</sub> or L<sub>A</sub>, B-weighted sound pressure level is written either as dB<sub>B</sub> or L<sub>B</sub>, and C-weighted sound pressure level is written either as dB<sub>C</sub> or L<sub>C</sub>. Unweighted sound pressure level is called "linear sound pressure level" and is often written as dB<sub>L</sub> or just L. Some sound measuring instruments use the letter "Z" as an indication of linear SPL.

|

| align="right" | 1.3

| align="right" | 176

|-

|1883 eruption of Krakatoa

|165&nbsp;km

| align="right" | 8

| align="right" | 172

|-

| .30-06 rifle being fired

| 1&nbsp;m to<br/>shooter's side

| align="right" | 7

| align="right" | 171

|-

|Firecracker

|0.5 m

|align="right" | 7

|align="right" | 171

|-

| Stun grenade

| Ambient

| align="right" |

| align="right" | 158–172

|-

| party balloon inflated to rupture

| At ear

| align="right" | 4.9×10<sup>3</sup>

| align="right" | 168

|-

| diameter balloon crushed to rupture

| 1 m

| align="right" | 8.9×10<sup>2</sup>

| align="right" | 153

|-

| party balloon inflated to rupture

| Ambient

| align="right" | 200

| align="right" | 140

|-

| Instantaneous peak workplace noise (C-weighted) which legally obligates use of hearing protection by workers in the EU

| 0.5 m

| align="right" | 63.2

| align="right" | 130

|-

| Vuvuzela horn

| 1 m

| align="right" | 20.0

| align="right" | 120

|-

| Threshold of pain

| At ear

| align="right" | 20–100

| align="right" | 120–134

|-

| Risk of instantaneous noise-induced hearing loss

| At ear

| align="right" | 20.0

| align="right" | 120

|-

| Maximum instantaneous peak (C-weighted) for amplified sound at children's venues/events complying with WHO's global safe listening standard

| Ambient

| align="right" | 20.0

| align="right" | 120

|-

| Jet engine

| 100–30 m

| align="right" | 6–200

| align="right" | 110–140

|-

| Two-stroke chainsaw

| 1 m

| align="right" | 6

| align="right" | 110

|-

| Jackhammer

| 1 m

| align="right" | 2.00

| align="right" | 100

|-

| Sound level limit (A-weighted, moving average over 15-minute interval) at safe listening venues/events per WHO global standard

| Ambient

| align="right" | 2.00

| align="right" | 100

|-

| NIOSH recommended exposure limit (REL) for workplace noise (15-minute average, A-weighted)

| Ambient

| align="right" | 1.0

| align="right" | 94

|-

| NIOSH REL for workplace noise (1-hour average, A-weighted)

| At ear

| align="right" | 0.36

| align="right" | 85

|-

| Hearing damage (over long-term exposure, need not be continuous)

| At ear

| align="right" | 0.36

| align="right" | 85

|-

| Vacuum cleaner, A-weighted (1981)

| At ear

| align="right" | 0.2

| align="right" | 80

|-

| Average level (A-weighted) at 40 hours per week (on a rolling basis) equivalent to the "sound allowance" for a "safe listening device" in "Mode&nbsp;2: WHO standard level for sensitive users (e.g. children)" per WHO/ITU-T Rec.&nbsp;H.870

| Ambient

| align="right" | 0.11

| align="right" | 75

|-

| EPA-identified maximum to protect against hearing loss and other disruptive effects from noise, such as sleep disturbance, stress, learning detriment, etc.

| Ambient

| align="right" | 0.06

| align="right" | 70

|-

| Passenger car at 30&nbsp;km/h (electric and combustion engines)

| 10 m

| align="right" | 0.04–0.06

| align="right" | 65–70

|-

| Normal conversation

| 1 m

| align="right" | 2–0.02

| align="right" | 40–60

|-

| Passenger car at 10&nbsp;km/h (combustion)

| At ear

| align="right" | 2.00

| align="right" | 0

|-

| Anechoic chamber, Orfield Labs, A-weighted

| Ambient

| align="right" | 6.8

| align="right" | −9.4

|-

| Anechoic chamber, University of Salford, A-weighted

| Ambient

| align="right" | 4.8

| align="right" | −12.4

|-

| Anechoic chamber, Microsoft, A-weighted

| Ambient

| align="right" | 1.90

| align="right" | −20.35

|}

See also

References

;General

  • Beranek, Leo L., Acoustics (1993), Acoustical Society of America, .
  • Daniel R. Raichel, The Science and Applications of Acoustics (2006), Springer New York, .
  • Sound Pressure and Sound Power, Two Commonly Confused Characteristics of Sound
  • Decibel (Loudness) Comparison Chart