# Sound pressure

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

## Sound pressure

Sound pressure is the pressure deviation from the local ambient 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 is the pascal (symbol: Pa). 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. In a sound wave, the complementary variable to sound pressure is the acoustic particle velocity. 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 medium. The local instantaneous sound intensity is the product of the sound pressure and the acoustic partical velocity and is, therefore, a vector quantity.

The sound pressure deviation p is:

$p = \frac{F}{A}$

F = force
A = area

The entire pressure ptotal is:

$p_{total} = p_0 + p \,$

p0 = local ambient pressure
p = sound pressure deviation

## Sound pressure level

Sound pressure level (SPL) or sound level Lp is a logarithmic measure of the rms pressure (force/area) of a particular noise relative to a reference noise source. It is usually measured in decibels (dB (SPL), dBSPL, or dBSPL).

$L_\mathrm{p}=10\, \log_{10}\left(\frac{{p}^2}{{p_0}^2}\right) =20\, \log_{10}\left(\frac{p}{p_0}\right)\mbox{ dB}$
where p0 is the reference sound pressure and p is the root-mean-square sound pressure being measured.

The commonly used reference sound pressure in air is p0 = 20 µPa (root-mean-square).

It can be useful to express sound pressure in this way when dealing with hearing, as the perceived loudness of a sound correlates roughly logarithmically to its sound pressure. See also Weber-Fechner law.

### Measuring sound pressure levels

When making measurements in air (and other gases), SPL is almost always expressed in decibels compared to a reference sound pressure of 20 µPa (micropascals), which is usually considered the threshold of human hearing (roughly the sound of a mosquito flying 3 metres away). Thus, most measurements of audio equipment will be made relative to this level. However, in other media, such as underwater, a reference level of 1 µPa is more often used. [1] These references are defined in ANSI S1.1-1994. [2] In general, it is necessary to know the reference level when comparing measurements of SPL. The unit dB (SPL) is often abbreviated to just "dB", which gives some the erroneous notion that a dB is an absolute unit by itself.

Since the human ear does not have a flat spectral response, sound pressure levels are often frequency weighted so that the measured level will match perceived sound level. The International Electrotechnical Commission (IEC) has defined several weighting schemes. A-weighting attempts to match the response of the human ear to noise, while C-weighting is used to measure peak sound levels. [3] If the actual, as opposed to weighted, SPL is desired, many instruments allow a "flat" or unweighted measurement to be made. See also Weighting filter.

When measuring the sound created by an object, it is important to measure the distance from the object as well, since the SPL decreases in distance from a point source with 1/r (and not with 1/r2</sub>, like sound intensity). It often varies in direction from the source, as well, so many measurements 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 p in N/m2 or Pa is:

$p = Zv = \frac{J}{v} = \sqrt{JZ}$
Z: acoustic impedance, sound impedance, or characteristic impedance, in Pa·s/m
v: particle velocity in m/s
J: acoustic intensity or sound intensity, in W/m2

Sound pressure p is connected to particle displacement (or particle amplitude) ξ, in m, by:

$\xi = \frac{v}{2 \pi f} = \frac{v}{\omega} = \frac{p}{Z \omega} = \frac{p}{ 2 \pi f Z}$

Sound pressure p:

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

normally in units of N/m2 = Pa.

where:

Symbol Units 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
J W/m² sound intensity
E W·s/mm³ sound energy density
Pac watts sound power or acoustic power
A m² Area

The distance law for the sound pressure p is inverse-proportional to the distance r of a punctual sound source.

$p \propto \frac{1}{r}$ (proportional)
$\frac{p_1} {p_2} = \frac{r_2}{r_1}$
$p_1 = p_{2} \cdot r_{2} \cdot \frac{1}{r_1}$

The assumption of 1/r² with the square is here wrong. Thats only right for sound intensity.

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.

### Examples of sound pressure and sound pressure levels

situation sound pressure (pascal) sound pressure level dB (SPL)
threshold of pain 100 Pa 134 dB
hearing damage during short term effect 20 Pa approx. 120 dB
jet, 100 m distant 6 – 200 Pa 110 – 140 dB
jack hammer, 1 m distant / discotheque 2 Pa approx. 100 dB
hearing damage during long-term effect 6×10−1 Pa approx. 90 dB
major road, 10 m distant 2×10−1 – 6×10−1 Pa 80 – 90 dB
passenger car, 10 m distant 2×10−2 – 2×10−1 Pa 60 – 80 dB
TV set at home level, 1 m distant 2×10−2 Pa ca. 60 dB
normal talking, 1 m distant 2×10−3 – 2×10−2 Pa 40 – 60 dB
very calm room 2×10−4 – 6×10−4 Pa 20 – 30 dB
leaves noise, calm breathing 6×10−5 Pa 10 dB
auditory threshold at 2 kHz 2×10−5 Pa 0 dB

### SPL in audio equipment

Most audio manufacturers use SPL to describe the efficiency of their speakers. The most common means is measuring the sound pressure level from the speaker with the measuring device placed directly in front of and one meter away from the source. Then a particular sound (usually white noise or pink noise) is played through the source at a particular intensity so that the source is consuming one watt of power. The SPL is then measured and the product labeled, something like "SPL: 93 dB 1 W/1 m". This measurement can also be represented as a strict efficiency ratio of audio output (sound power) to electrical input (electrical power), but this is far less common. This method of rating speakers using SPL is often deceiving because most speakers produce very different SPLs at different frequencies of sound, often varying as much as ±10 dB throughout the speaker's usable frequency range (it generally varies less in higher quality speakers). The SPL quoted by the manufacturer is often an average over a particular range.