# Sound pressure

Sound quantities

The sound pressure or alternating sound pressure , symbol p ("pressure"), is the most important sound field quantity in sound engineering and acoustics . The SI unit of sound pressure, like pressure, is the pascal with the unit symbol Pa. The sound pressure level in dB can be calculated from the rms value of the sound pressure . ${\ displaystyle {\ tilde {p}}}$

## definition

Sound pressure refers to the pressure fluctuations in a compressible sound transmission medium (usually air ) that occur when sound propagates . These pressure fluctuations are converted by the eardrum as a sensor into movements for hearing sensation.

The sound pressure p is the alternating pressure (an alternating quantity ) which is superimposed on the static pressure p 0 ( air pressure ) of the surrounding medium. Here is the alternating sound pressure

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

with the area A -acting force F per surface area of A .

The following applies to the entire pressure p tot :

${\ displaystyle p _ {\ mathrm {ges}} = p_ {0} + p \,}$

The sound pressure p (alternating sound pressure ) is usually many orders of magnitude smaller than the static air pressure. Since a pressure cannot be linked to any direction specification, it is a scalar quantity. From a mathematical point of view, the sound pressure as a function of the coordinates in three-dimensional space is therefore a scalar field .

Furthermore, in the case of sinusoidal signals, it is specified as an effective value

${\ displaystyle p _ {\ mathrm {eff}} = {\ tilde {p}} = {\ frac {\ hat {p}} {\ sqrt {2}}} \ quad \,}$

common. The sound pressure amplitude, on the other hand, is the peak value of the sound pressure.

If the sound is a tone , i.e. a harmonic oscillation (often also referred to as a " sine oscillation") with only one frequency , then the following applies to the time dependence of the sound pressure: ${\ displaystyle f}$

${\ displaystyle p (t) = {\ hat {p}} \ sin (2 \ pi ft) = {\ hat {p}} \ sin (\ omega t) \,}$

where the sound pressure amplitude and ω is the angular frequency . ${\ displaystyle {\ hat {p}}}$ ${\ displaystyle \ omega = 2 \ pi \, f \,}$

## Distance dependency

The effective value of the sound pressure in the free field is inversely proportional to the distance r from a (point) sound source (1 / r law, distance law ): ${\ displaystyle {\ tilde {p}}}$

${\ displaystyle {\ tilde {p}} \ sim {\ frac {1} {r}}}$
${\ displaystyle {\ frac {{\ tilde {p}} _ {2}} {{\ tilde {p}} _ {1}}} = {\ frac {r_ {1}} {r_ {2}}} \,}$
${\ displaystyle {\ tilde {p}} _ {2} = {\ tilde {p}} _ {1} {\ frac {r_ {1}} {r_ {2}}} \,}$

${\ displaystyle {\ tilde {p}} _ {1} \,}$= Sound pressure at a distance = sound pressure at a distance${\ displaystyle r_ {1} \,}$
${\ displaystyle {\ tilde {p}} _ {2} \,}$${\ displaystyle r_ {2} \,}$

(Note: The quadratic sound energy quantities , such as the sound intensity , decrease with 1 / r 2 over the distance in the case of point sources of sound .) As you can see here, in addition to the indication of the measured sound pressure, it is essential to assess the strength of a sound source the specification of the position of the measuring point as distance r from the sound source is necessary.

In reverberant surroundings, the 1 / r law only applies to a limited extent:

• In the direct field of the sound source, i.e. outdoors and where the direct sound D outweighs the room sound R, the 1 / r law applies .
• Outside the immediate direct field, where the reflections have an impact on the total sound pressure, the 1 / r law only applies to a limited extent.
• Outside the reverberation radius r H , that is the distance from the sound source at which the direct sound D is just as strong as the room sound R, the sound pressure remains essentially constant as the distance from the sound source increases, as it is mainly dependent on the reflections of the Walls is determined.

## Connection with other acoustic quantities

In a plane wave , the sound pressure is linked to the acoustic parameters sound impedance , sound power , sound velocity and sound intensity as follows: ${\ displaystyle p}$ ${\ displaystyle Z}$ ${\ displaystyle P _ {\ mathrm {ak}}}$ ${\ displaystyle v}$ ${\ displaystyle I}$

${\ displaystyle p = Z \; v = {\ frac {I} {v}} = {\ sqrt {I \; Z}} = {\ frac {P _ {\ mathrm {ak}}} {v \; A }} = {\ sqrt {\ frac {P _ {\ mathrm {ak}} \; Z} {A}}} = {\ xi \; Z \; \ omega} = {\ frac {a \; Z} { \ omega}} = c \; {\ sqrt {\ rho \; E}}}$.

Where:

symbol units meaning
${\ displaystyle p}$ Pa Sound pressure
${\ displaystyle f}$ Hz frequency
${\ displaystyle \ xi}$ m Sound deflection
${\ displaystyle c}$ m / s Speed ​​of sound
${\ displaystyle v}$ m / s Speed ​​of sound
${\ displaystyle \ omega}$ 1 / s Angular frequency
${\ displaystyle \ rho}$ kg / m 3 Air density (density of the medium)
${\ displaystyle Z = c \; \ rho}$ N · s / m 3 Characteristic acoustic impedance, acoustic field impedance
${\ displaystyle a}$ m / s 2 Sound acceleration
${\ displaystyle I}$ W / m 2 Sound intensity
${\ displaystyle E}$ W s / m 3 Sound energy density
${\ displaystyle P _ {\ mathrm {ak}}}$ W. Sound power
${\ displaystyle A}$ m 2 Transmitted surface

## Table: Sound pressure and sound pressure level of various sound sources

### Sound pressure in air

For comparison
static air pressure at sea level: approx. 100 kPa
Sound source and situation
(distance)
Sound pressure (effective value) (in Pascal) ${\ displaystyle {\ tilde {p}}}$

Sound pressure
level L p
dB re 20 µPa
M1 Garand rifle (1 m) 5000 168
Jet plane (30 m) 600 150
Pain threshold 100 134
Hearing damage from short-term exposure 20th from 120
Jet plane (100 m) 6 ... 200 110 ... 140
Jackhammer (1 m); discotheque 2 100
Long-term exposure
to hearing damage more than 8 hours a day
0.6 from 90
Main road (10 m) 0.2 ... 0.6 80 ... 90
Car (10 m) 0.02 ... 0.2 60 ... 80
TV at room volume (1 m) 0.02 approx. 60
normal conversation (1 m) 2 ... 6 · 10 −3 40 ... 50
very quiet room 2 ... 6 · 10 −4 20 ... 30
Rustling leaves, calm breathing 6 · 10 −5 10
Hearing threshold at 1 kHz 2 · 10 −5 0

### Sound pressure in water

For comparison
static pressure at sea level on the water surface: approx. 100 kPa
in 100 m water depth: approx. 1100 kPa
in 5 km water depth: approx. 51 100 kPa
Sound source and situation
(distance)
Sound pressure (in pascal)${\ displaystyle {\ tilde {p}}}$
Sound pressure
level L p
dB re 1 µPa
military sonar (1 m) 10 6 240
Diver's hearing threshold at 1 kHz 2.2 · 10 −3 67

## literature

• Hans Breuer: dtv-Atlas Physik , Volume 1. Mechanics, acoustics, thermodynamics, optics . dtv, Munich 1996, ISBN 3-423-03226-X
• Heinrich Kuttruff: Acoustics . Hirzel, Stuttgart 2004, ISBN 3-7776-1244-8
• Gerhard Müller, Michael Möser: Paperback of technical acoustics . 3. revised Edition. Springer, Berlin 2003, ISBN 3-540-41242-5
• Ivar Veit: Technical acoustics . Vogel-Verlag, Würzburg 2005, ISBN 3-8343-3013-2