# Speed ​​of sound

Sound quantities

The speed of sound , symbol (also ), indicates the rate of change at which the air particles (or particles of the sound transmission medium) oscillate around their position of rest ; i.e. the instantaneous speed of a vibrating particle. The associated logarithmic quantity is the sound velocity level . ${\ displaystyle {\ vec {v}}}$${\ displaystyle {\ vec {u}}}$

The speed of sound depends mainly on the 'frequency' (pitch or sound energy of a noise or other complex waveforms) and the sound pressure level of a measured sound event, as well as the prevailing pressure and the inertia of the medium (air, water, solids, etc.) .).

The speed of sound must not be confused with the speed of sound  c , i.e. the phase speed of the sound waves or the group speed in the transmission medium , although these are also measured in m / s.

## Definition, related quantities

In acoustics , the speed of sound is a vectorial sound field quantity , which is often referred to as “speed” for short and is specified in m / s. It is calculated as the time derivative of the deflection ( sound deflection or sound deflection ) of the particle: ${\ displaystyle {\ vec {\ xi}}}$

${\ displaystyle {\ vec {v}} = {\ frac {\ mathrm {d} {\ vec {\ xi}}} {\ mathrm {d} t}} = {\ dot {\ vec {\ xi}} } \,}$.

If the deflection follows z. B. a sine oscillation ( tone ), then the speed of sound follows its time derivative, a cosine oscillation. In this case, the speed of sound is always zero ( zero crossing ) when the deflection from the rest position (in positive or negative direction) is greatest, and vice versa.

The time derivative of the speed of sound is in turn the sound acceleration :

${\ displaystyle {\ vec {a}} = {\ dot {\ vec {v}}} \,}$.

The fast potential is mainly used for mathematical calculations .

## Sound velocity level

Since the speed of sound is a vector quantity, the rms value is often formed from the absolute value or the components of the vector for numerical values . This means that the level can be specified in decibels (sound velocity level): ${\ displaystyle {\ tilde {v}}}$

${\ displaystyle L_ {v} = 10 \ lg \ left ({\ frac {\ tilde {v}} {v_ {0}}} \ right) ^ {2} \ mathrm {dB} = 20 \ lg \ left ( {\ frac {\ tilde {v}} {v_ {0}}} \ right) \ mathrm {dB}.}$

v 0 is the reference value . In Europe the reference value is

${\ displaystyle v_ {0} = 5 {,} 0 \ cdot 10 ^ {- 8} \, {\ frac {\ mathrm {m}} {\ mathrm {s}}}}$

common. This value roughly corresponds to the speed of sound in a plane wave in air with a sound pressure level of 0 dB (effective value of the sound pressure = ). Sometimes, however, a reference value of is also used. ${\ displaystyle 2 \ cdot 10 ^ {- 5} \, \ mathrm {Pa}}$${\ displaystyle v_ {0} = 1 \, {\ tfrac {\ mathrm {m}} {\ mathrm {s}}}}$

## Connection with other quantities

The speed of sound is linked to the sound pressure via the Euler equation . This is z. B. is used to determine the sound intensity , i.e. the product of the sound velocity and sound pressure. In the case of a plane progressing wave, the sound velocity and sound pressure phase are the same.

The speed of sound v in m / s for plane progressing sound waves is:

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

The symbols stand for the following quantities:

symbol units meaning
v m / s Speed ​​of sound
p Pascal = N / m² Sound pressure
Z = c · ρ N · s / m 3 Characteristic acoustic impedance, acoustic field impedance
I. W / m 2 Sound intensity
ξ m , meter Sound deflection
${\ displaystyle \ omega}$= 2 · · f${\ displaystyle \ pi}$ rad / s Angular frequency
a m / s 2 Sound acceleration
ρ kg / m 3 Air density , density of the air (of the medium)
f Hertz , 1 / s frequency
E. W · s / m 3 Sound energy density
P ak W , watt Sound power
A. m 2 Transmitted surface
c m / s Speed ​​of sound

## Measurement

The measurement of the speed of sound turns out to be difficult because a membrane such as is used in microphones , would have to follow the movement of the air particles inertially and therefore have to be practically massless.