# Sound intensity

Sound quantities
• Sound deflection ${\ displaystyle \ xi}$ • Sound pressure ${\ displaystyle p}$ • Sound pressure level ${\ displaystyle L_ {p}}$ • Sound energy density ${\ displaystyle E}$ • Sound energy ${\ displaystyle W}$ • Sound flow ${\ displaystyle q}$ • Speed ​​of sound ${\ displaystyle c _ {\ text {S}}}$ • Acoustic impedance ${\ displaystyle Z}$ • Sound intensity ${\ displaystyle I}$ • Sound power ${\ displaystyle P _ {\ text {ak}}}$ • Speed ​​of sound ${\ displaystyle v}$ • Fast sound amplitude ${\ displaystyle v}$ The sound intensity (symbol I ), which is one of the sound energy quantities , describes the sound power that passes through a sound-penetrated area per unit area. The associated logarithmic quantity is the sound intensity level . Two-microphone technology (see below) is mostly used to measure the sound intensity . Sometimes the sound intensity is also referred to as the sound energy flux density.

## definition

The “energy flow” in sound fields can be described punctiformly with the sound intensity. The sound intensity indicates how big the "energy layer" is in a point in space and in which direction the energy spreads there. It is calculated from the product of the sound velocity v and the sound pressure p . The sound intensity, like the sound velocity, is a directed quantity:

${\ displaystyle {\ vec {I}} = p \; {\ vec {v}} \,}$ . (1)

By integrating the sound intensity over the area under consideration, one obtains the sound power that passes through this area, with only the portions perpendicular to the surface having an influence on the determination of the sound power for each piece of surface. Mathematically, this relationship corresponds to the scalar product of the sound intensity vector with an area vector, where the area vector is oriented perpendicular to the respective area:

${\ displaystyle P_ {ak} = \ int _ {A} ^ {} {\ vec {I}} \ cdot d {\ vec {A}} \,}$ The unit of intensity is W / m². The sound intensity is a sound energy quantity . In contrast to this, the sound pressure is a sound field quantity .

## Sound intensity level

It is also common to specify the amount of sound intensity as the sound intensity level L I in decibels  (dB):

${\ displaystyle L_ {I} = 10 \ lg \ left ({\ frac {| {\ vec {I}} |} {I_ {0}}} \ right) \, \ mathrm {dB}}$ with the standardized reference value I 0 = 10 −12 W / m².

## Plane wave and spherical wave

In the sound field of a plane wave, the sound intensity results from the product of the effective values ​​of sound pressure p and sound velocity v .

${\ displaystyle I = {\ frac {1} {T}} \ int \ limits _ {0} ^ {T} p (t) \ cdot v (t) \, \ mathrm {d} t \,}$ For a spherical sound source, the following applies to the intensity as a function of the distance r :

${\ displaystyle I (r) = {\ dfrac {P_ {ak}} {A}} = {\ dfrac {P_ {ak}} {4 \ cdot \ pi \ cdot r ^ {2}}} \,}$ Here P ak is the sound power and A the spherical surface of an imaginary sphere with the radius r . The following applies:

${\ displaystyle I \ sim {\ dfrac {1} {r ^ {2}}} \,}$ ${\ displaystyle {\ dfrac {I_ {2}} {I_ {1}}} = {\ dfrac {{r_ {1}} ^ {2}} {{r_ {2}} ^ {2}}} \, }$ ${\ displaystyle I_ {2} = I_ {1} \ cdot {\ dfrac {{r_ {1}} ^ {2}} {{r_ {2}} ^ {2}}} \,}$ ${\ displaystyle I_ {1} \,}$ = Sound intensity at a smaller distance = sound intensity at a greater distance${\ displaystyle r_ {1} \,}$ ${\ displaystyle I_ {2} \,}$ ${\ displaystyle r_ {2} \,}$ Thus, the sound intensity as a sound energy quantity in the free field decreases with 1 / r 2 the distance from a point sound source, whereas the sound pressure as a sound field quantity only decreases with 1 / r the distance from a point sound source ( distance law ).

The sound intensity is used ${\ displaystyle I \,}$ • to describe a sound field at any point
• as an intermediate step to determine the sound power passing through a surface
• as an intermediate step in measuring the sound power of a sound source.

The sound intensity I in W / m² for a plane progressing wave is:

${\ displaystyle I = p \ cdot v = {\ dfrac {p ^ {2}} {Z}} = Z \ cdot v ^ {2} = \ xi ^ {2} \ cdot \ omega ^ {2} \ cdot Z = {\ dfrac {a ^ {2} \ cdot Z} {\ omega ^ {2}}} = E \ cdot c = {\ dfrac {P_ {ak}} {A}} \,}$ . (9)

The symbols stand for the following quantities:

symbol units meaning
I. W / m 2 Sound intensity
p Pascal = N / m² Sound pressure
v m / s Speed ​​of sound
Z = c ρ N · s / m 3 Characteristic acoustic impedance, acoustic field impedance
ρ kg / m 3 Air density , density of the air (of the medium)
a m / s 2 Sound acceleration
ξ m , meter Sound deflection
${\ displaystyle \ omega}$ = 2 · f${\ displaystyle \ pi}$ rad / s Angular frequency
f hertz 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 of the sound intensity with the two-microphone technique

An intensity measuring probe must supply signals from which two field variables, the sound pressure p (t) and the fast component v n (t), can be determined at the measuring location . The measurement of the sound pressure can be done easily with a measuring microphone. Determining the speed of sound is more difficult. For this purpose z. B. miniature ultrasonic transmitters and receivers can be used. These are arranged close together in the measuring direction. The frequency change of the ultrasonic signal that occurs in the receiver signal due to the Doppler effect can then be used as a measure of the speed of sound.

Most common, however, is to make use of the relationship between pressure and speed described in the Euler equation . The sound velocity component v n in a certain spatial direction n can then be calculated as follows:

${\ displaystyle v_ {n} (t) = - {\ frac {1} {\ rho}} _ {0} \ int {\ frac {dp (t)} {dn}} dt \,}$ . (10)

Since it is not possible to determine the differential quotient of the sound pressure with simple means , the sound pressure is measured and the difference quotient is formed at two closely adjacent locations whose connecting line lies in spatial direction n . This method is known as the two-microphone technique. The probe used here consists of two microphones specially selected for their phase behavior, which are arranged next to one another at a small distance Δ r and records the sound pressures p A and p B received at the two locations .

The sound velocity component v n in direction n can now be calculated analogously to Eq. (10), determine as follows

${\ displaystyle v_ {n} (t) = - {\ frac {1} {\ rho}} _ {0} \ int {\ frac {p_ {B} (t) -p_ {A} (t)} { \ Delta r}} dt \,}$ . (11)

The sound pressure p ( t ) is calculated as the mean value of p A ( t ) and p B ( t ):

${\ displaystyle p (t) = {\ frac {p_ {A} (t) + p_ {B} (t)} {2}} \,}$ . (12)

If one sets Eq. (11) and (12) in Eq. (1), the result for the sound intensity is:

${\ displaystyle I_ {n} (t) = - {\ frac {p_ {A} (t) + p_ {B} (t)} {2 \ rho _ {0} \ Delta r}} \ int [p_ { B} (t) -p_ {A} (t)] dt \,}$ . (10)

The distance Δ r between the microphones determines the frequency range of the probe. Therefore, spacers of different lengths are used for different frequency ranges.

In the case of low frequencies, i.e. large wavelengths, larger microphone distances are selected in order not to let the pressure differences between the two microphones become too small and thus to keep the measurement errors small due to different signal propagation times and measurement accuracies in the two measurement channels. At high frequencies, however, smaller microphone distances are set. Here the frequency range is limited by the fact that the difference quotient for determining the speed of sound compared to the differential quotient no longer provides sufficiently accurate results from a certain frequency.