# Sound pressure level

Sound quantities

The sound pressure level ( English Sound Pressure Level and often abbreviated as SPL ) is a logarithmic quantity used to describe the strength of a sound event . The sound pressure level is a calculated quantity based on the rms value of the sound pressure .

It belongs to the sound field quantities . Often the sound pressure level, although then physically ambiguous, simply sound called.

## definition

The sound pressure level L p (symbol L of Engl. Level , level ' ; with index p of Engl. Pressure , pressure' ) describes the logarithmic ratio of the squared root mean square value of the sound pressure (symbol with the unit Pa for Pascal ) of a sound event to the square of the reference value p 0 . The result is marked with the auxiliary unit of measurement decibel . ${\ displaystyle {\ tilde {p}}}$

${\ displaystyle L _ {\ mathrm {p}} = 10 \, \ log _ {10} \ left ({\ frac {{\ tilde {p}} ^ {2}} {{p_ {0}} ^ {2 }}} \ right) \, \ mathrm {dB} = 20 \, \ log _ {10} \ left ({\ frac {\ tilde {p}} {p_ {0}}} \ right) \, \ mathrm {dB}}$.
Sound pressure level meter with numerical display; in front the measurement microphone

The reference value for airborne noise was set at the beginning of the 20th century at p 0 = 20 µPa = 2 · 10 −5 Pa . This sound pressure was assumed to be the hearing threshold of human hearing at a frequency ofkHz . Although it later turned out that this value was set a little too low for 1 kHz, it was retained as a reference value . A reference value of 1 µPa is specified for specifying a sound pressure level in water and other media. The sound pressure level can assume both positive (sound pressure is greater than the reference value) and negative (sound pressure is less than the reference value) values ​​as a level variable . ${\ displaystyle p_ {0}}$

However, the maximum sound pressure [Pa] cannot be determined from the sound pressure level [dB], because: The sound pressure level is calculated from the effective value (standard deviation, RMS) of the pressure fluctuations. And the amplitude of the signal can no longer be determined from a standard deviation without further assumptions.

## Measurement

Sound pressure level is measured with microphones . The measurable level range does not start significantly below 0 dB and ends at around 150 to 160 dB.

The upper limit is based on the fact that the laws of linear acoustics are only applicable if the air pressure fluctuations are significantly smaller than the atmospheric pressure. The adiabatic changes of state generated by the pressure fluctuations in the air and thus the relationships between the sound field sizes only then behave linearly.

Strictly speaking, however, it is not the sound level that is measured by the microphone, but the rms value of the sound pressure. From this the sound level in the measuring device is then calculated in dB.

The directional characteristic of measurement microphones for determining sound pressure is generally spherical. For so-called binaural recordings are artificial heads used. One speaks of a binaural sound pressure level when a total level is formed from the two sound pressure levels of the left and right ears. For this size the designation bspl (has in psychoacoustics b inaural s ound P ressure l evel ) established. The formation of the BSPL is carried out according to the so-called 6 dB loudness law using the following formula:

${\ displaystyle \ mathrm {BSPL} = 6 \ cdot \ log _ {2} \ left (2 ^ {\ frac {L _ {\ mathrm {l}}} {6}} + 2 ^ {\ frac {L _ {\ mathrm {r}}} {6}} \ right) \, \ mathrm {dB}}$

In this formula, which is only valid in the diffuse field, the quantities L l and L r stand for the sound pressure levels that are measured on the left and right artificial head ears.

## Human perception

The sound pressure level is a technical and not a psychoacoustic parameter. It is only possible to draw conclusions about the perceived loudness of the sound pressure level to a very limited extent. In general it can be said that an increase or decrease in the sound pressure level also tends to produce a louder or quieter perception of sound. Above a volume level of 40 phons (with a 1 kHz sinus tone this corresponds to a sound pressure level of 40 dB), the loudness perception follows Stevens' power law and a difference of 10 phons is perceived as a doubling of the loudness. Below 40 phons, a slight change in the volume level leads to the feeling of doubling the loudness.

The detectability of changes in the sound pressure level depends on the output level. “As the sound pressure increases, the hearing becomes more and more sensitive to changes in the amplitude of sinusoidal tones. At a low level of 20 dB, the just perceptible degree of modulation is around 10%. At a level of 100 dB, it reaches about 1%. "

High sound pressure levels cause discomfort and pain . The discomfort threshold depends strongly on the type and origin of the noise or noise; the pain threshold is between 120 dB and 140 dB depending on the frequency composition of the noise. If the hearing is exposed to sound pressure in the area of ​​the pain threshold, permanent hearing damage can be expected even with only a short exposure time.

Perception: curves of equal volume according to the applicable ISO 226 (2003) (red) and 40-phon curve of the original standard (blue)

## Scored measurement

The dependence on the perceived volume and sound pressure level is heavily dependent on frequency. This frequency dependency is itself in turn dependent on the sound pressure level, which means that there are different frequency dependencies for different levels. If statements are to be made about the perception of a sound event, the frequency spectrum of the sound pressure must therefore be considered - a frequency evaluation takes place using filters marked with letters A to G, which take this level and frequency-dependent perception into account.

In addition, the course over time has an influence on perception. This is taken into account with the evaluation of the top values.

In order to be able to quantify the perceived loudness, there are the quantities of weighted sound pressure level , volume level and loudness . Loudness level (unit of measure: phon ) and loudness (unit of measure: sone ) are psychoacoustic quantities, i.e. they describe the perception of sound, not its physical properties. The definition of such quantities is only possible through psychoacoustic experiments (listening tests). The weighted sound pressure level is, in turn, a simplified representation that was derived from these findings: The weighted sound pressure level is determined by breaking down a measured spectrum in the frequency range into narrow-band parts and weighting (“weighting”) them according to the frequency dependence of the perception. The weighted total level then results from the energetic summation of these weighted partial levels. To identify this, the frequency filter used is added as an index to the formula symbol and often also after the dB specification in brackets, e.g. B. L pA  = 35 dB (A) when using the A filter. In a roughly simplified manner, the frequency filters assume the same and constant loudness of each frequency group contained in the sound. In addition, the isophones (curves of equal volume) determined for this loudness for sinusoidal individual tones are used for the narrow bands. The evaluated sound pressure level thus provides a psychoacoustically flawed, but still usable and standardized consideration of the frequency dependence of human volume perception and is decisive for acoustic limit values ​​in almost every legal provision and every standard. Which frequency filter (A, B, C or D) is best used depends on the level of the overall noise, as each of these filters is based on a different isophone. Regardless of the overall level, the A level has predominantly prevailed; however, there are also national differences on this issue.

Digital sound level meters can usually also display the psychoacoustic variables loudness and volume level. These two values ​​are continuously calculated from the measured spectrum.

## Continuous sound level

The equivalent continuous sound level is the sound pressure level averaged over the measurement time. The equivalent continuous sound level is usually used to determine the weighted sound pressure level . ${\ displaystyle L _ {\ text {eq}}}$

A distinction is made between the following averages:

• Energy-equivalent averaging according to DIN 45 641 ( )${\ displaystyle L _ {\ text {eq}}}$
• Averaging according to DIN 45 643 in accordance with the aircraft noise law ( ). This averaging takes into account the frequency, duration and strength of the individual aircraft noise events.${\ displaystyle L _ {\ text {eq4}}}$

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

Situation or sound source Distance from the
sound source
or measuring location
Sound pressure
(effective value)
${\ displaystyle {\ tilde {p}}}$
unweighted
sound pressure level
L p
Loudest possible sound Ambient air pressure 101 325 Pa 194 dB
Jet 30 m 630 Pa 150 dB
Rifle shot 1 m 200 Pa 140 dB
Pain threshold at the ear 100 Pa 134 dB
Hearing damage from
short-term exposure
at the ear from 20 Pa 120 dB
Fighter plane 100 m 6.3-200 Pa 110-140 dB
Pneumatic hammer / discotheque 1 m 2 Pa 100 dB

Long-term exposure to hearing damage
at the ear from 360 mPa 85 dB
Main thoroughfare 10 m 200-630 mPa 80-90 dB
Car 10 m 20-200 mPa 60-80 dB
TV at
room volume
1 m 20 mPa 60 dB
Speaking human
(normal conversation)
1 m 2-20 mPa 40-60 dB
Very quiet room at the ear 200-630 µPa 20-30 dB
Rustling leaves,
calm breathing
at the ear 63.2 µPa 10 dB
Hearing threshold at 2 kHz at the ear 20 µPa 0 dB

At higher sound pressure levels, distortions occur because the temperature of the medium becomes pressure-dependent due to adiabatic compression. Pressure maxima then spread faster than the pressure minima, which is why sinusoidal modulations increasingly distort in a sawtooth shape at higher sound pressure levels. At particularly high sound pressures, one speaks of shock waves.

## Dependence on the measuring distance

In emission measurements is investigated, which sound causes a specific sound source (eg. B. Measurement of the noise radiated by an aircraft of a particular type). Since the sound pressure level always depends on the distance to the source of the sound, in addition to the indication of the measured level, the distance r at which the measurement was carried out is absolutely necessary for emission measurements.

In the case of immission measurements, on the other hand, the sound pressure level is measured at the place where it affects people. One example is the measurement of the sound pressure level in a house that is in the approach path of an airport. With immission measurements, the number of existing sound sources and their distance from the measuring point are irrelevant.

As an alternative, when measuring emissions at the source of interference, the sound power level is often specified, which is independent of distance and room, as it expresses the entire sound power of the source in question, radiated in all directions . The sound pressure level that is generated at a certain distance from the source of noise emitting interference can be calculated directly from the sound power level. In this calculation, however, the local conditions of the scene for which the calculation is to apply must be taken into account.

In the case of point sources of sound (and generally of sources radiating uniformly in all spatial directions), the sound pressure level decreases by around 6 dB per doubling of the distance, i.e. to half the sound pressure. This results from the fact that the sound pressure is inversely proportional to the distance r from the sound source according to the so-called distance law ( 1 / r law ). Mathematically, this relationship can be easily understood from the calculation formula for the sound pressure:

{\ displaystyle {\ begin {aligned} \ Delta L & = L_ {2} -L_ {1} = {\ left (10 \, \ cdot \, \ log _ {10} {\ left ({\ frac {p_ { 2}} {p_ {0}}} \ right)} ^ {2} \, - \, 10 \, \ cdot \, \ log _ {10} {\ left ({\ frac {p_ {1}} { p_ {0}}} \ right)} ^ {2} \ right)} \, \ mathrm {dB} \\ & = 10 \, \ cdot \, \ log _ {10} {\ left ({\ frac { p_ {2}} {p_ {0}}} {\ frac {p_ {0}} {p_ {1}}} \ right)} ^ {2} \, \ mathrm {dB} \\ & = 10 \, \ cdot \, \ log _ {10} {\ left ({\ frac {p_ {2}} {p_ {1}}} \ right)} ^ {2} \, \ mathrm {dB} \\\ end { aligned}}}

So if, according to the 1 / r law , p 2 / p 1 = r 1 / r 2 , then for a doubling of the distance (i.e. r 2 = 2 r 1 ):

${\ displaystyle \ Delta L = 10 \, \ cdot \, \ log _ {10} {\ left ({\ frac {1} {2}} \ right)} ^ {2} \, \ mathrm {dB} = 20 \, \ cdot \, \ log _ {10} {\ left ({\ frac {1} {2}} \ right)} \, \ mathrm {dB} = -20 \, \ cdot \, \ log _ {10} {\ left (2 \ right)} \, \ mathrm {dB} = -6 {,} 021 \, \ mathrm {dB} \ approx -6 \, \ mathrm {dB}}$

It is sometimes claimed that the sound pressure decreases with 1 / r 2 . However, this only applies to square sizes, such as sound intensity or sound energy . Here, too, doubling the distance results in a level difference of 6 dB, since these energetic variables, in contrast to sound pressure, are not squared again in the level calculation formula.

## Addition of the sound pressure level of several sound sources

### Incoherent sound sources

When adding incoherent sound sources, the correct total level results from the energetic addition of the sound sources involved. Level values ​​in decibels can not simply be added up. If only the sound pressure levels of the individual sound sources to be added are available, the squared sound pressures (which are proportional to the energy) must first be calculated from them. This process is called "de-logarithmizing" (analogous to "logarithmizing" when calculating a level).

For the total sound pressure level of n incoherently radiating sources, the following applies:

${\ displaystyle L _ {\ Sigma} = 10 \, \ cdot \, \ log _ {10} \ left ({\ frac {p_ {1} ^ {2} + p_ {2} ^ {2} + \ cdots + p_ {n} ^ {2}} {p_ {0} ^ {2}}} \ right) = 10 \, \ cdot \, \ log _ {10} \ left (\ left ({\ frac {p_ {1 }} {p_ {0}}} \ right) ^ {2} + \ left ({\ frac {p_ {2}} {p_ {0}}} \ right) ^ {2} + \ cdots + \ left ( {\ frac {p_ {n}} {p_ {0}}} \ right) ^ {2} \ right)}$

From the calculation formula for the sound pressure level it follows directly that:

${\ displaystyle \ left ({\ frac {p_ {i}} {p_ {0}}} \ right) ^ {2} = 10 ^ {\ frac {L_ {i}} {10}}, \ qquad i = 1,2, \ cdots, n}$

or

${\ displaystyle {\ frac {p_ {i}} {p_ {0}}} = 10 ^ {\ frac {L_ {i}} {20}}, \ qquad i = 1,2, \ cdots, n}$

Inserting this into the equation for calculating the total sound level results in the addition formula sought:

${\ displaystyle L _ {\ Sigma} = 10 \, \ cdot \, \ log _ {10} \ left (10 ^ {\ frac {L_ {1}} {10}} + 10 ^ {\ frac {L_ {2 }} {10}} + \ cdots +10 ^ {\ frac {L_ {n}} {10}} \ right) \, \ mathrm {dB}}$

Special case of equally strong incoherent sound sources

At a certain location, two equally strong sound sources each generate the same sound pressure, i.e. H. also the same sound pressure level. When adding such incoherent sources, the above equation for calculating the total sound pressure level is simplified as follows:

{\ displaystyle {\ begin {aligned} L _ {\ Sigma} & = 10 \, \ cdot \, \ log _ {10} \ left (10 ^ {\ frac {L_ {1}} {10}} + 10 ^ {\ frac {L_ {2}} {10}} + \ cdots +10 ^ {\ frac {L_ {n}} {10}} \ right) \, \ mathrm {dB}, \ qquad L_ {1} = L_ {2} = \ cdots = L_ {n} \\ & = 10 \, \ cdot \, \ log _ {10} \ left (n \ cdot 10 ^ {\ frac {L_ {n}} {10}} \ right) \, \ mathrm {dB} \\ & = 10 \, \ cdot \ left (\ log _ {10} (n) + \ log _ {10} \ left (10 ^ {\ frac {L_ {n }} {10}} \ right) \ right) \, \ mathrm {dB} \\ & = 10 \, \ cdot \, \ log _ {10} (n) \, \ mathrm {dB} + L_ {n } \ end {aligned}}}

For n = 2 equally strong, incoherent sound sources, z. B. a level increase of 10 · log 10 (2) dB = 3.01 dB compared to the case that only one source is available. For n = 10 there is a level increase of 10 dB.

### Coherent sound sources

The addition of the sound pressure level of coherent sound sources cannot be accomplished by simple energetic addition. Rather, interference occurs between the sound signals from the various sources . The calculation of the sound pressure level at a specific location is possible by applying the superposition principle :

Depending on how the phase differences of the various sounds are at the point under consideration, the total sound is amplified or weakened. Maximum gain z. B. occurs when the distance covered by the various sounds is a whole multiple of the wavelength. In the case of equally strong, coherent sound sources, the level at these points of maximum amplification increases by doubling the number of sources by 6 dB.

At points whose distance to both sources differs by half a wavelength or an odd multiple thereof, the sound is partially canceled. In the special case of equally strong sources, the extinction is complete, i. H. the level goes against . At all other points in the room the level assumes values ​​which lie between the maximum and the minimum. ${\ displaystyle - \ infty \, \ mathrm {dB}}$

For point-type sound sources in the free field, an analytical calculation of the level depending on the measurement location is easy to carry out. In closed rooms, on the other hand, the reflections create a complex sound field that can only be calculated numerically assuming simplifications.

One method for active noise reduction is the generation of so-called anti- noise . The interference effect that occurs between coherent sound signals is exploited profitably: A sound signal with the same time sequence and the same magnitude spectrum as the interference sound, but with a phase spectrum shifted by 180 ° compared to the interference sound, just extinguishes it. In order to cancel the interfering sound at every point in the room, the anti-phase signal would have to be sent to a loudspeaker located at the location of the interference source. No sound at all would then be emitted. Since in practice there can never be different sound sources in exactly the same place, it is either possible to emit "anti-noise" in such a way that it cancels out the background noise at a certain point. However, if the listener moves away from this point, the cancellation works worse or not at all because the transit time differences between interfering and anti-noise and thus the phase shifts change. Another possibility is to feed headphones with the amplified, anti-phase signal from a microphone arranged on them. In both cases, there is the problem in practice that high frequencies can only be partially or completely canceled: Due to their short wavelength, even minimal deviations in the transit time differences can lead to significant phase shifts. These can be caused by inaccuracies in the geometric positions (sound source, anti-sound source, listener), processing times of the signal processor used or temperature fluctuations in the air.

## swell

1. D. W. Robinson and L. S. Whittle, Acustica, Vol. 10 (1960), pp. 74-80
2. E. Zwicker, R. Feldtkeller: The ear as a message receiver . S. Hirzel, Stuttgart 1967.
3. Designation e.g. B. Noise Guard