Sound pressure

In acoustics, sound pressure or acoustic pressure is the pressure difference between the ambient (average or equilibrium) atmospheric pressure and that of a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone. The SI unit of sound pressure is the pascal (Pa).

Mathematical definition

A sound wave in a transmission medium causes a deviation (sound pressure, a dynamic pressure) in the local ambient pressure, a static pressure.

Sound pressure, denoted p, is defined by

where ptotal is the total pressure, pstat is the static pressure.

Sound measurements

Sound intensity

In a sound wave, the complementary variable to sound pressure is the particle velocity. Together, they determine the sound intensity of the wave.

Sound intensity, denoted I and measured in W·m−2 in SI units, is defined by

where p is the sound pressure, v is the particle velocity.

Acoustic impedance

Acoustic impedance, denoted Z and measured in Pa·m−3·s in SI units, is defined by

where is the Laplace transform of sound pressure, is the Laplace transform of sound volume flow rate.

Specific acoustic impedance, denoted z and measured in Pa·m−1·s in SI units, is defined by

where is the Laplace transform of sound pressure, is the Laplace transform of particle velocity.

Particle displacement

The particle displacement of a progressive sine wave is given by

where is the amplitude of the particle displacement, is the phase shift of the particle displacement, k is the angular wavevector, ω is the angular frequency.

It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave x are given by

where vm is the amplitude of the particle velocity, is the phase shift of the particle velocity, pm is the amplitude of the acoustic pressure, is the phase shift of the acoustic pressure.

Taking the Laplace transforms of v and p with respect to time yields

Since , the amplitude of the specific acoustic impedance is given by

Consequently, the amplitude of the particle displacement is related to that of the acoustic velocity and the sound pressure by

Inverse-proportional law

When measuring the sound pressure created by a sound source, it is important to measure the distance from the object as well, since the sound pressure of a spherical sound wave decreases as 1/r from the centre of the sphere (and not as 1/r2, like the sound intensity):

This relationship is an inverse-proportional law.

If the sound pressure p1 is measured at a distance r1 from the centre of the sphere, the sound pressure p2 at another position r2 can be calculated:

The inverse-proportional law for sound pressure comes from the inverse-square law for sound intensity:

Indeed,

where is the particle velocity, is the convolution operator, z−1 is the convolution inverse of the specific acoustic impedance, hence the inverse-proportional law:

Sound pressure level

Sound pressure level (SPL) or acoustic pressure level (APL) is a logarithmic measure of the effective pressure of a sound relative to a reference value.

Sound pressure level, denoted Lp and measured in dB,"Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units", IEC 60027-3 Ed. 3.0, International Electrotechnical Commission, 19 July 2002. is defined by:

where p is the root mean square sound pressure, p0 is a reference sound pressure, is the neper, is the bel, is the decibel.

The commonly used reference sound pressure in air isRoss Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.

which is often considered as the threshold of human hearing (roughly the sound of a mosquito flying 3 m away). The proper notations for sound pressure level using this reference are or , but the suffix notations , , dBSPL, and dBSPL are very common, even if they are not accepted by the SI.Thompson, A. and Taylor, B. N. Sec. 8.7: "Logarithmic quantities and units: level, neper, bel", Guide for the Use of the International System of Units (SI) 2008 Edition, NIST Special Publication 811, 2nd printing (November 2008), SP811 PDF.

Most sound-level measurements will be made relative to this reference, meaning will equal an SPL of . In other media, such as underwater, a reference level of is used. These references are defined in ANSI S1.1-2013.

The main instrument for measuring sound levels in the environment is the sound level meter. Most sound level meters provide readings in A, C, and Z-weighted decibels and must meet international standards such as IEC 61672-2013.

Examples The lower limit of audibility is defined as SPL of , but the upper limit is not as clearly defined. While ( or ) is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere (i. e., if the thermodynamic properties of the air are disregarded; in reality, the sound waves become progressively non-linear starting over 150 dB), larger sound waves can be present in other atmospheres or other media, such as underwater or through the Earth.

Ears detect changes in sound pressure. Human hearing does not have a flat spectral sensitivity (frequency response) relative to frequency versus amplitude. Humans do not perceive low- and high-frequency sounds as well as they perceive sounds between 3,000 and 4,000 Hz, as shown in the equal-loudness contour. Because the frequency response of human hearing changes with amplitude, three weightings have been established for measuring sound pressure: A, B and C.

In order to distinguish the different sound measures, a suffix is used: A-weighted sound pressure level is written either as dBA or LA, B-weighted sound pressure level is written either as dBB or LB, and C-weighted sound pressure level is written either as dBC or LC. Unweighted sound pressure level is called "linear sound pressure level" and is often written as dBL or just L. Some sound measuring instruments use the letter "Z" as an indication of linear SPL.

Distance The distance of the measuring microphone from a sound source is often omitted when SPL measurements are quoted, making the data useless, due to the inherent effect of the inverse proportional law. In the case of ambient environmental measurements of "background" noise, distance need not be quoted, as no single source is present, but when measuring the noise level of a specific piece of equipment, the distance should always be stated. A distance of one metre (1 m) from the source is a frequently used standard distance. Because of the effects of reflected noise within a closed room, the use of an anechoic chamber allows sound to be comparable to measurements made in a free field environment.

According to the inverse proportional law, when sound level Lp1 is measured at a distance r1, the sound level Lp2 at the distance r2 is

Multiple sources The formula for the sum of the sound pressure levels of n incoherent radiating sources is

Inserting the formulas

in the formula for the sum of the sound pressure levels yields

Examples of sound pressure

+ Examples of sound pressure in air at standard atmospheric pressure Source of sound Distance Sound pressure level (Pa) (dB) Shock wave (distorted sound waves > 1 atm; waveform valleys are clipped at zero pressure) >1 >191 Simple open-ended thermoacoustic device 1.3 1761883 eruption of Krakatoa165 km 8 172 .30-06 rifle being fired 1 m toshooter's side 7 171Firecracker0.5 m 7 171 Stun grenade Ambient 158–172 party balloon inflated to rupture At ear 4.9×103 168 diameter balloon crushed to rupture At ear 1.8×103 159 party balloon inflated to rupture 0.5 m 1.4×103 157 diameter balloon popped with a pin At ear 1.1×103 155 LRAD 1000Xi Long Range Acoustic Device 1 m 8.9×102 153 party balloon inflated to rupture 1 m 730 151 Jet engine 1 m 630 150 diameter balloon crushed to rupture 0.95 m 450 147 diameter balloon popped with a pin 1 m 280 143 Maximum instantaneous peak (C-weighted) for amplified sound at "safe listening" venues/events per WHO's global standard Ambient 200 140 Instantaneous peak workplace noise (C-weighted) which legally obligates use of hearing protection by workers in the EU At ear 140 137 Instantaneous peak workplace noise (C-weighted) which legally obligates employers to offer hearing protectors to workers in the EU At ear 112 135 Loudest human voice 1 inch 110 135 TrumpetRecording Brass & Reeds. 0.5 m 63.2 130 Vuvuzela horn 1 m 20.0 120 Threshold of painRealistic Maximum Sound Pressure Levels for Dynamic Microphones – Shure. At ear 20–100 120–134 Risk of instantaneous noise-induced hearing loss At ear 20.0 120 Maximum instantaneous peak (C-weighted) for amplified sound at children's venues/events complying with WHO's global safe listening standard Ambient 20.0 120 Jet engine 100–30 m 6–200 110–140 Two-stroke chainsaw 1 m 6 110 Jackhammer 1 m 2.00 100 Sound level limit (A-weighted, moving average over 15-minute interval) at safe listening venues/events per WHO global standard Ambient 2.00 100 NIOSH recommended exposure limit (REL) for workplace noise (15-minute average, A-weighted) At ear 2.00 100 NIOSH REL for workplace noise (30-minute average, A-weighted) At ear 1.42 97 Sound level limit (A-weighted, moving average over 15-minute interval) at children's venues/events per WHO's "safe listening" global standard Ambient 1.0 94 NIOSH REL for workplace noise (1-hour average, A-weighted) At ear 1.0 94 NIOSH REL for workplace noise (2-hour average, A-weighted) At ear 0.71 91 Sound level limit (A-weighted, moving average over 15-minute interval) at venues/events for "young children" per WHO's safe listening global standard Ambient 0.63 90 NIOSH REL for workplace noise (4-hour average, A-weighted) At ear 0.50 88 NIOSH recommended exposure limit (REL) for workplace noise (average over 8-hour workday, A-weighted) At ear 0.36 85 Hearing damage (over long-term exposure, need not be continuous) At ear 0.36 85 Workplace noise level (8-hour daily average, A-weighted) that legally obligates use of hearing protection by workers in the EU At ear 0.36 85 Vacuum cleaner, A-weighted (1981) 1.8 m 0.36 85 Workplace noise level (8-hour daily average, A-weighted) that legally obligates employers to offer hearing protectors to workers in the EU At ear 0.2 80 Average level (A-weighted) at 40 hours per week (on a rolling basis) equivalent to the "sound allowance" for a "safe listening device" in "Mode 1: WHO standard level for adults" per WHO/ITU-T Rec. H.870 At ear 0.2 80 Average level (A-weighted) at 40 hours per week (on a rolling basis) equivalent to the "sound allowance" for a "safe listening device" in "Mode 2: WHO standard level for sensitive users (e.g. children)" per WHO/ITU-T Rec. H.870 At ear 0.11 75 Television (A-weighted) Ambient 0.11 75 EPA-identified maximum to protect against hearing loss and other disruptive effects from noise, such as sleep disturbance, stress, learning detriment, etc. Ambient 0.06 70 Passenger car at 30 km/h (electric and combustion engines) 10 m 0.04–0.06 65–70 Normal conversation 1 m 2–0.02 40–60 Passenger car at 10 km/h (combustion) 10 m 13 56Passenger car at 10 km/h (electric)10 m 6 50 Very calm room Ambient 20–30 Light leaf rustling, calm breathing Ambient 6 10 Auditory threshold at 1 kHz At ear 2.00 0 Anechoic chamber, Orfield Labs, A-weighted Ambient 6.8 −9.4 Anechoic chamber, University of Salford, A-weighted Ambient 4.8 −12.4 Anechoic chamber, Microsoft, A-weighted Ambient 1.90 −20.35

Sound pressure

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