Sound_pressure - Wikiparency

# Sound pressure > *For the logarithmic unit itself, see [[Decibel]]. For the power carried per square meter, see [[Sound_intensity|Sound intensity]].* **Sound pressure** is the local deviation of pressure from the ambient [[Atmosphere_of_Earth|atmospheric]] or hydrostatic value caused by a passing [[Sound|sound]] wave. It is measured in pascals, and because the ear and the instruments that imitate it respond over a range of more than a million to one in pressure, it is usually reported as a **sound pressure level** in [[Decibel|decibels]] relative to 20 micropascals. The pressures involved are tiny: the loudest sound a person can endure without pain changes the pressure of the air by about one part in 5,000. Sound pressure is the quantity a [[Microphone|microphone]] and a [[Sound_level_meter|sound level meter]] actually sense. From it follow the other field quantities of acoustics: the [[Particle_velocity|particle velocity]] through the [[Acoustic_impedance|acoustic impedance]] of the medium, the displacement of the air, and the [[Sound_intensity|intensity]], which is the product of pressure and velocity. In open air, pressure from a small source falls as the inverse of the distance, which the decibel turns into a fixed 6 dB loss for every doubling of distance. The framework microsim *Sound pressure and the decibel: walking away from a source* draws the spherical wavefronts around a small source; the reader sets the source power, walks a listener out to 30 meters while a meter reads the level, and switches on a second source to see how levels add. ## Mathematical definition Sound pressure is defined as the difference between the instantaneous pressure at a point and the pressure that would exist there without the sound, `p = p_total − p_0`. In the [[Acoustic_wave_equation|acoustic wave equation]] it is the small field variable that obeys the [[Wave_equation|wave equation]], while the ambient pressure `p_0` is a constant background. For the simple traveling [[Sine_wave|sine wave]] used in the textbooks, `p(x, t) = p_max sin(kx − ωt)`, so the pressure swings equally above and below ambient and averages to zero.[^up17-1] Because the average is zero, a single number for "how much pressure" has to be an average of the square. The root-mean-square value `p_rms` is the standard one; for a pure tone `p_rms = p_max / √2`, so a tone with a 1.41 Pa peak has a 1.00 Pa rms pressure. Peak pressure is still quoted for impulses such as gunshots and blasts, where the waveform is far from a sine and the rms depends on how long the averaging window is. The unit is the pascal (1 Pa = 1 N/m²). Compared with standard [[Standard_temperature_and_pressure|atmospheric pressure]] of 101,325 Pa the numbers are small. A pressure of 1 Pa rms, which is loud, is a fluctuation of about 10 parts per million, and the 20 µPa threshold of hearing is two parts in ten billion. Sound in air is therefore almost always a small-signal, linear phenomenon: [[Superposition_principle|superposition]] holds, pressures from different sources add as simple numbers, and the speed of sound does not depend on the amplitude.[^up17-4] ## Sound measurements ### Sound intensity A sound wave carries energy as well as pressure. The [[Sound_intensity|sound intensity]] is the time-averaged power crossing one square meter, `I = ⟨p u⟩`, the product of the pressure and the particle velocity `u` at the same point. For a plane progressive wave the two are in phase and in fixed ratio, which gives the textbook form `I = p_rms² / (ρc)`, or equivalently `I = p_max² / (2ρc)` in terms of the pressure amplitude, where ρ is the [[Density|density]] of the medium and c the [[Speed_of_sound|speed of sound]].[^up17-3] Intensity grows as the square of pressure: doubling the pressure quadruples the power per square meter. ### Acoustic impedance The ratio of sound pressure to particle velocity is the specific [[Acoustic_impedance|acoustic impedance]], `z = p / u`. For a plane wave in an unbounded medium it is simply the characteristic impedance ρc. For air at 20 °C, ρ ≈ 1.20 kg/m³ and c ≈ 343 m/s give ρc ≈ 413 rayl (Pa·s/m), the value the microsim uses. [[Water]] at 1,000 kg/m³ and 1,480 m/s has ρc ≈ 1.48 million rayl, about 3,600 times more. The same pressure in water therefore moves the medium 3,600 times more slowly and carries 3,600 times less intensity, which is one reason underwater levels cannot be compared directly with levels in air. ### Particle displacement The air itself moves back and forth by a distance `ξ = u / ω` for a tone of angular frequency ω. A worked number shows how small this is. At 1 Pa rms and 1 kHz, the particle velocity is 1 / 413 ≈ 2.4 mm/s and the displacement amplitude is about 0.39 µm, less than the wavelength of visible light. At the threshold of hearing, 20 µPa, the same arithmetic gives about 8 picometers, a small fraction of the diameter of a nitrogen molecule; the textbook describes air molecules at threshold as vibrating "over a distance of less than one molecular diameter."[^up17-3] The ear is a pressure detector of extraordinary sensitivity, and so is a measurement microphone, whose diaphragm responds to pressure rather than to displacement. ## Inverse-proportional law A small source radiating power P equally in all directions into free space spreads that power over spheres of growing area `4πr²`. With no absorption, the intensity at distance r is `I = P / (4πr²)`, the [[Inverse-square_law|inverse-square law]], and since intensity goes as pressure squared, the pressure itself falls as `1/r`.[^up17-3][^us17] Sound pressure is therefore called inversely proportional to distance, while intensity is inversely proportional to the square of distance. The decibel turns these ratios into steps of equal size. Doubling the distance halves the pressure, and `20 log10(1/2) ≈ −6.02 dB`, whatever the starting distance. The Minnesota Pollution Control Agency's noise guide puts the rule in plain terms: "When the distance is doubled from a point source, the sound level decreases six decibels."[^mpca] A long line source such as a busy highway spreads over cylinders rather than spheres, and the same guide gives 3 dB per doubling for it.[^mpca] A worked number from the microsim's defaults: a source radiating 1 mW of acoustic power gives, at 4 m, an intensity of 10⁻³ / (4π · 16) ≈ 4.97 × 10⁻⁶ W/m², an rms pressure of √(4.97 × 10⁻⁶ × 413) ≈ 0.045 Pa, and a level of about 67 dB. At 8 m the level is 61 dB; at 16 m, 55 dB. The law has limits. Close to a real source of finite size, in the near field, pressure and velocity are not in phase and the simple `1/r` fails. Far away, air absorption removes energy on top of spreading, most strongly at high frequencies ([[Acoustic_attenuation|acoustic attenuation]]). Indoors, reflections from walls build a diffuse field whose level stops falling beyond a certain distance ([[Reverberation|reverberation]]). The microsim deliberately shows only the free-field part. *Try: drag listener distance r from 1 m to 2 m to 4 m and watch the SPL readout drop by the same 6 dB each time (the "at 2r" readout always says −6.02 dB); raise source power by a factor of ten and see every level rise by 10 dB; tick second source 3 m away and see the total bar rise by 3 dB where the two contributions are equal.* ## Sound pressure level The **sound pressure level** is `L_p = 20 log10(p_rms / p_0)` dB, with reference pressure `p_0 = 20 µPa` in air.[^iso1683] The factor is 20 rather than 10 because the [[Decibel|decibel]] is defined as ten times the logarithm of a power ratio, and power goes as pressure squared. Twenty micropascals was chosen to sit near the threshold of hearing at 1 kHz, so 0 dB is roughly the quietest audible tone. In water the reference pressure is 1 µPa by convention,[^iso18405] which by itself makes an underwater level 20 log10(20) ≈ 26 dB larger than the in-air level for the same pressure, before any impedance difference is considered. Textbooks often introduce the related sound intensity level, `β = 10 log10(I / I_0)` with `I_0 = 10⁻¹² W/m²`.[^up17-3][^us17] In air at room conditions the two scales agree to about 0.1 dB, because `(20 µPa)² / 413 rayl ≈ 0.97 × 10⁻¹² W/m²`, so the tables of familiar sound levels can be read as either. Measured levels are usually frequency-weighted. A sound level meter's A-weighting filter de-emphasizes low and very high frequencies; in the MPCA guide's words, "The A-weighting network is used to duplicate the sensitivity of the human ear."[^mpca] Levels weighted this way are written dB(A). The shape of the ear's own sensitivity is mapped by the [[Equal-loudness_contour|equal-loudness contours]] discussed under [[Hearing|hearing]]. ### Examples Two reference points anchor the scale. A pressure of exactly 1 Pa rms is 20 log10(1 / 20 × 10⁻⁶) ≈ 94.0 dB, which is why acoustic calibrators commonly produce a 94 dB tone at 1 kHz. The microsim's default listener, 4 m from a 1 mW source, receives 0.045 Pa and reads 67 dB, a little below the textbook's noisy office at 70 dB. ### Distance Between two distances in free field the levels differ by `L_2 = L_1 − 20 log10(r_2 / r_1)`. Ten times the distance costs 20 dB, a hundred times 40 dB. A source of 1 W acoustic power produces an intensity level of about 109 dB at 1 m, 89 dB at 10 m and 69 dB at 100 m, if nothing but spreading acts. ### Multiple sources Sounds from independent sources, such as two machines or two voices, add their mean-square pressures, which means their intensities, not their decibel values. Two equal sources give twice the power and a level 10 log10(2) ≈ 3 dB higher: 60 dB plus 60 dB is 63 dB, and the MPCA guide states the same rule as "A doubling of sound energy yields an increase of three decibels."[^mpca] Unequal sources barely add: 60 dB plus 50 dB is 60.4 dB. Two coherent sources driven by the same signal are different: their pressures add with phase, so the level can rise by up to 6 dB at some points and fall toward silence at others, the pattern studied as [[Wave_interference|wave interference]]. The microsim's field picture freezes one instant of that pattern, while its meter reports the time-averaged power sum. ## Examples of sound pressure The table converts the familiar sound levels listed in the OpenStax physics text into rms sound pressures with `p = 20 µPa × 10^(L/20)`. The levels are the textbook's (sound intensity levels, which agree with sound pressure levels in air within the rounding shown); the pressures are computed here.[^up17-3] | Source or effect (textbook example) | Level (dB) | rms sound pressure | |---|---|---| | Threshold of hearing at 1,000 Hz | 0 | 20 µPa | | Rustle of leaves | 10 | 63 µPa | | Whisper at 1 m | 20 | 200 µPa | | Average home | 40 | 2 mPa | | Normal conversation | 60 | 20 mPa | | Noisy office, busy traffic | 70 | 63 mPa | | Inside a heavy truck; damage from prolonged exposure | 90 | 0.63 Pa | | Noisy factory, siren at 30 m | 100 | 2 Pa | | Loud rock concert; threshold of pain | 120 | 20 Pa | | Jet airplane at 30 m | 140 | 200 Pa | | Bursting of eardrums | 160 | 2,000 Pa | The same source notes that several government agencies and health associations recommend not exceeding 85 dB for eight-hour daily exposure without hearing protection.[^up17-3] At the top of the scale, a sine wave whose trough reaches zero absolute pressure has a peak amplitude equal to atmospheric pressure, 101,325 Pa; expressed as a level of peak pressure that is about 194 dB. Beyond that point the rarefaction cannot go lower than a vacuum, the wave distorts, and the linear description of sound pressure no longer applies. ## Minnesota *This section is specific to Wikitube.* [[Minnesota]] sets statewide noise standards in terms of sound level statistics rather than single readings. Under Minnesota Rules part 7030.0040, adopted under Minnesota Statutes section 116.07, the level is measured in dB(A) over a one-hour survey, and two statistics are limited: L10, the level exceeded 10 percent of the time, and L50, the level exceeded half the time. Daytime runs from 7:00 a.m. to 10:00 p.m.; nighttime from 10:00 p.m. to 7:00 a.m. Noise area classification 1 covers households, lodging, religious, medical and educational uses, among others.[^mpca] | Noise area classification | Daytime L10 | Daytime L50 | Nighttime L10 | Nighttime L50 | |---|---|---|---|---| | 1 (residential and similar) | 65 | 60 | 55 | 50 | | 2 | 70 | 65 | 70 | 65 | | 3 | 80 | 75 | 80 | 75 | The inverse-distance law makes these numbers easy to use in siting. A steady point source that measures 62 dB(A) at 25 m must be moved to about 100 m, four times as far and 12 dB quieter, to meet the nighttime L50 of 50 dB(A) at a home, if spreading is the only loss. ## See also - [[Decibel]] - [[Sound_intensity]] - [[Sound]] - [[Acoustic_attenuation]] - [[Reverberation]] - [[Hearing]] - [[Sound_level_meter]] ## References [^up17-1]: OpenStax (2016). *University Physics Volume 1*. Ling, S. J.; Sanny, J.; Moebs, W. Rice University. Chapter 17 "Sound," §17.1 "Sound Waves," Eq. 17.1–17.2, pp. 808–810. https://openstax.org/details/books/university-physics-volume-1 (Portal Books 077). [^up17-4]: OpenStax (2016). *University Physics Volume 1*. §17.4 "Normal Modes of a Standing Sound Wave," p. 830 (pressures from two sources "add and subtract like simple numbers"). Portal Books 077. [^up17-3]: OpenStax (2016). *University Physics Volume 1*. §17.3 "Sound Intensity," Eq. 17.8–17.12 and Table 17.2 "Sound Intensity Levels and Intensities," pp. 818–823. Portal Books 077. [^us17]: *Ultrasound Physics and its Application in Medicine* (2024). Chapter 1 "Basic Principles of Ultrasound," §1.7 "Quantifying Ultrasound," p. 8 and Fig. 1-4 (inverse-square law; intensity level re I₀ = 10⁻¹² W/m²). Portal Books 091. [^iso1683]: International Organization for Standardization (2015). *ISO 1683:2015 Acoustics — Preferred reference values for acoustical and vibratory levels*. Geneva: ISO. https://www.iso.org/standard/64648.html (reference sound pressure 20 µPa in gases). [^iso18405]: International Organization for Standardization (2017). *ISO 18405:2017 Underwater acoustics — Terminology*. Geneva: ISO. https://www.iso.org/standard/62406.html (reference sound pressure 1 µPa in water). [^mpca]: Minnesota Pollution Control Agency (2015). *A Guide to Noise Control in Minnesota* (p-gen6-01, revised November 2015). St. Paul: MPCA. Noise standards table (Minn. R. 7030.0040), daytime and nighttime definitions, L10/L50 definitions, distance and source-addition rules, A-weighting. https://www.pca.state.mn.us/sites/default/files/p-gen6-01.pdf  **Microsim — three.js (Wikitube framework):** *Sound pressure* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Sound_pressure.html" data-title="Sound pressure"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Sound_pressure.json`; part of the [[PORTAL_Acoustics|Acoustics portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Sound_pressure) : [Wikitube](https://en.wikitube.io/wiki/Sound_pressure) - skeleton pinned to revision 1372840241 (2026-09-11).