Sound pressure

Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average, or equilibrium) atmospheric pressure, caused by 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).^[1]

Mathematical definition

Sound pressure diagram:
1. silence;
2. audible sound;
3. atmospheric pressure;
4. sound pressure

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

p_\mathrm{total} = p_\mathrm{stat} + p,

where

• p_total is the total pressure;
• p_stat 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

\mathbf I = p \mathbf v,

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^[2]

Z(s) = \frac{\hat{p}(s)}{\hat{Q}(s)},

where

• \hat{p}(s) is the Laplace transform of sound pressure;
• \hat{Q}(s) 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^[2]

z(s) = \frac{\hat{p}(s)}{\hat{v}(s)},

where

• \hat{p}(s) is the Laplace transform of sound pressure;
• \hat{v}(s) is the Laplace transform of particle velocity.

Particle displacement

The particle displacement of a progressive sine wave is given by

\delta(\mathbf{r},\, t) = \delta_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0}),

where

• ξ_m is the amplitude of the particle displacement;
• \varphi_{\delta, 0} 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

v(\mathbf{r},\, t) = \frac{\partial \delta}{\partial t} (\mathbf{r},\, t) = \omega \delta_\mathrm{m} \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0} + \frac{\pi}{2}\right) = v_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{v, 0}),

p(\mathbf{r},\, t) = -\rho c^2 \frac{\partial \delta}{\partial x} (\mathbf{r},\, t) = \rho c^2 k_x \delta_\mathrm{m} \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0} + \frac{\pi}{2}\right) = p_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{p, 0}),

where

• v_m is the amplitude of the particle velocity;
• \varphi_{v, 0} is the phase shift of the particle velocity;
• p_m is the amplitude of the acoustic pressure;
• \varphi_{p, 0} is the phase shift of the acoustic pressure.

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

\hat{v}(\mathbf{r},\, s) = v_\mathrm{m} \frac{s \cos \varphi_{v,0} - \omega \sin \varphi_{v,0}}{s^2 + \omega^2},

\hat{p}(\mathbf{r},\, s) = p_\mathrm{m} \frac{s \cos \varphi_{p,0} - \omega \sin \varphi_{p,0}}{s^2 + \omega^2}.

Since \varphi_{v,0} = \varphi_{p,0}, the amplitude of the specific acoustic impedance is given by

z_\mathrm{m}(\mathbf{r},\, s) = |z(\mathbf{r},\, s)| = \left|\frac{\hat{p}(\mathbf{r},\, s)}{\hat{v}(\mathbf{r},\, s)}\right| = \frac{p_\mathrm{m}}{v_\mathrm{m}} = \frac{\rho c^2 k_x}{\omega}.

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

\delta_\mathrm{m} = \frac{v_\mathrm{m}}{\omega},

\delta_\mathrm{m} = \frac{p_\mathrm{m}}{\omega z_\mathrm{m}(\mathbf{r},\, s)}.

Inverse-proportional law

When measuring the sound pressure created by an object, 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):^[3]

p(r) \propto \frac{1}{r}.

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:

p_2 = \frac{r_1}{r_2}\,p_1.

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

I(r) \propto \frac{1}{r^2}.

Indeed,

I(r) = p(r)v(r) = p(r)[p * z^{-1}](r) \propto p^2(r),

where

• * is the convolution operator;
• z ⁻¹ is the convolution inverse of the specific acoustic impedance,

hence the inverse-proportional law:

p(r) \propto \frac{1}{r}.

The sound pressure may vary in direction from the centre of the sphere as well, so measurements at different angles may be necessary, depending on the situation. An obvious example of a sound source whose spherical sound wave varies in level in different directions is a bullhorn.

Sound pressure level

Sound pressure level (SPL) or acoustic pressure level is a logarithmic measure of the effective pressure of a sound relative to a reference value.
Sound pressure level, denoted L_p and measured in dB, is defined by^[4]

L_p = \ln\!\left(\frac{p}{p_0}\right)\!~\mathrm{Np} = 2 \log_{10}\!\left(\frac{p}{p_0}\right)\!~\mathrm{B} = 20 \log_{10}\!\left(\frac{p}{p_0}\right)\!~\mathrm{dB},

where

• p is the root mean square sound pressure;^[5]
• p₀ is the reference sound pressure;
• 1 Np = 1 is the neper;
• 1 B = (1/2) ln(10) is the bel;
• 1 dB = (1/20) ln(10) is the decibel.

The commonly used reference sound pressure in air is^[6]

p_0 = 20~\mathrm{\mu Pa},

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 L_{p/(20 μPa)} or L_p (re 20 μPa), but the suffix notations dB SPL, dB(SPL), dBSPL, or dB_SPL are very common, even if they are not accepted by the SI.^[7]

Most sound level measurements will be made relative to this reference, meaning 1 Pa will equal an SPL of 94 dB. In other media, such as underwater, a reference level of 1 μPa is used.^[8] These references are defined in ANSI S1.1-1994.^[9]

Examples

The lower limit of audibility is defined as SPL of 0 dB, but the upper limit is not as clearly defined. While 1 atm (194 dB Peak or 191 dB SPL) is the largest pressure variation an undistorted sound wave can have in Earth's atmosphere, larger sound waves can be present in other atmospheres or other media such as under water, or through the Earth.

Equal-loudness contour

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 near 2,000 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. A-weighting applies to sound pressures levels up to 55 dB, B-weighting applies to sound pressures levels between 55 dB and 85 dB, and C-weighting is for measuring sound pressure levels above 85 dB.

In order to distinguish the different sound measures a suffix is used: A-weighted sound pressure level is written either as dB_A or L_A. B-weighted sound pressure level is written either as dB_B or L_B, and C-weighted sound pressure level is written either as dB_C or L_C. Unweighted sound pressure level is called "linear sound pressure level" and is often written as dB_L 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. 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 for sound to be comparable to measurements made in a free field environment.

According to the inverse proportional law, when sound level L_p1 is measured at a distance r₁, the sound level L_p2 at the distance r₂ is

L_{p_2} = L_{p_1} + 20 \log_{10}\!\left( \frac{r_1}{r_2} \right)\!~\mathrm{dB}.

Multiple sources

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

L_\Sigma = 10 \log_{10}\!\left(\frac},\quad i = 1,\, 2,\, \ldots,\, n,

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

L_\Sigma = 10 \log_{10}\!\left(10^{\frac{L_1}{10\,\mathrm{dB}}} + 10^{\frac{L_2}{10\,\mathrm{dB}}} + \ldots + 10^{\frac{L_n}{10\,\mathrm{dB}}} \right)\!~\mathrm{dB}.

Examples of sound pressure

*All values listed are the effective sound pressure unless otherwise stated.

See also

• Acoustics
• Phon (unit)
• Loudness
• Sone (unit)
• Sound level meter
• Stevens' power law
• Weber–Fechner law, especially The case of sound

References

General

• Beranek, Leo L., Acoustics (1993), Acoustical Society of America, ISBN 0-88318-494-X.
• Daniel R. Raichel, The Science and Applications of Acoustics (2006), Springer New York, ISBN 1441920803.

External links

• Sound Pressure and Sound Power, Effect and Cause
• Conversion of Sound Pressure to Sound Pressure Level and Vice Versa
• Table of Sound Levels, Corresponding Sound Pressure and Sound Intensity
• Ohm's Law as Acoustic Equivalent, Calculations
• A Table of SPL Values
• Relationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave
• Sound Pressure and Sound Power, Two Commonly Confused Characteristics of Sound
• How Many Decibels Is Twice as Loud? Sound Level Change and the Respective Factor of Sound Pressure or Sound Intensity
• Decibel (Loudness) Comparison Chart