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 $${\displaystyle p_{\text{total}}=p_{\text{stat}}+p,}$$ where

• p_total is the total pressure,
• p_stat is the static pressure.

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 $${\displaystyle \mathbf {I} =p\mathbf {v} ,}$$ where

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

Acoustic impedance, denoted Z and measured in Pa·m⁻³·s in SI units, is defined by^[2] $${\displaystyle Z(s)={\frac {{\hat {p}}(s)}{{\hat {Q}}(s)}},}$$ where

• ${\displaystyle {\hat {p}}(s)}$ is the Laplace transform of sound pressure,^{[citation needed]}
• ${\displaystyle {\hat {Q}}(s)}$ 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] $${\displaystyle z(s)={\frac {{\hat {p}}(s)}{{\hat {v}}(s)}},}$$ where

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

The particle displacement of a progressive sine wave is given by $${\displaystyle \delta (\mathbf {r} ,t)=\delta _{\text{m}}\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}),}$$ where

• ${\displaystyle \delta _{\text{m}}}$ is the amplitude of the particle displacement,
• ${\displaystyle \varphi _{\delta ,0}}$ 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 $${\displaystyle v(\mathbf {r} ,t)={\frac {\partial \delta }{\partial t}}(\mathbf {r} ,t)=\omega \delta _{\text{m}}\cos \left(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}+{\frac {\pi }{2}}\right)=v_{\text{m}}\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{v,0}),}$$ $${\displaystyle p(\mathbf {r} ,t)=-\rho c^{2}{\frac {\partial \delta }{\partial x}}(\mathbf {r} ,t)=\rho c^{2}k_{x}\delta _{\text{m}}\cos \left(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}+{\frac {\pi }{2}}\right)=p_{\text{m}}\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{p,0}),}$$ where

• v_m is the amplitude of the particle velocity,
• ${\displaystyle \varphi _{v,0}}$ is the phase shift of the particle velocity,
• p_m is the amplitude of the acoustic pressure,
• ${\displaystyle \varphi _{p,0}}$ is the phase shift of the acoustic pressure.

Taking the Laplace transforms of v and p with respect to time yields $${\displaystyle {\hat {v}}(\mathbf {r} ,s)=v_{\text{m}}{\frac {s\cos \varphi _{v,0}-\omega \sin \varphi _{v,0}}{s^{2}+\omega ^{2}}},}$$ $${\displaystyle {\hat {p}}(\mathbf {r} ,s)=p_{\text{m}}{\frac {s\cos \varphi _{p,0}-\omega \sin \varphi _{p,0}}{s^{2}+\omega ^{2}}}.}$$

Since ${\displaystyle \varphi _{v,0}=\varphi _{p,0}}$, the amplitude of the specific acoustic impedance is given by $${\displaystyle z_{\text{m}}(\mathbf {r} ,s)=|z(\mathbf {r} ,s)|=\left|{\frac {{\hat {p}}(\mathbf {r} ,s)}{{\hat {v}}(\mathbf {r} ,s)}}\right|={\frac {p_{\text{m}}}{v_{\text{m}}}}={\frac {\rho c^{2}k_{x}}{\omega }}.}$$

Consequently, the amplitude of the particle displacement is related to that of the acoustic velocity and the sound pressure by $${\displaystyle \delta _{\text{m}}={\frac {v_{\text{m}}}{\omega }},}$$ $${\displaystyle \delta _{\text{m}}={\frac {p_{\text{m}}}{\omega z_{\text{m}}(\mathbf {r} ,s)}}.}$$

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):^[3] $${\displaystyle p(r)\propto {\frac {1}{r}}.}$$

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: $${\displaystyle p_{2}={\frac {r_{1}}{r_{2}}}\,p_{1}.}$$

The inverse-proportional law for sound pressure comes from the inverse-square law for sound intensity: $${\displaystyle I(r)\propto {\frac {1}{r^{2}}}.}$$ Indeed, $${\displaystyle I(r)=p(r)v(r)=p(r)\left[p*z^{-1}\right](r)\propto p^{2}(r),}$$ where

• ${\displaystyle v}$ is the particle velocity,
• ${\displaystyle *}$ is the convolution operator,
• z⁻¹ is the convolution inverse of the specific acoustic impedance,

hence the inverse-proportional law: $${\displaystyle p(r)\propto {\frac {1}{r}}.}$$

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 L_p and measured in dB,^[4] is defined by:^[5] $${\displaystyle L_{p}=\ln \left({\frac {p}{p_{0}}}\right)~{\text{Np}}=2\log _{10}\left({\frac {p}{p_{0}}}\right)~{\text{B}}=20\log _{10}\left({\frac {p}{p_{0}}}\right)~{\text{dB}},}$$ where

• p is the root mean square sound pressure,^[6]
• p₀ is a reference sound pressure,
• 1 Np is the neper,
• 1 B = (⁠1/2⁠ ln 10) Np is the bel,
• 1 dB = (⁠1/20⁠ ln 10) Np is the decibel.

The commonly used reference sound pressure in air is^[7]

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, and dB_SPL are very common, even if they are not accepted by the SI.^[8]

Most sound-level measurements will be made relative to this reference, meaning 1 Pa will equal an SPL of ${\displaystyle 20\log _{10}\left({\frac {1}{2\times 10^{-5}}}\right)~{\text{dB}}\approx 94~{\text{dB}}}$. In other media, such as underwater, a reference level of 1 μPa is used.^[9] These references are defined in ANSI S1.1-2013.^[10]

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.

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)^[11][12] 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.^[13]

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 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.^[13]

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.^[13]

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 $${\displaystyle L_{p_{2}}=L_{p_{1}}+20\log _{10}\left({\frac {r_{1}}{r_{2}}}\right)~{\text{dB}}.}$$

The formula for the sum of the sound pressure levels of n incoherent radiating sources is $${\displaystyle L_{\Sigma }=10\log _{10}\left({\frac {p_{1}^{2}+p_{2}^{2}+\dots +p_{n}^{2}}{p_{0}^{2}}}\right)~{\text{dB}}=10\log _{10}\left[\left({\frac {p_{1}}{p_{0}}}\right)^{2}+\left({\frac {p_{2}}{p_{0}}}\right)^{2}+\dots +\left({\frac {p_{n}}{p_{0}}}\right)^{2}\right]~{\text{dB}}.}$$

Inserting the formulas $${\displaystyle \left({\frac {p_{i}}{p_{0}}}\right)^{2}=10^{\frac {L_{i}}{10~{\text{dB}}}},\quad i=1,2,\ldots ,n}$$ in the formula for the sum of the sound pressure levels yields $${\displaystyle L_{\Sigma }=10\log _{10}\left(10^{\frac {L_{1}}{10~{\text{dB}}}}+10^{\frac {L_{2}}{10~{\text{dB}}}}+\dots +10^{\frac {L_{n}}{10~{\text{dB}}}}\right)~{\text{dB}}.}$$

• Acoustics – Branch of physics involving mechanical waves
• Phon – Logarithmic unit of loudness level
• Loudness – Subjective perception of sound pressure
• Sone – Unit of perceived loudness
• Sound level meter – Device for acoustic measurements
• Stevens's power law – Empirical relationship between actual and perceived changed intensity of stimulus
• Weber–Fechner law – Related laws in the field of psychophysics