Sound 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,
• ${\displaystyle {\hat {Q}}(s)}$ 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

$${\displaystyle \delta _{\text{m}}={\frac {v_{\text{m}}}{\omega }},}$$

$${\displaystyle \delta _{\text{m}}={\frac {p_{\text{m}}}{\omega z_{\text{m}}(\mathbf {r} ,s)}}.}$$

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)[10][11] 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.[12]

Equal-loudness contour, showing sound-pressure-vs-frequency at different perceived loudness levels

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.[12]

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.[12]

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

$${\displaystyle L_{\Sigma }=10\log _{10}\left(10^{\frac {L_{1}}{10~{\text{dB}}}}+10^{\frac {L_{2}}{10~{\text{dB}}}}+\dots +10^{\frac {L_{n}}{10~{\text{dB}}}}\right)~{\text{dB}}.}$$