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). 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 formula_1 where 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 in SI units, is defined by formula_2 where Acoustic impedance. "Acoustic impedance", denoted "Z" and measured in Pa·m·s in SI units, is defined by formula_3 where "Specific acoustic impedance", denoted "z" and measured in Pa·m·s in SI units, is defined by formula_6 where Particle displacement. The "particle displacement" of a "progressive sine wave" is given by formula_9 where It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave "x" are given  by formula_12 formula_13 where Taking the Laplace transforms of "v" and "p" with respect to time yields formula_16 formula_17 Since formula_18, the amplitude of the specific acoustic impedance is given by formula_19 Consequently, the amplitude of the particle displacement is related to that of the acoustic velocity and the sound pressure by formula_20 formula_21 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/"r", like the sound intensity): formula_22 This relationship is an "inverse-proportional law". If the sound pressure "p" is measured at a distance "r" from the centre of the sphere, the sound pressure "p" at another position "r" can be calculated: formula_23 The inverse-proportional law for sound pressure comes from the inverse-square law for sound intensity: formula_24 Indeed, formula_25 where hence the inverse-proportional law: formula_22 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" and measured in dB, is defined by formula_28 where The commonly used reference sound pressure in air is 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, or dB are very common, even if they are not accepted by the SI. 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 dB or L. B-weighted sound pressure level is written either as dB or L, and C-weighted sound pressure level is written either as dB or L. Unweighted sound pressure level is called "linear sound pressure  level" and is often written as dB 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 square law, which summarily states that doubling the distance between the source and receiver results in dividing the measurable effect by four. 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 "L" is measured at a distance "r", the sound level "L" at the distance "r" is formula_29 Multiple sources. The formula for the sum of the sound pressure levels  of "n" incoherent radiating sources is formula_30 Inserting the formulas formula_31 in the formula for the sum of the sound pressure levels yields formula_32