Sound pressure


Sound pressure or acoustic pressure is the local pressure deviation from the ambient 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.

Mathematical definition

A sound wave in a transmission medium causes a deviation in the local ambient pressure, a static pressure.
Sound pressure, denoted p, is defined by
where
  • ptotal is the total pressure,
  • pstat 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
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
where
Specific acoustic impedance, denoted z and measured in Pa·m−1·s in SI units, is defined by
where
  • is the Laplace transform of sound pressure,
  • is the Laplace transform of particle velocity.

    Particle displacement

The particle displacement of a progressive sine wave is given by
where
It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave x are given by
where
  • vm is the amplitude of the particle velocity,
  • is the phase shift of the particle velocity,
  • pm is the amplitude of the acoustic pressure,
  • is the phase shift of the acoustic pressure.
Taking the Laplace transforms of v and p with respect to time yields
Since, the amplitude of the specific acoustic impedance is given by
Consequently, the amplitude of the particle displacement is related to that of the acoustic velocity and the sound pressure by

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 :
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:
The inverse-proportional law for sound pressure comes from the inverse-square law for sound intensity:
Indeed,
where
  • is the particle velocity,
  • is the convolution operator,
  • z−1 is the convolution inverse of the specific acoustic impedance,
hence the inverse-proportional law:

Sound pressure level

Sound pressure level or acoustic pressure level is a logarithmic measure of the effective pressure of a sound relative to a reference value.
Sound pressure level, denoted Lp and measured in dB, is defined by:
where
The commonly used reference sound pressure in air is
which is often considered as the threshold of human hearing. The proper notations for sound pressure level using this reference are or, but the suffix notations,, dBSPL, and dBSPL 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 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 underwater or through the Earth.
Ears detect changes in sound pressure. Human hearing does not have a flat spectral sensitivity 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 dBA or LA, B-weighted sound pressure level is written either as dBB or LB, and C-weighted sound pressure level is written either as dBC or LC. Unweighted sound pressure level is called "linear sound pressure level" and is often written as dBL 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 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 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 Lp1 is measured at a distance r1, the sound level Lp2 at the distance r2 is

Multiple sources

The formula for the sum of the sound pressure levels of n incoherent radiating sources is
Inserting the formulas
in the formula for the sum of the sound pressure levels yields

Examples of sound pressure