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Particle velocity is the velocity of a particle (real or imagined) in a
medium Medium may refer to: Science and technology Aviation * Medium bomber, a class of war plane * Tecma Medium, a French hang glider design Communication * Media (communication), tools used to store and deliver information or data * Medium ...
as it transmits a
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
. The
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
of particle velocity is the metre per second (m/s). In many cases this is a
longitudinal wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal waves ...
of
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
as with sound, but it can also be a
transverse wave In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example o ...
as with the vibration of a taut string. When applied to a sound wave through a medium of a fluid like air, particle velocity would be the physical speed of a parcel of fluid as it moves back and forth in the direction the sound wave is travelling as it passes. Particle velocity should not be confused with the speed of the
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
as it passes through the medium, i.e. in the case of a sound wave, particle velocity is not the same as the
speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as we ...
. The wave moves relatively fast, while the particles oscillate around their original position with a relatively small particle velocity. Particle velocity should also not be confused with the velocity of individual molecules, which depends mostly on the temperature and molecular mass. In applications involving sound, the particle velocity is usually measured using a logarithmic
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a ...
scale called particle velocity level. Mostly pressure sensors (microphones) are used to measure sound pressure which is then propagated to the velocity field using
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
.


Mathematical definition

Particle velocity, denoted \mathbf v, is defined by :\mathbf v = \frac where \delta is the
particle displacement Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. The SI unit of particle displacement is the metre (m). In m ...
.


Progressive sine waves

The particle displacement of a ''progressive
sine wave A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
'' is given by :\delta(\mathbf,\, t) = \delta_\mathrm \cos(\mathbf \cdot \mathbf - \omega t + \varphi_), where *\delta_\mathrm is the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of the particle displacement; *\varphi_ is the
phase shift In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it ...
of the particle displacement; *\mathbf is the angular wavevector; *\omega is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
. 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,\, t) = \frac = \omega \delta \cos\!\left(\mathbf \cdot \mathbf - \omega t + \varphi_ + \frac\right) = v_\mathrm \cos(\mathbf \cdot \mathbf - \omega t + \varphi_), :p(\mathbf,\, t) = -\rho c^2 \frac = \rho c^2 k_x \delta \cos\!\left(\mathbf \cdot \mathbf - \omega t + \varphi_ + \frac\right) = p_\mathrm \cos(\mathbf \cdot \mathbf - \omega t + \varphi_), where *v_\mathrm is the amplitude of the particle velocity; *\varphi_ is the phase shift of the particle velocity; *p_\mathrm is the amplitude of the acoustic pressure; *\varphi_ is the phase shift of the acoustic pressure. Taking the Laplace transforms of v and p with respect to time yields :\hat(\mathbf,\, s) = v_\mathrm \frac, :\hat(\mathbf,\, s) = p_\mathrm \frac. Since \varphi_ = \varphi_, the amplitude of the specific acoustic impedance is given by :z_\mathrm(\mathbf,\, s) = , z(\mathbf,\, s), = \left, \frac\ = \frac = \frac. Consequently, the amplitude of the particle velocity is related to those of the particle displacement and the sound pressure by :v_\mathrm = \omega \delta_\mathrm, :v_\mathrm = \frac.


Particle velocity level

Sound velocity level (SVL) or acoustic velocity level or particle velocity level is a
logarithmic measure In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used ...
of the effective particle velocity of a sound relative to a reference value.
Sound velocity level, denoted ''L''''v'' and measured in dB, is defined by"Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units"
''IEC 60027-3 Ed. 3.0'', International Electrotechnical Commission, 19 July 2002.
:L_v = \ln\!\left(\frac\right)\!~\mathrm = 2 \log_\!\left(\frac\right)\!~\mathrm = 20 \log_\!\left(\frac\right)\!~\mathrm, where *''v'' is the
root mean square In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
particle velocity; *''v''0 is the ''reference particle velocity''; * is the
neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. A ...
; * is the
bel BEL can be an abbreviation for: * The ISO 3166-1 alpha-3 country code for Belgium * ''BEL'' or bell character in the C0 control code set * Belarusian language, in the ISO 639-2 and SIL country code lists * Bharat Electronics Limited, an Indian sta ...
; * is the
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a ...
. The commonly used reference particle velocity in air isRoss Roeser, Michael Valente, ''Audiology: Diagnosis'' (Thieme 2007), p. 240. :v_0 = 5 \times 10^~\mathrm. The proper notations for sound velocity level using this reference are or , but the notations , , dBSVL, or dBSVL are very common, even though they are not accepted by the SI.Thompson, A. and Taylor, B. N. sec 8.7, "Logarithmic quantities and units: level, neper, bel", ''Guide for the Use of the International System of Units (SI) 2008 Edition'', NIST Special Publication 811, 2nd printing (November 2008), SP81
PDF
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See also

*
Sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
* Sound particle *
Particle displacement Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. The SI unit of particle displacement is the metre (m). In m ...
*
Particle acceleration In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the particle acceleration or sound acceleration with the symbol a in metre/second2. In acoustics or physics, acceleration (symbol: ''a'') is def ...


References

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External links


Ohm's Law as Acoustic Equivalent. CalculationsRelationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound WaveThe particle Velocity Can Be Directly Measured with a MicroflownParticle velocity measured with Weles Acoustics sensor - working principleAcoustic Particle-Image Velocimetry. Development and Applications
Acoustics Sound Sound measurements Physical quantities