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Particle displacement or displacement amplitude is a
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared to ...
of
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. The
SI unit The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of units of measurement, system of measurement. It is the only system ...
of particle displacement is the
metre The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
(m). In most cases this is a longitudinal wave of pressure (such as
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 the br ...
), but it can also be a transverse wave, such as the
vibration Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the os ...
of a taut string. In the case of a
sound wave 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 the ...
travelling through air, the particle displacement is evident in the
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
s of air
molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s with, and against, the direction in which the sound wave is travelling. A particle of the medium undergoes displacement according to the particle velocity of the sound wave traveling through the medium, while the sound wave itself moves at 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 elasticity (solid mechanics), elastic medium. More simply, the speed of sound is how fast vibrations travel. At , the speed of sound in a ...
, equal to in air at .


Mathematical definition

Particle displacement, denoted δ, is given by :\mathbf \delta = \int_ \mathbf v\, \mathrmt where v is the particle velocity.


Progressive sine waves

The particle displacement of a ''progressive
sine wave A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic function, periodic wave whose waveform (shape) is the trigonometric function, trigonometric sine, sine function. In mechanics, as a linear motion over time, this is ''simple ...
'' is given by :\delta(\mathbf,\, t) = \delta \sin(\mathbf \cdot \mathbf - \omega t + \varphi_), where *\delta is the amplitude of the particle displacement; *\varphi_ is the phase shift of the particle displacement; *\mathbf is the angular wavevector; *\omega is the
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
. 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 \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 \cos(\mathbf \cdot \mathbf - \omega t + \varphi_), where *v is the amplitude of the particle velocity; *\varphi_ is the phase shift of the particle velocity; *p 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 \frac, :\hat(\mathbf,\, s) = p \frac. Since \varphi_ = \varphi_, the amplitude of the specific acoustic impedance is given by :z(\mathbf,\, s) = , z(\mathbf,\, s), = \left, \frac\ = \frac = \frac. Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by :\delta = \frac, :\delta = \frac.


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 the br ...
* Sound particle * Particle velocity * Particle acceleration


References and notes

Related Reading: * * *{{cite book , last = Barron , first = Randall F. , title = Industrial noise control and acoustics , publisher = CRC Press , date = January 2003 , location = NYC, New York , pages = 79, 82, 83, 87 , url =https://books.google.com/books?id=k1tXPl2hC-cC&q=instantaneous+particle+displacement&pg=PA82 , isbn =978-0-8247-0701-9


External links


Acoustic Particle-Image Velocimetry. Development and ApplicationsRelationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave
Acoustics Sound Sound measurements Physical quantities