Particle displacement or displacement amplitude is a
measurement of
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
of the movement of a
sound particle
In the context of particle displacement and velocity, a sound particle is an imaginary infinitesimal volume of a medium that shares the movement of the medium in response to the presence of sound at a specified point or in a specified region. Sou ...
from its
equilibrium position in a medium as it transmits a sound wave.
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 displacement is the
metre
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its prefi ...
(m). In most 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 (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'' b ...
), 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 of t ...
, such as the
vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, su ...
of a taut string. In the case of a
sound wave travelling through
air
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
, 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 held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
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
Particle velocity is the velocity of a particle (real or imagined) in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can ...
of the sound wave traveling through the medium, while the sound wave itself moves at the
speed of sound, equal to in air at .
Mathematical definition
Particle displacement, denoted
δ, is given by
[
]
:
where v is the
particle velocity
Particle velocity is the velocity of a particle (real or imagined) in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can ...
.
Progressive sine waves
The particle displacement of a ''progressive
sine wave'' is given by
:
where
*
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;
*
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;
*
is the
angular wavevector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
;
*
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 tim ...
.
It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave ''x'' are given by
:
:
where
*
is the amplitude of the particle velocity;
*
is the phase shift of the particle velocity;
*
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 those of the particle velocity and the sound pressure by
:
:
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'' b ...
*
Sound particle
In the context of particle displacement and velocity, a sound particle is an imaginary infinitesimal volume of a medium that shares the movement of the medium in response to the presence of sound at a specified point or in a specified region. Sou ...
*
Particle velocity
Particle velocity is the velocity of a particle (real or imagined) in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can ...
*
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 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