Particle displacement
Encyclopedia
Particle displacement or particle amplitude (represented in mathematics
by the lower-case Greek letter ξ) is a measurement
of distance
of the movement of a particle from its equilibrium position in a medium as it transmits a wave
.
In most cases this is a longitudinal wave
of pressure (such as sound
), but it can also be a transverse wave
, such as the vibration
of a taut string. In the case of a sound wave travelling through air, the particle displacement is evident in the oscillation
s of air molecule
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 wave traveling through the medium, while the sound wave itself moves at the speed of sound
, equal to 343 m/s in air at 20 °C.
The instantaneous particle displacement ξ for a wave is:
If the wave is a standing wave or a traveling wave containing a single frequency, the particle displacement is:
This expression for
undergoes simple harmonic oscillation, and as such is usually expressed as an rms
time average.
Particle displacement for a traveling wave containing a single frequency can be represented in terms of other measurements:
where in the above equation, the quantities
may be taken throughout as rms time-averages (or all as maximum values). The single frequency traveling wave has acoustic impedance equal to the characteristic impedance, 
. Further representations for 
can be found from the above equations using the replacement 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
by the lower-case Greek letter ξ) is a measurement
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...
of distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
of the movement of a particle from its equilibrium position in a medium as it transmits a wave
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
.
In most cases this is a longitudinal wave
Longitudinal wave
Longitudinal waves, as known as "l-waves", are waves that have the same direction of vibration as their direction of travel, which means that the movement of the medium is in the same direction as or the opposite direction to the motion of the wave. Mechanical longitudinal waves have been also...
of pressure (such as sound
Sound
Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...
), but it can also be a transverse wave
Transverse wave
A transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer...
, such as the vibration
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
of a taut string. In the case of a sound wave travelling through air, the particle displacement is evident in the oscillation
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
s of air molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
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
Particle velocity is the velocity v of a particle in a medium as it transmits a wave. In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string....
of the wave traveling through the medium, while the sound wave itself moves at the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
, equal to 343 m/s in air at 20 °C.
The instantaneous particle displacement ξ for a wave is:
If the wave is a standing wave or a traveling wave containing a single frequency, the particle displacement is:
This expression for
undergoes simple harmonic oscillation, and as such is usually expressed as an rmsRMS
-Science and technology:* Root mean square, a concept encapsulating the "average", in some sense, of a quantity. Frequently encountered in statistics, the physical sciences and electronics...
time average.
Particle displacement for a traveling wave containing a single frequency can be represented in terms of other measurements:
where in the above equation, the quantities
may be taken throughout as rms time-averages (or all as maximum values). The single frequency traveling wave has acoustic impedance equal to the characteristic impedance, . Further representations for can be found from the above equations using the replacement .| Symbol | Units | Meaning |
|---|---|---|
| ξ | m, meters | Particle displacement |
| v | m/s | particle velocity Particle velocity Particle velocity is the velocity v of a particle in a medium as it transmits a wave. In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string.... |
| ω = 2πf | radians/s Second The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock.... |
angular frequency Angular frequency In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity... |
| f | Hz, hertz Hertz The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications.... |
frequency Frequency Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency... |
| p | Pa, pascal Pascal (unit) The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre... s |
sound pressure Sound pressure Sound pressure or acoustic pressure is the local pressure deviation from the ambient atmospheric pressure caused by a sound wave. Sound pressure can be measured using a microphone in air and a hydrophone in water... |
| Z0 = c · ρ | N·s Second The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock.... /m3 |
characteristic impedance Characteristic impedance The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm... |
| Z = p / v | N·s Second The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock.... /m3 |
acoustic impedance Acoustic impedance The acoustic impedance at a particular frequency indicates how much sound pressure is generated by a given air vibration at that frequency. The acoustic impedance Z is frequency dependent and is very useful, for example, for describing the behaviour of musical wind instruments... |
| c | m/s Second The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock.... |
Speed of sound Speed of sound The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds.... |
| ρ | kg Kilogram The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water... /m3 |
Density of air Density of air The density of air, ρ , is the mass per unit volume of Earth's atmosphere, and is a useful value in aeronautics and other sciences. Air density decreases with increasing altitude, as does air pressure. It also changes with variances in temperature or humidity... |
| I | W Watt The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:... /m2 |
sound intensity Sound intensity Sound intensity or acoustic intensity is defined as the sound power Pac per unit area A. The usual context is the noise measurement of sound intensity in the air at a listener's location.-Acoustic intensity:... |
| E | W Watt The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:... ·s Second The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock.... /m3 |
sound energy density Sound energy density The sound energy density or sound density is an adequate measure to describe the sound field at a given point as a sound energy value... |
| Pac | W, watt Watt The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:... s |
sound power or acoustic power |
| A | m2 | Area Area Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat... |
| a | m/s Second The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock.... 2 |
Particle acceleration 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 is defined as the rate of change of velocity. It is thus a vector... |