Phase in sinusoidal functions or in waves has two different, but closely related meanings. One is the initial angle of a sinusoidal function at its origin and is sometimes called phase offset. Another usage is the fraction of the wave cycle which has elapsed relative to the origin.^[1]

Formula[link]

The phase of an oscillation or wave refers to a sinusoidal function such as the following:

Failed to parse (Missing texvc executable; please see math/README to configure.): x(t) = A \ \cos( 2 \pi f t + \phi )\,

Failed to parse (Missing texvc executable; please see math/README to configure.): y(t) = A \ \sin( 2 \pi f t + \phi ) = A\cdot \cos( 2 \pi f t + \phi - \pi/2),\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): A\, , Failed to parse (Missing texvc executable; please see math/README to configure.): f\, , and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

are constant parameters called the amplitude, frequency, and phase of the sinusoid.  These functions are periodic with period Failed to parse (Missing texvc executable; please see math/README to configure.): T = 1/f\,

, and they are identical except for a displacement of Failed to parse (Missing texvc executable; please see math/README to configure.): T/4\,

along the Failed to parse (Missing texvc executable; please see math/README to configure.): t\,
axis.  The term phase can refer to several different things:

• It can refer to a specified reference, such as Failed to parse (Missing texvc executable; please see math/README to configure.): \cos( 2 \pi f t)\,

, in which case we would say the phase of Failed to parse (Missing texvc executable; please see math/README to configure.): x(t)\,

is Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

, and the phase of Failed to parse (Missing texvc executable; please see math/README to configure.): y(t)\,

is Failed to parse (Missing texvc executable; please see math/README to configure.): \phi - \pi/2\,

.

• It can refer to Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

, in which case we would say Failed to parse (Missing texvc executable; please see math/README to configure.): x(t)\,

and Failed to parse (Missing texvc executable; please see math/README to configure.): y(t)\,
have the same phase but are relative to their own specific references.

• In the context of communication waveforms, the time-variant angle  Failed to parse (Missing texvc executable; please see math/README to configure.): 2 \pi f t + \phi,\,

  or its modulo Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi

value, is referred to as instantaneous phase, often just phase.  

Phase shift[link]

Illustration of phase shift. The horizontal axis represents an angle (phase) that is increasing with time.

Phase shift is any change that occurs in the phase of one quantity, or in the phase difference between two or more quantities.^[1]

Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

is sometimes referred to as a phase shift or phase offset, because it represents a "shift" from zero phase.  But a change in Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,
is also referred to as a phase shift.  

For infinitely long sinusoids, a change in Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

is the same as a shift in time, such as a time delay. If Failed to parse (Missing texvc executable; please see math/README to configure.): x(t)\,
is delayed (time-shifted) by Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix} \frac{1}{4} \end{matrix}\,
of its cycle, it becomes:

Failed to parse (Missing texvc executable; please see math/README to configure.): x(t - \begin{matrix} \frac{1}{4} \end{matrix}T) \,	Failed to parse (Missing texvc executable; please see math/README to configure.): = A\cdot \cos(2 \pi f (t - \begin{matrix} \frac{1}{4} \end{matrix}T) + \phi) \,
	Failed to parse (Missing texvc executable; please see math/README to configure.): = A\cdot \cos(2 \pi f t - \begin{matrix}\frac{\pi }{2} \end{matrix} + \phi ),\,

whose "phase" is now Failed to parse (Missing texvc executable; please see math/README to configure.): \phi - \begin{matrix}\frac{\pi }{2} \end{matrix}.

It has been shifted by Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix}\frac{\pi }{2} \end{matrix}

radians.

[edit] Phase difference

In-phase waves

Out-of-phase waves

Left: the real part of a plane wave moving from top to bottom. Right: the same wave after a central section underwent a phase shift, for example, by passing through a glass of different thickness than the other parts. (The illustration on the right ignores the effect of diffraction, which would make the waveform continuous away from material interfaces and would add increasing distortions with distance.).

Phase difference is the difference, expressed in electrical degrees or time, between two waves having the same frequency and referenced to the same point in time.^[1] Two oscillators that have the same frequency and different phases have a phase difference, and the oscillators are said to be out of phase with each other. The amount by which such oscillators are out of step with each other can be expressed in degrees from 0° to 360°, or in radians from 0 to 2π. If the phase difference is 180 degrees (π radians), then the two oscillators are said to be in antiphase. If two interacting waves meet at a point where they are in antiphase, then destructive interference will occur. It is common for waves of electromagnetic (light, RF), acoustic (sound) or other energy to become superposed in their transmission medium. When that happens, the phase difference determines whether they reinforce or weaken each other. Complete cancellation is possible for waves with equal amplitudes.

Time is sometimes used (instead of angle) to express position within the cycle of an oscillation.

• A phase difference is analogous to two athletes running around a race track at the same speed and direction but starting at different positions on the track. They pass a point at different instants in time. But the time difference (phase difference) between them is a constant - same for every pass since they are at the same speed and in the same direction. If they were at different speeds (different frequencies), the phase difference is undefined and would only reflect different starting positions. Technically, phase difference between two entities at various frequencies is undefined and does not exist.
• Time zones are also analogous to phase differences.

In-phase and quadrature (I&Q) components[link]

The term in-phase is also found in the context of communication signals:

Failed to parse (Missing texvc executable; please see math/README to configure.): A(t)\cdot \sin[2\pi ft + \phi(t)] = I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \underbrace{\sin\left(2\pi ft + \begin{matrix} \frac{\pi}{2} \end{matrix} \right)}_{\cos(2\pi ft)}

and:

Failed to parse (Missing texvc executable; please see math/README to configure.): A(t)\cdot \cos[2\pi ft + \phi(t)] = I(t)\cdot \cos(2\pi ft) + Q(t)\cdot \underbrace{\cos\left(2\pi ft + \begin{matrix} \frac{\pi}{2} \end{matrix}\right)}_{-\sin(2\pi ft)},

where Failed to parse (Missing texvc executable; please see math/README to configure.): \ f\,

represents a carrier frequency, and

Failed to parse (Missing texvc executable; please see math/README to configure.): I(t)\ \stackrel{\text{def}}{=}\ A(t)\cdot \cos\left(\phi(t)\right), \,

Failed to parse (Missing texvc executable; please see math/README to configure.): Q(t)\ \stackrel{\text{def}}{=}\ A(t)\cdot \sin\left(\phi(t)\right).\,

Failed to parse (Missing texvc executable; please see math/README to configure.): A(t)\,

and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi(t)\,
represent possible modulation of a pure carrier wave, e.g.:  Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(2\pi ft)\,
(or Failed to parse (Missing texvc executable; please see math/README to configure.): \cos(2\pi ft).\,

) The modulation alters the original Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\,

(or Failed to parse (Missing texvc executable; please see math/README to configure.): \cos\,

) component of the carrier, and creates a (new) Failed to parse (Missing texvc executable; please see math/README to configure.): \cos\,

(or Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\,

) component, as shown above. The component that is in phase with the original carrier is referred to as the in-phase component. The other component, which is always 90° (Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix} \frac{\pi}{2} \end{matrix}

radians) "out of phase", is referred to as the quadrature component.

Phase coherence[link]

Coherence is the quality of a wave to display well defined phase relationship in different regions of its domain of definition.

In physics, quantum mechanics ascribes waves to physical objects. The wave function is complex and since its square modulus is associated with the probability of observing the object, the complex character of the wave function is associated to the phase. Since the complex algebra is responsible for the striking interference effect of quantum mechanics, phase of particles is therefore ultimately related to their quantum behavior.

Phase compensation[link]

Phase compensation is the correction of phase error (i.e., the difference between the actually needed phase and the obtained phase). A phase compensation is required to obtain stability in an opamp. A capacitor/RC network is usually used in the phase compensation to keep a phase margin. A phase compensator subtracts out an amount of phase shift from a signal which is equal to the amount of phase shift added by switching one or more additional amplifier stages into the amplification signal path.

See also[link]

• Instantaneous phase
• Lissajous curve
• Phase angle
• Phase cancellation
• Phase problem
• Phase velocity
• Polarity
• Polarization

References[link]

External links[link]

Wikimedia Commons has media related to: Phase (waves)

• Relationship of phase difference and time-delay
• Phase angle, phase difference, time delay, and frequency
• ECE 209: Sources of Phase Shift — Discusses the time-domain sources of phase shift in simple linear time-invariant circuits.
• Phase Difference Java Applet

Look up phase in Wiktionary, the free dictionary.

Phase or phases may refer to:

In science[link]

In physics[link]

• Phase (matter), a physically distinctive form of a substance, such as the solid, liquid, and gaseous states of ordinary matter
  • Phase transition is the transformation of a thermodynamic system from one phase to another
• Phase (separation science)
  • Mobile phase and Stationary phase in chromatography
  • Aqueous phase and organic phase in liquid-liquid extraction
• Phase (waves), the initial angle of a sinusoidal function at its origin
• Polyphase system, a means of distributing alternating current electric power in multiple conducting wires with definite phase offsets
  • Single-phase electric power
  • Three-phase electric power
    • Three-phase, the mathematics of three-phase electric power
• Phase problem, the loss of information (the phase) from a physical measurement
• Phase factor, a complex scalar used in quantum mechanics
• Continuous Fourier transform, the angle of a complex coefficient representing the phase of one sinusoidal component
• Instantaneous phase, generalization for both cyclic and non-cyclic phenomena

In other sciences[link]

• Archaeological phase, a discrete period of occupation at an archaeological site
• Lunar phase, the appearance of the Moon as viewed from the Earth
• Planetary phase, the appearance of the illuminated section of a planet
• In biology, a part of the cell cycle in which cells divide and reproduce
• Color phase, in biology, a group of individuals within a species with a particular coloration
• Phase (syntax), in linguistics, a cyclic domain (proposed by Noam Chomsky)

In entertainment[link]

• Phase (band), an alternative rock band
• "Phases", a box set of albums by The Who
• "Phases" (Buffy episode), an episode of the TV series Buffy the Vampire Slayer
• Phases (.hack), fictional monsters in the .hack franchise
• Phase IV, a 1974 science fiction movie directed by Saul Bass
• Phase, an incarnation of the DC Comics character usually known as Phantom Girl
• Phaze, an alternate world in Piers Anthony's Apprentice Adept series

Other uses[link]

• Phase 10, a card game created by Fundex Games
• Phase (video game), a music game for the iPod developed by Harmonix Music Systems
• Phase (combat) usually a period of combat within a larger military operation
• A musical composition using Steve Reich's phasing technique

See also[link]

• Phasing (disambiguation)
• Faze (disambiguation)
• Phaser (disambiguation)
• Phasor (disambiguation)
• FASOR (disambiguation)

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

Surface waves in water

In physics, a wave is a disturbance or oscillation that travels through spacetime, accompanied by a transfer of energy. 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 transport. They consist, instead, of oscillations or vibrations around almost fixed locations. Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave.

There are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed. The deformation reverses itself owing to restoring forces resulting from its deformation. For example, sound waves propagate via air molecules colliding with their neighbors. When air molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave.

The second type of wave, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations in electrical and magnetic fields generated by charged particles, and can therefore travel through a vacuum. These types of waves vary in wavelength, and include radio waves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.

A wave can be transverse or longitudinal depending on the direction of its oscillation. Transverse waves occur when a disturbance creates oscillations perpendicular (at right angles) to the propagation (the direction of energy transfer). Longitudinal waves occur when the oscillations are parallel to the direction of propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse.

General features[link]

A single, all-encompassing definition for the term wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value. However, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon to be called a wave results in a fuzzy border line.

The term wave is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium (Hall 1980, p. 8). However, this notion is problematic for a standing wave (for example, a wave on a string), where energy is moving in both directions equally, or for electromagnetic (e.g., light) waves in a vacuum, where the concept of medium does not apply and interaction with a target is the key to wave detection and practical applications. There are water waves on the ocean surface; gamma waves and light waves emitted by the Sun; microwaves used in microwave ovens and in radar equipment; radio waves broadcast by radio stations; and sound waves generated by radio receivers, telephone handsets and living creatures (as voices), to mention only a few wave phenomena.

It may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a mechanical rather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, or elasticity, become therefore crucial in describing acoustic (as distinct from optic) wave processes. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved. For example, in the case of air: vortices, radiation pressure, shock waves etc.; in the case of solids: Rayleigh waves, dispersion; and so on.

Other properties, however, although usually described in terms of origin, may be generalized to all waves. For such reasons, wave theory represents a particular branch of physics that is concerned with the properties of wave processes independently of their physical origin.^[1] For example, based on the mechanical origin of acoustic waves, a moving disturbance in space–time can exist if and only if the medium involved is neither infinitely stiff nor infinitely pliable. If all the parts making up a medium were rigidly bound, then they would all vibrate as one, with no delay in the transmission of the vibration and therefore no wave motion. This is impossible because it would violate general relativity. On the other hand, if all the parts were independent, then there would not be any transmission of the vibration and again, no wave motion. Although the above statements are meaningless in the case of waves that do not require a medium, they reveal a characteristic that is relevant to all waves regardless of origin: within a wave, the phase of a vibration (that is, its position within the vibration cycle) is different for adjacent points in space because the vibration reaches these points at different times.

Similarly, wave processes revealed from the study of waves other than sound waves can be significant to the understanding of sound phenomena. A relevant example is Thomas Young's principle of interference (Young, 1802, in Hunt 1992, p. 132). This principle was first introduced in Young's study of light and, within some specific contexts (for example, scattering of sound by sound), is still a researched area in the study of sound.

Mathematical description of one-dimensional waves[link]

Wave equation[link]

Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling

Wavelength λ, can be measured between any two corresponding points on a waveform

• in the Failed to parse (Missing texvc executable; please see math/README to configure.): x

direction in space. E.g., let the positive Failed to parse (Missing texvc executable; please see math/README to configure.): x
direction be to the right, and the negative Failed to parse (Missing texvc executable; please see math/README to configure.): x
direction be to the left.

• with constant amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): u

• with constant velocity Failed to parse (Missing texvc executable; please see math/README to configure.): v

, where Failed to parse (Missing texvc executable; please see math/README to configure.): v

is

• • independent of wavelength (no dispersion)
  • independent of amplitude (linear media, not nonlinear).^[2]
• with constant waveform, or shape

This wave can then be described by the two-dimensional functions

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t) = F(x - v \ t)

 (waveform Failed to parse (Missing texvc executable; please see math/README to configure.): F
traveling to the right)

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t) = G(x + v \ t)

 (waveform Failed to parse (Missing texvc executable; please see math/README to configure.): G
traveling to the left)

or, more generally, by d'Alembert's formula:^[3]

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x,t)=F(x-vt)+G(x+vt). \,

representing two component waveforms Failed to parse (Missing texvc executable; please see math/README to configure.): F

and Failed to parse (Missing texvc executable; please see math/README to configure.): G
traveling through the medium in opposite directions. This wave can also be represented by the partial differential equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{v^2}\frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}. \,

General solutions are based upon Duhamel's principle.^[4]

Wave forms[link]

Sine, square, triangle and sawtooth waveforms.

The form or shape of F in d'Alembert's formula involves the argument x − vt. Constant values of this argument correspond to constant values of F, and these constant values occur if x increases at the same rate that vt increases. That is, the wave shaped like the function F will move in the positive x-direction at velocity v (and G will propagate at the same speed in the negative x-direction).^[5]

In the case of a periodic function F with period λ, that is, F(x + λ − vt) = F(x − vt), the periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying periodically in space with period λ (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F(x − v(t + T)) = F(x − vt) provided vT = λ, so an observation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v.^[6]

Amplitude and modulation[link]

Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.

The amplitude of a wave may be constant (in which case the wave is a c.w. or continuous wave), or may be modulated so as to vary with time and/or position. The outline of the variation in amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written in the form:^[7][8][9]

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t) = A(x, \ t)\sin (kx - \omega t + \phi) \ ,

where Failed to parse (Missing texvc executable; please see math/README to configure.): A(x,\ t)

is the amplitude envelope of the wave, Failed to parse (Missing texvc executable; please see math/README to configure.): k
is the wavenumber and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi
is the phase. If the group velocity Failed to parse (Missing texvc executable; please see math/README to configure.): v_g
(see below) is wavelength-independent, this equation can be simplified as:[10]

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t) = A(x - v_g \ t)\sin (kx - \omega t + \phi) \ ,

showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the group velocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation.^[10][11]

Phase velocity and group velocity[link]

Frequency dispersion in groups of gravity waves on the surface of deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity.

There are two velocities that are associated with waves, the phase velocity and the group velocity. To understand them, one must consider several types of waveform. For simplification, examination is restricted to one dimension.

This shows a wave with the Group velocity and Phase velocity going in different directions.

The most basic wave (a form of plane wave) may be expressed in the form:

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi (x, \ t) = A e^{i \left( kx - \omega t \right)} \ ,

which can be related to the usual sine and cosine forms using Euler's formula. Rewriting the argument, Failed to parse (Missing texvc executable; please see math/README to configure.): kx-\omega t = \left(\frac{2\pi}{\lambda}\right)(x - vt) , makes clear that this expression describes a vibration of wavelength Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda = \frac{2\pi}{k}

traveling in the x-direction with a constant phase velocity Failed to parse (Missing texvc executable; please see math/README to configure.): v_p = \frac{\omega}{k}\,

.^[12]

The other type of wave to be considered is one with localized structure described by an envelope, which may be expressed mathematically as, for example:

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi (x, \ t) = \int_{-\infty} ^{\infty}\ dk_1 \ A(k_1)\ e^{i\left(k_1x - \omega t \right)} \ ,

where now A(k₁) (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peak in a region of wave vectors Δk surrounding the point k₁ = k. In exponential form:

Failed to parse (Missing texvc executable; please see math/README to configure.): A = A_o (k_1) e^ {i \alpha (k_1)} \ ,

with A_o the magnitude of A. For example, a common choice for A_o is a Gaussian wave packet:^[13]

Failed to parse (Missing texvc executable; please see math/README to configure.): A_o (k_1) = N\ e^{-\sigma^2 (k_1-k)^2 / 2} \ ,

where σ determines the spread of k₁-values about k, and N is the amplitude of the wave.

The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ(k₁), and where it varies rapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ.^[12] However, an exception occurs at the location where the argument φ of the exponential varies slowly. (This observation is the basis for the method of stationary phase for evaluation of such integrals.^[14]) The condition for φ to vary slowly is that its rate of change with k₁ be small; this rate of variation is:^[12]

Failed to parse (Missing texvc executable; please see math/README to configure.): \left . \frac{d \varphi }{d k_1} \right | _{k_1 = k } = x - t \left . \frac{d \omega}{dk_1}\right | _{k_1 = k } +\left . \frac{d \alpha}{d k_1}\right | _{k_1 = k } \ ,

where the evaluation is made at k₁ = k because A(k₁) is centered there. This result shows that the position x where the phase changes slowly, the position where ψ is appreciable, moves with time at a speed called the group velocity:

Failed to parse (Missing texvc executable; please see math/README to configure.): v_g = \frac{d \omega}{dk} \ .

The group velocity therefore depends upon the dispersion relation connecting ω and k. For example, in quantum mechanics the energy of a particle represented as a wave packet is E = ħω = (ħk)²/(2m). Consequently, for that wave situation, the group velocity is

Failed to parse (Missing texvc executable; please see math/README to configure.): v_g = \frac {\hbar k}{m} \ ,

showing that the velocity of a localized particle in quantum mechanics is its group velocity.^[12] Because the group velocity varies with k, the shape of the wave packet broadens with time, and the particle becomes less localized.^[15] In other words, the velocity of the constituent waves of the wave packet travel at a rate that varies with their wavelength, so some move faster than others, and they cannot maintain the same interference pattern as the wave propagates.

Sinusoidal waves[link]

Sinusoidal waves correspond to simple harmonic motion.

Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (or harmonic wave or sinusoid) with an amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): u

described by the equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x, \ t)= A \sin (kx - \omega t + \phi) \ ,

where

• Failed to parse (Missing texvc executable; please see math/README to configure.): A

is the maximum amplitude of the wave, maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle. In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.

• Failed to parse (Missing texvc executable; please see math/README to configure.): x

is the space coordinate

• Failed to parse (Missing texvc executable; please see math/README to configure.): t

is the time coordinate

• Failed to parse (Missing texvc executable; please see math/README to configure.): k

is the wavenumber

• Failed to parse (Missing texvc executable; please see math/README to configure.): \omega

is the angular frequency

• Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

is the phase.

The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a wave on a string) have an amplitude expressed as a distance (e.g., meters), longitudinal mechanical waves (e.g., sound waves) use units of pressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field (e.g., volts/meter).

The wavelength Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda

is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber Failed to parse (Missing texvc executable; please see math/README to configure.): k

, the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation

Failed to parse (Missing texvc executable; please see math/README to configure.): k = \frac{2 \pi}{\lambda}. \,

The period Failed to parse (Missing texvc executable; please see math/README to configure.): T

is the time for one complete cycle of an oscillation of a wave. The frequency Failed to parse (Missing texvc executable; please see math/README to configure.): f
is the number of periods per unit time (per second) and is typically measured in hertz. These are related by:

Failed to parse (Missing texvc executable; please see math/README to configure.): f=\frac{1}{T}. \,

In other words, the frequency and period of a wave are reciprocals.

The angular frequency Failed to parse (Missing texvc executable; please see math/README to configure.): \omega

represents the frequency in radians per second. It is related to the frequency or period by

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega = 2 \pi f = \frac{2 \pi}{T}. \,

The wavelength Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda

of a sinusoidal waveform traveling at constant speed Failed to parse (Missing texvc executable; please see math/README to configure.): v
is given by:[16]

Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda = \frac{v}{f},

where Failed to parse (Missing texvc executable; please see math/README to configure.): v

is called the phase speed (magnitude of the phase velocity) of the wave and Failed to parse (Missing texvc executable; please see math/README to configure.): f
is the wave's frequency.

Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, the incoming wave undulates with a varying local wavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.^[17]

Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems, in the presence of dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude, making it easy to analyze.^[18] Due to the Kramers–Kronig relations, a linear medium with dispersion also exhibits loss, so the sine wave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon the medium.^[19] The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.^[20][21]

The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves that exist for a limited span in space and duration in time. Fortunately, an arbitrary wave shape can be decomposed into an infinite set of sinusoidal waves by the use of Fourier analysis. As a result, the simple case of a single sinusoidal wave can be applied to more general cases.^[22][23] In particular, many media are linear, or nearly so, so the calculation of arbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using the superposition principle to find the solution for a general waveform.^[24] When a medium is nonlinear, the response to complex waves cannot be determined from a sine-wave decomposition.

Plane waves[link]

Standing waves[link]

Standing wave in stationary medium. The red dots represent the wave nodes

A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions.

The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut, where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node. Halfway between two nodes there is an antinode, where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time.

• 

  One-dimensional standing waves; the fundamental mode and the first 6 overtones.

• 

  A two-dimensional standing wave on a disk; this is the fundamental mode.

• 

  A standing wave on a disk with two nodal lines crossing at the center; this is an overtone.

Physical properties[link]

Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism

Waves exhibit common behaviors under a number of standard situations, e.g.,

Transmission and media[link]

Waves normally move in a straight line (i.e. rectilinearly) through a transmission medium. Such media can be classified into one or more of the following categories:

• A bounded medium if it is finite in extent, otherwise an unbounded medium
• A linear medium if the amplitudes of different waves at any particular point in the medium can be added
• A uniform medium or homogeneous medium if its physical properties are unchanged at different locations in space
• An anisotropic medium if one or more of its physical properties differ in one or more directions
• An isotropic medium if its physical properties are the same in all directions

Absorption[link]

Reflection[link]

When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line normal to the surface equals the angle made by the reflected wave and the same normal line.

Interference[link]

Waves that encounter each other combine through superposition to create a new wave called an interference pattern. Important interference patterns occur for waves that are in phase.

Refraction[link]

Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results.

Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the size of the phase velocity changes. Typically, refraction occurs when a wave passes from one medium into another. The amount by which a wave is refracted by a material is given by the refractive index of the material. The directions of incidence and refraction are related to the refractive indices of the two materials by Snell's law.

Diffraction[link]

A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads after emerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or opening is comparable to the wavelength of the wave.

Polarization[link]

A wave is polarized if it oscillates in one direction or plane. A wave can be polarized by the use of a polarizing filter. The polarization of a transverse wave describes the direction of oscillation in the plane perpendicular to the direction of travel.

Longitudinal waves such as sound waves do not exhibit polarization. For these waves the direction of oscillation is along the direction of travel.

Dispersion[link]

Schematic of light being dispersed by a prism. Click to see animation.

A wave undergoes dispersion when either the phase velocity or the group velocity depends on the wave frequency. Dispersion is most easily seen by letting white light pass through a prism, the result of which is to produce the spectrum of colours of the rainbow. Isaac Newton performed experiments with light and prisms, presenting his findings in the Opticks (1704) that white light consists of several colours and that these colours cannot be decomposed any further.^[25]

Mechanical waves[link]

Waves on strings[link]

The speed of a wave traveling along a vibrating string ( v ) is directly proportional to the square root of the tension of the string ( T ) over the linear mass density ( μ ):

Failed to parse (Missing texvc executable; please see math/README to configure.): v=\sqrt{\frac{T}{\mu}}, \,

where the linear density μ is the mass per unit length of the string.

Acoustic waves[link]

Acoustic or sound waves travel at speed given by

Failed to parse (Missing texvc executable; please see math/README to configure.): v=\sqrt{\frac{B}{\rho_0}}, \,

or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed of sound).

Water waves[link]

• Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.
• Sound—a mechanical wave that propagates through gases, liquids, solids and plasmas;
• Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect;
• Ocean surface waves, which are perturbations that propagate through water.

Seismic waves[link]

Shock waves[link]

Other[link]

• Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves^[26]

• Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.

Electromagnetic waves[link]

(radio, micro, infrared, visible, uv)

An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. In the 19th century, James Clerk Maxwell showed that, in vacuum, the electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed of light. From this emerged the idea that light is an electromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), giving rise to various types of radiation such as radio waves, microwaves, infrared, visible light, ultraviolet and X-rays.

Quantum mechanical waves[link]

The Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of this equation are wave functions which can be used to describe the probability density of a particle.

A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.^[27]

de Broglie waves[link]

Louis de Broglie postulated that all particles with momentum have a wavelength

Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda = \frac{h}{p},

where h is Planck's constant, and p is the magnitude of the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a de Broglie wavelength of about 10⁻¹³ m.

A wave representing such a particle traveling in the k-direction is expressed by the wave function:

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi (\mathbf{r}, \ t=0) =A\ e^{i\mathbf{k \cdot r}} \ ,

where the wavelength is determined by the wave vector k as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda = \frac {2 \pi}{k} \ ,

and the momentum by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf p = \hbar \mathbf{k} \ .

However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet,^[28] a waveform often used in quantum mechanics to describe the wave function of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value.

In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet.^[29] Gaussian wave packets also are used to analyze water waves.^[30]

For example, a Gaussian wavefunction ψ might take the form:^[31]

Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(x,\ t=0) = A\ \exp \left( -\frac{x^2}{2\sigma^2} + i k_0 x \right) \ ,

at some initial time t = 0, where the central wavelength is related to the central wave vector k₀ as λ₀ = 2π / k₀. It is well known from the theory of Fourier analysis,^[32] or from the Heisenberg uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the more localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian.^[33] Given the Gaussian:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = e^{-x^2 / (2\sigma^2)} \ ,

the Fourier transform is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tilde{ f} (k) = \sigma e^{-\sigma^2 k^2 / 2} \ .

The Gaussian in space therefore is made up of waves:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \ \tilde{f} (k) e^{ikx} \ dk \ ;

that is, a number of waves of wavelengths λ such that kλ = 2 π.

The parameter σ decides the spatial spread of the Gaussian along the x-axis, while the Fourier transform shows a spread in wave vector k determined by 1/σ. That is, the smaller the extent in space, the larger the extent in k, and hence in λ = 2π/k.

Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles

Gravitational waves[link]

Researchers believe that gravitational waves also travel through space, although gravitational waves have never been directly detected. Not to be confused with gravity waves, gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.

WKB method[link]

In a nonuniform medium, in which the wavenumber k can depend on the location as well as the frequency, the phase term kx is typically replaced by the integral of k(x)dx, according to the WKB method. Such nonuniform traveling waves are common in many physical problems, including the mechanics of the cochlea and waves on hanging ropes.

References[link]

See also[link]

Sources[link]

• Campbell, M. and Greated, C. (1987). The Musician’s Guide to Acoustics. New York: Schirmer Books.
• French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics series). Nelson Thornes. ISBN 0-393-09936-9. OCLC 163810889. 
• Hall, D. E. (1980). Musical Acoustics: An Introduction. Belmont, California: Wadsworth Publishing Company. ISBN 0-534-00758-9. .
• Hunt, F. V. (1992) (^{[dead link]}). Origins in Acoustics. New York: Acoustical Society of America Press. http://asa.aip.org/publications.html#pub17. .
• Ostrovsky, L. A.; Potapov, A. S. (1999). Modulated Waves, Theory and Applications. Baltimore: The Johns Hopkins University Press. ISBN 0-8018-5870-4. .
• Vassilakis, P.N. (2001). Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance. Doctoral Dissertation. University of California, Los Angeles.

External links[link]

Wikimedia Commons has media related to: Wave

Look up wave in Wiktionary, the free dictionary.

• Interactive Visual Representation of Waves
• Science Aid: Wave properties—Concise guide aimed at teens
• Simulation of diffraction of water wave passing through a gap
• Simulation of interference of water waves
• Simulation of longitudinal traveling wave
• Simulation of stationary wave on a string
• Simulation of transverse traveling wave
• Sounds Amazing—AS and A-Level learning resource for sound and waves
• chapter from an online textbook
• Simulation of waves on a string
• of longitudinal and transverse mechanical wave
• MIT OpenCourseWare 8.03: Vibrations and Waves Free, independent study course with video lectures, assignments, lecture notes and exams.

PHASE 2 (aka Lonny Wood)^[1] is one of the most influential and well known New York City aerosol artists. Mostly active in the 1970s, Phase 2 is generally credited with originating the "bubble letter" style of aerosol writing, also known as "softies".^[2] He was also influential in the early hip-hop scene.

Graffiti career[link]

Phase 2 is from The Bronx, and attended DeWitt Clinton High School along with a number of other early graffiti artists. Many famous graffiti writers of the early 1970s would meet at a doughnut shop across from the school called the Coffee Shop before heading down to the subway station at 149th Street and Grand Concourse to watch tagged trains on the IRT line pass by.^[3] Phase 2 was mentored in graffiti by his friend and neighbor Thomas Lee aka Lee 163d!, one of the pioneers of graf writing in The Bronx.^[2]

He began writing in late 1971 under the name Phase 2, a moniker which had a rather mundane provenance. As Phase 2 would later recall, "the previous year we'd given this party. We were getting ready to give another one and I said, 'We'll call this one Phase Two.' I don't know why, but I was stuck on the name. It had meaning for me. I started writing 'Phase 2.'"^[4]

Part of the appeal of graffiti writing for Phase 2 was that it allowed him to get his "name" known yet remain anonymous.^[5] He noted later that tagging provided disadvantaged urban teens "the only significant vehicle to represent their 'existence.'"^[6]

It was in late 1972 that Phase 2 first used an early version of the "bubble letter" or "softie", a style of graffiti writing which would become extremely influential and is considered a "giant leap" in the art form.^[2] The puffed-out, marshmallow-like letters drawn by Phase 2 were soon copied by other artists who added their own variations. Phase himself quickly embellished on his original form, creating and naming dozens of varieties of softies such as "phasemagorical phantastic" (bubble letters with stars), "bubble cloud", and "bubble drip."^[7] He is also credited with pioneering the use of arrows in graf writing around this same time.^[8] Hip-hop journalist Jeff Chang has noted that Phase 2's canvasses from 1973 have "been widely recognized as defining the early genre."^[9]

Over time Phase's work become more and more complex, moving far away from the simple tags of the early 70s to "hieroglyphical calligraphic abstraction."^[5] Chang points out that much of Phase's work involved "deconstructing the letter", transforming characters in the alphabet "into hard lines, third eyes, horns, drills, spikes, Egyptian pharaohs and dogs, pure geometrics."^[9] Another New York graffiti artist, Vulcan, remarked that "one of the things about Phase is that he was the only person at the time whose name could roll by ten times and each piece was different. That's what you noticed about his [work]."^[5]

In 1974 Phase 2 joined the newly created United Graffiti Artists, a professional graffiti collective which began to attract media attention. He was featured in an important essay on graffiti art by Richard Goldstein which appeared in New York magazine and inspired a new generation of graffiti artists.^[2]

In the 1980s, Phase 2 began publishing International Graffiti Times, the first zine about graffiti writing.^[10]

Influence in hip-hop[link]

Unlike some other pioneers of New York City graffiti, Phase 2 had a prominent role in The South Bronx hip-hop scene in the early 1980s. He also continues to be referenced in hip-hop songs.

Phase participated in the legendary hip-hop shows organized by Kool Lady Blue during the summer of 1982 at the Roxy nightclub in the Chelsea neighborhood of Manhattan. These shows brought together the top DJ's, MC's, breakers, and aerosol artist from The South Bronx and introduced hip-hop music and culture to the downtown punk and new wave scenes. Phase 2 designed the flyers for these events and often did aerosol pieces live on stage.^[11] He was also part of the first "international" hip-hop tour when stars from the Roxy performances toured in England and France in November of that year.^[12]

Phase 2 was one of the few aerosol artists to be involved in the musical side of hip-hop culture as well. He had a background as a DJ in the very early days of hip-hop, though he never made a name for himself in that role.^[13] In 1982, as part of his involvement with the Roxy scene, Phase released two rap singles. "Beach Boy" was a collaboration with Barry Michael Cooper, who would later co-write the script for New Jack City. "The Roxy" featured the Bill Laswell-led group Material and Grandmixer D.ST, though Phase 2 would later remark that he was disappointed in the song and felt that it "wasn't done properly."^[13][14]

Phase was also an early b-boy and claims that his dance crew pioneered the uprock (or "battle rock") style of dance despite claims that it originated in Brooklyn.^[13] He was thus actively involved in all of the traditional "four elements" of hip-hop culture.

Though he did not have a role in the production, Phase 2 did apparently influence the classic early hip-hop movie Wild Style. In the DVD commentary for the film, director Charlie Ahearn explained that, when thinking about the key character named "Phade", he had Phase 2 in mind (either to actually play the part or simply as a model) because Phase was a legendary graffiti writer from the past who was also involved in the hip-hop scene, as was the character of Phade. The role would ultimately be played by Fab 5 Freddy, himself a graffiti artist who along with Ahearn was the major creative force behind Wild Style. Phase 2 did take on an official role in another early hip-hop film when he worked as a graffiti consultant on the 1984 movie Beat Street.^[15]

In his 1995 song "Out for Fame" - an homage to graffiti artists and culture - KRS-One implores his audience "in the name of Phase 2" and fellow Bronx graffiti legend Stay High to "grab your cans and hit the streets." Several years later Mos Def mentioned Phase 2 on his widely respected debut album Black on Both Sides, specifically on the track "Hip Hop", in which he noted that hip-hop itself was "all city like Phase 2" - presumably a reference to the ubiquity of Phase 2's graffiti pieces on trains throughout the city during the early 1970s. In 1996, he appeared in the song I Messaggeri Pt.I contained in italian artist Neffa's album I Messaggeri della Dopa.

References[link]

Works cited[link]

• George, Nelson Buppies, B-boys, Baps & Bohos: Notes on Post-Soul Black Culture. Da Capo Press. 2001. ISBN 0-306-81027-1. 
• Chang, Jeff (2005). Can't Stop Won't Stop: A History of the Hip-Hop Generation. Picador. ISBN 0-312-42579-1.
• Castleman, Craig (1982). Getting Up: Subway Graffiti in New York. MIT Press. ISBN 0-262-53051-1. 

External links[link]

• Adam Mansbach interview with PHASE 2
• "Writer of the Storm", Raw Vision interview with PHASE 2

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Karl Rudolph Koenig

Karl Rudolph Koenig (German: Rudolf Koenig; November 26, 1832 – October 2, 1901), known by himself and others as Rudolph Koenig, was a German physicist, chiefly concerned with acoustic phenomena.

Biography[link]

Sound analyser with 8 resonator balls, by Koenig, 1880, Teylers Instrument Room

Koenig was born in Königsberg (Province of Prussia), and studied at the University of Königsberg in his native town.

About 1852 he went to Paris, and became apprentice to the famous violin-maker, Jean Baptiste Vuillaume (1798-1875), and some six years later he started business on his own account. He called himself a "maker of musical instruments," but the instrument for which his name is best known is the tuning fork. Koenig's work speedily gained a high reputation among physicists for accuracy and general excellence. From this business Koenig derived his livelihood for the rest of his life. One of his last catalogs had 262 different items.

He was, however, very far from being a mere tradesman. Acoustical research was his real interest, and to that he devoted all the time and money he could spare from his business. In the 1860s, the Koenig sound analyzer revolutionized musical and scientific worlds by demonstrated visually that musical notes and voices were in fact made up of simple sounds.^[1] An exhibit which he sent to the London Exhibition of 1862 gained a gold medal, and at the Philadelphia Exposition at 1876 great admiration was expressed for a tonometric apparatus of his manufacture. This consisted of about 670 tuning-forks, of as many different pitches, extending over four octaves, and it afforded a perfect means for testing, by enumeration of the beats, the number of vibrations producing any given note and for accurately tuning any musical instrument. An attempt was made to secure this apparatus for the University of Pennsylvania, and Koenig was induced to leave it behind him in America on the assurance that it would be purchased; but, ultimately, the money not being forthcoming, the arrangement fell through, to his great disappointment.

Some of the forks he disposed of to the University of Toronto and the remainder he used as a nucleus for the construction of a still more elaborate tonometer. While the range of the old apparatus was only between 128 and 4096 Hz, the lowest fork of the new one vibrated at only 16 Hz, while the highest gave a sound too high to be perceptible to the human ear.

Koenig will also be remembered as the inventor and constructor of many other beautiful pieces of apparatus for the investigation of acoustical problems, among which may be mentioned his wave-sirens, the first of which was shown at Philadelphia in 1876. His original work dealt, among other things, with Wheatstone's sound-figures, the characteristic notes of the different vowels, a manometric flame apparatus, a vibration microscope, among others; but perhaps the most important of his researches are those devoted to the phenomena produced by the interference of two tones, in which he disputed the views of Helmholtz as to the existence of summation and difference tones.

He died in Paris.

References[link]

External links[link]

• Koenig's Manometric Flame Apparatus
• Koenig's Vibration Microscope
• Koenig's Fourier Analyser
• Biography and bibliography in the Virtual Laboratory of the Max Planck Institute for the History of Science

Source[link]

 This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). Encyclopædia Britannica (11th ed.). Cambridge University Press.