The wind flows towards a mountain and produces a first oscillation (A) followed by more waves. the following waves will have lower amplitude because of the natural damping. Lenticular clouds stuck on top of the flow (A) and (B) will appear immobile despite the strong wind.

In meteorology, lee waves are atmospheric standing waves. The most common form is mountain waves, which are atmospheric internal gravity waves. These were discovered in 1933 by two German glider pilots, Hans Deutschmann and Wolf Hirth, above the Riesengebirge.^[1][2] They are periodic changes of atmospheric pressure, temperature and orthometric height in a current of air caused by vertical displacement, for example orographic lift when the wind blows over a mountain or mountain range. They can also be caused by the surface wind blowing over an escarpment or plateau,^[3] or even by upper winds deflected over a thermal updraft or cloud street.

The vertical motion forces periodic changes in speed and direction of the air within this air current. They always occur in groups on the lee side of the terrain that triggers them. Usually a turbulent vortex, with its axis of rotation parallel to the mountain range, is generated around the first trough; this is called a rotor. The strongest lee waves are produced when the lapse rate shows a stable layer above the obstruction, with an unstable layer above and below.^[3]

Clouds[link]

Both lee waves and the rotor may be indicated by specific wave cloud formations if there is sufficient moisture in the atmosphere, and sufficient vertical displacement to cool the air to the dew point. Waves may also form in dry air without cloud markers.^[3] Wave clouds do not move downwind as clouds usually do, but remain fixed in position relative to the obstruction that forms them.

A wave window over the Bald Eagle Valley of central Pennsylvania as seen from a glider looking north. The wind flow is from upper left to lower right. The Allegheny Front is under the left edge of the window, the rising air is at the right edge, and the distance between them is 3-4 km.

• Around the crest of the wave, adiabatic expansion cooling can form a cloud in shape of a lens (lenticularis). Multiple lenticular clouds can be stacked on top of each other if there are alternating layers of relatively dry and moist air aloft.
• The rotor may generate cumulus or cumulus fractus in its upwelling portion, also known as a "roll cloud". The rotor cloud looks like a line of cumulus. It forms on the lee side and parallel to the ridge line. Its base is near the height of the mountain peak, though the top can extend well above the peak and can merge with the lenticular clouds above. Rotor clouds have ragged leeward edges and are dangerously turbulent.^[3]
• A foehn wall cloud may exist at the lee side of the mountains, however this is not a reliable indication of the presence of lee waves.
• A pileus or cap cloud, similar to a lenticular cloud, may form above the mountain or cumulus cloud generating the wave.
• Adiabatic compression heating in the trough of each wave oscillation may also evaporate cumulus or stratus clouds in the airmass, creating a "wave window" or "Foehn gap".

Aviation[link]

Lee waves provide a possibility for gliders to gain altitude or fly long distances when soaring. World record wave flight performances for speed, distance or altitude have been made in the lee of the Sierra Nevada, Alps, Patagonic Andes, and Southern Alps mountain ranges.^[4] The Perlan Project is working to demonstrate the viability of climbing above the tropopause in an unpowered glider using lee waves, making the transition into stratospheric standing waves. They did this for the first time on August 30, 2006 in Argentina, climbing to an altitude of 50,671 ft (15,447 m).^[5] The Mountain Wave Project of the Organisation Scientifique et Technique du Vol à Voile focusses on analysis and classification of lee waves and associated rotors.^[6][7][8]

The conditions favoring strong lee waves suitable for soaring are:

• A gradual increase in windspeed with altitude
• Wind direction within 30° of perpendicular to the mountain ridgeline
• Strong low-altitude winds in a stable atmosphere
• Ridgetop winds of at least 20 knots

The rotor turbulence may be harmful for other small aircraft such as balloons, hang gliders and para gliders. It can even be a hazard for large aircraft; the phenomenon is believed responsible for many aviation accidents and incidents including the in-flight break up of BOAC Flight 911, a Boeing 707, near Mt. Fuji, Japan in 1966, and the in-flight separation of an engine on an Evergreen International Airlines Boeing 747 cargo jet near Anchorage, Alaska in 1993.^[9]

The rising air of the wave, which allows gliders to climb to great heights, can also result in high altitude upset in jet aircraft trying to maintain level cruising flight in lee waves. Rising, descending or turbulent air in or above the lee waves can cause overspeed or stall, resulting in mach tuck and loss of control, especially when the aircraft is operated near the "coffin corner".

The 2012 Norwegian C-130 crash into the peak of the Swedish Kebnekaise was most likely caused by Lee waves.

Sailing[link]

Bermuda rigged leisure sailing boats fear indeed Lee waves and could in worst case appear behind high islands like in the Aegean Sea (Greece) or in Dalmatia (Adriatic) where relatively small islands can be very high and good steady wind, like the Meltemi/Etesian. The wind on the lee side of the island can go downwards. Feared because it causes instability in controlling the boat and in worst case push the sail under water that could lead to a disastrous Capsizing. That is not logically possible with side winds and normal sailboats with Bermuda rig, because the side wind push in the sail, disappears when the sail is horizontal. Many use the expression Downburst for this extreme sailing weather condition but it is certainly more related to Lee waves.

Other varieties of atmospheric waves[link]

There are a variety of distinctive types of waves which form under different atmospheric conditions.

• Wind shear can also create waves. This occurs when an atmospheric inversion separates two layers with a marked difference in wind direction. If the wind encounters distortions in the inversion layer caused by thermals coming up from below, it will create significant shear waves in the lee of the distortions that can be used for soaring.^[10]

• Hydraulic jump induced waves are a type of wave that forms when there exists a lower layer of air which is dense, yet thin relative to the size of the mountain. After flowing over the mountain, a type of shock wave forms at the trough of the flow, and a sharp vertical discontinuity called the hydraulic jump forms which can be several times higher than the mountain. The hydraulic jump is similar to a rotor in that it is very turbulent, yet it is not as spatially localized as a rotor. The hydraulic jump itself acts as an obstruction for the stable layer of air moving above it, thereby triggering wave. Hydraulic jumps can distinguished by their towering roll clouds, and have been observed on the Sierra Nevada range^[11] as well as mountain ranges in southern California.

• 

  

  Hydrostatic wave (schematic drawing)
  Hydrostatic waves are vertically propagating waves which form over spatially large obstructions. In hydrostatic equilibrium, the pressure of a fluid can depend only on altitude, not on horizontal displacement. Hydrostatic waves get their name from the fact that they approximately obey the laws of hydrostatics, i.e. pressure amplitudes vary primarily in the vertical direction instead of the horizontal. Whereas conventional, non-hydrostatic waves are characterized by horizontal undulations of lift and sink, largely independent of altitude, hydrostatic waves are characterized by undulations of lift and sink at different altitudes over the same ground position.

• Kelvin–Helmholtz instability can occur when velocity shear is present within a continuous fluid or when there is sufficient velocity difference across the interface between two fluids.

• Rossby waves (or planetary waves) are large-scale motions in the atmosphere whose restoring force is the variation in Coriolis effect with latitude.

See also[link]

• Gravity wave
• Nor'west arch

References[link]

Further reading[link]

• Grimshaw, R., (2002). Environmental Stratified Flows. Boston: Kluwer Academic Publishers.
• Jacobson, M., (1999). Fundamentals of Atmospheric Modeling. Cambridge: Cambridge University Press.
• Nappo, C., (2002). An Introduction to Atmospheric Gravity Waves. Boston: Academic Press.
• Pielke, R., (2002). Mesoscale Meteorological Modeling. Boston: Academic Press.
• Turner, B., (1979). Buoyancy Effects in Fluids. Cambridge: Cambridge University Press.
• Whiteman, C., (2000). Mountain Meteorology. Oxford Oxfordshire: Oxford University Press.

External links[link]

• http://www.inglaner.com/meteorologia_onda.htm chronological collection of meteorological data, satpics and cloud images of mountain waves in Bariloche, Argentina. In Spanish
• http://www.mountain-wave-project.com official website

Wikimedia Commons has media related to: Lee waves

Look up lee, Lee, leeward, or alee in Wiktionary, the free dictionary.

Lee may refer to:

Given name[link]

• Lee (given name), a unisex English given name
• Lee (Hebrew name), a Hebrew given name

Surname[link]

• Lee (English name), a common English surname
• Li (李) or Lee, a ubiquitous Chinese surname (李, pin. Lǐ) originally meaning "plum"
• Lee (Korean name) or Rhee, a common Korean surname (Hanja 李, Hangul 리 or 이) derived from the Chinese
• Lý (surname), Ly, or Lee, a common Vietnamese surname (Chữ Nôm 李) derived from the Chinese
• Li (surname) or Lee, several other East Asian surnames

Place and river names[link]

Great Britain[link]

• The Lee, Buckinghamshire, parish and village name, formally known as Lee
• Lee, Devon
• Lee-on-the-Solent
• Lee, London
• Lee District (Metropolis)
• River Lee - alternative name for River Lea

United States of America[link]

• Lee (RTA Blue Line Rapid Transit station), a station on the RTA Blue Line in Cleveland, Ohio
• Lee, Florida
• Lee, Illinois
• Lee, Indiana
• Lee, Maine
• Lee, Massachusetts
• Lee, Nevada
• Lee, New Hampshire
• Lee, New York
• Lees Station, Tennessee, also called Lee
• Lee Vining, California
• Fort Lee, New Jersey
• Fort Lee, Virginia
• Mount Lee, in California

Other countries[link]

• Electoral district of Lee, South Australia
• Lée, France
• River Lee (Ireland)

Enterprises[link]

• Lee Enterprises (LEE on NYSE), publicly traded newspaper holding company including St. Louis Post Dispatch
• Lee (jeans), a clothing brand
• Thomas H. Lee Partners, a private equity firm founded by, but after 2006 not associated with, Thomas H. Lee
• Lee Equity Partners, a private equity firm founded by Thomas H. Lee after leaving Thomas H. Lee Partners in 2006
• Lee Tires, a division of Goodyear
• Lee University, a private liberal arts university in Cleveland, Tennessee, United States

Other uses[link]

• Tropical Storm Lee (2011)
• M3 Lee, a medium tank used during World War II
• Lee side, the side of a ship which is downwind, the side away from windward
• Lee waves, periodic changes of pressure in a stream of air when the wind moves over mountains

See also[link]

• List of persons with the surname Lee
• List of people with the Korean family name Lee
• LEA (disambiguation)
• Leah (disambiguation)
• Le (disambiguation)
• LEE (disambiguation)
• Lees (fermentation)
• Leigh (disambiguation)
• Ly (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.

The X Factor finalists (UK series 7)}} The seventh UK series of The X Factor was broadcast on ITV between 21 August and 12 December 2010. The final 12 were declared on 3 October 2010. Four wildcards were announced on the first live show of the finals on 9 October 2010, bringing the number of finalists up to 16.^[1]

Boys[link]

The Boys category was mentored by Dannii Minogue. Contestants in this category are males aged 16 to 28. The eight candidates were Karl Brown, Matt Cardle, Nicolò Festa, Aiden Grimshaw, Marlon McKenzie, Tom Richards,, Paije Richardson and John Wilding. Minogue chose:

Matt Cardle[link]

Matt Cardle (born 15 April 1983)^[2] is a former painter and decorator from Little Maplestead, Essex.^[3] Born to parents David and Jennifer Cardle, he was diagnosed with Wilms' tumour, a form of cancer when he was a toddler. He recovered after one of his kidneys was removed at age 2.^[4] Before applying to The X Factor, he was the frontman and acoustic guitar player of the band Seven Summers^[5] alongside Alex Baker (drums), Jon Holland (bass) and Neillo (guitars). The band had alternative rock, pop, urban, and folk influences and released a debut album^[6] on 22 January 2010 with their debut single being "Picture of You".^[7]

Cardle auditioned for The X Factor with "You Know I'm No Good",^[3] sang "The First Time Ever I Saw Your Face" at bootcamp,^[8] "If I Were a Boy" at judges' houses.^[9] In the first week, Cardle sang "When Love Takes Over". Show bosses asked him to perform a "secret gig" at the Five Bells pub, in Colne Engaine providing footage for the Saturday ITV show.^[10] In week two, he sang "Just the Way You Are", and in week 3 "...Baby One More Time". After reaching the final two singers, it was revealed that his would be winning single would be a cover of Biffy Clyro's "Many of Horror" retitled as "When We Collide". He was announced overall winner of the competition during the final held on 12 December 2010. His debut album Letters, was released in the UK on October 17, 2011.^[11]

Nicolò Festa[link]

Nicolò Festa is a 21 year old student from Treviso, Italy.^[12][13] He is the son of Michele Festa, deputy commissioner of the police flying squad in Verona, who apprehended notorious Mafia boss Felice Maniero after his 1994 escape from prison.^[12][13][14] He studied Languages at Ca' Foscari University of Venice and then moved to London a few months before applying for The X Factor. His father said his son's dream is to remain living there.^[12] He auditioned with "A Song for You" after which Louis Walsh called him "a diva in the making".^[15] At bootcamp, he sang "Life on Mars?", and made it through to the next round.^[16] He said that his fame was "inevitable" before performing "New York" in the judges' houses stage and was picked for the live shows by Minogue.^[17] For the first live show, he sang, "Just Dance" and was eliminated in the series' first double elimination, after receiving the fewest public votes. In answer to host Dermot O'Leary's question how he felt being the first one to exit, he replied: "I feel like crap".^[18] In April 2011 he began collaborating with independent label Aztec Records writing and recording new songs.^[19]

Aiden Grimshaw[link]

Aiden Grimshaw performing on The X Factor Live Tour (2011)

Aiden Grimshaw (born 4 December 1991) is a student from North Shore, Blackpool.^[20] He attended the Michael Hall Theatre School.^[20] Before auditioning for The X Factor, he played Tom Holmes in an episode of CBBC show Half Moon Investigations.^[21][22] He auditioned with "Gold Digger",^[23] prompting Cowell to say "Now that's what I call a pop star!".^[23] He then auditioned with "This Year's Love" during Bootcamp,^[16] and "Cannonball" at Judge's Houses.^[24] Dannii Minogue selected Grimshaw as one of her three contestants to continue in the competition for the live shows. For the first live show he sang "Mad World", which the judges praised. In the second live show he sang "Jealous Guy". In the third live show, his performance of "Diamonds Are Forever" was praised by the judges. In live show 4 he sang a slower rendition of Michael Jackson's "Thriller". Grimshaw sang "Nothing Compares 2 U" in week 5, and in week 6, Grimshaw sang "Rocket Man". Grimshaw landed in the final showdown that week, along with Katie Waissel. While he was saved by Minogue and Walsh in the judges vote, Cole and Cowell had voted for Waissel to stay, thus leaving the votes deadlocked. The public vote revealed that Grimshaw received the fewest votes and he was eliminated.^[25] His debut album, Misty Eye, is due for release on 13 August 2012 via RCA Records. It will be preceded by his debut single, "Is This Love", which is due for release on June 3.^[26]. The music video for Grimshaw's new single was released on YouTube on 30 April 2012. The video, though being that of Grimshaw's song, features Grimshaw himself for only approximately 2 seconds, as an 'extra'. He is in a corridor of a club walking past two of the video's central actors. The song was written by Grimshaw, Jarrad Rogers & Athlete frontman Joel Pott.^[27]

Paije Richardson[link]

Paije Richardson (born 15 November 1990) is a cinema usher and former fashion student from Islington, London.^[28] He has recently finished a BTEC National Diploma in Art and Design (Fashion and Clothing) at City and Islington College.^[28] Richardson also had a cameo appearance in Harry Potter and the Order of the Phoenix. He appeared again in Harry Potter and the Deathly Hallows – Part 2 playing an extra in the Battle of Hogwarts.^[29] He first auditioned with "Fly Me to the Moon" but Walsh was the only judge to enthuse over his performance, Cowell compared him to a wedding singer and joined Cole in giving him a no.^[30] Walsh went backstage to ask him to come back at the end of the day with another song.^[30] When Richardson returned he performed "It's A Man's Man's Man's World",^[30] to which he received a yes from all three judges, with Cowell complimenting his taste in music.^[30] At the judges houses stage he performed "True Colors" ^[24] but failed to make it into the initial top 12.^[9] He was then chosen by Minogue as her wildcard.^[1] For the first live show he sang "Killing Me Softly with His Song", in the second live show he sang "If I Ain't Got You" and in the third live show he sang "Ain't Nobody", and in week 4 he sang Back To Black by Amy Winehouse. After performing "Let It Be" in week 7, Richardson was in the final showdown for the first time, along with Cher Lloyd. Only Minogue voted to save him, and he was eliminated.^[31] After the series finished, the voting figures revealed that Richardson had actually received more votes from the public than Lloyd in week 7. On April 25, 2012, Richardson unvieled his debut single, "Bang Bang", which is due for release on July 16, 2012. His debut album was funded via donations made through Pledge Music. It will be released during September 2012.^[32]

Girls[link]

The Girls category was mentored by Cheryl Cole. Contestants in this category are females aged 16 to 28. The eight candidates were Keri Arrindell, Annastasia Baker, Treyc Cohen, Rebecca Ferguson, Cher Lloyd, Gamu Nhengu, Raquel Thomas and Katie Waissel. Cole chose:

Treyc Cohen[link]

Treyc Cohen, 26, from Stonydelph in Tamworth^[33] is a call-centre worker^[34] and former music student, who quit studying at the University of Wolverhampton in the summer.^[33] In the 2009 series, Cohen was in the over-25s category, getting through to the final twenty-four, but was eliminated at the judges' houses stage.^[35] This series, Cohen was placed in the girls' category (after the age limit was changed to 28) and she once again was through to the judges' houses, where she sang "Ave Maria".^[33] She was not picked to be in the initial top 12 acts,^[9] but Cole chose her as a wildcard for the live finals.^[1]

Cohen faced controversy after it was revealed that she signed a management deal with Artimis Music Management Ltd^[36] that landed her a contract in October 2009 with Birmingham-based Ajoupa Records, that released her debut single entitled "A Time to Be Heard",^[37] which was also the debut single of Ajoupa Records itself. It was released on 14 December 2009 featuring the a cappella group The Black Voices.^[38] The rules of The X Factor strictly forbid record deals while a contestant is on the show, and the show's legal team worked to free her from her contract.^[37]

Her rendition of "One" in the first live show brought her rave comments including Cowell saying: "I am amazed that we overlooked you in this competition. I didn't like what you're wearing, but honestly that's not an issue, because by a far mile, that was the best vocal for the entire competition thus far".^[39] Cohen went on to sing "Purple Rain" in the second week, progressing onto the third week, and sang "Whole Lotta Love". However, Cohen landed in the final showdown with John Adeleye, but was able to progress onto the fourth week after her performance of "One Night Only" convinced the judges to keep her in the competition. Cohen then went on to sing "Relight My Fire", and got through to the fifth week. On week 5, Cohen sung "I Don't Want to Miss a Thing". Cohen was once again in the bottom two, against Katie Waissal, and sang "Un-Break My Heart". Cheryl Cole, being both the girls' mentor, refused to send home either act, and after offering to go last to send the result to deadlock, the majority of votes was used. Only Minogue was in Treyc's favour, thus eliminating her from the competition and finishing tenth overall.^[40][41] After the series ended, it was revealed that Cohen received more votes than Waissel, meaning if Cole had been allowed to take the vote to deadlock, Waissel would not have been eliminated.

From 24 June 2011, Cohen is due to star as Dorothy Gale in a West Yorkshire Playhouse production of The Wiz; a re-telling of the 1978 film The Wiz.^[42]

Rebecca Ferguson[link]

Ferguson performing on the X Factor Live Tour

Rebecca Ferguson, (born 21 July 1986)^[43] is a student and single mother to Lillie May (5) and Karl (4) from Anfield, Liverpool.^[44] She is a qualified legal secretary, having studied at Hugh Baird College, Bootle.^[45] She had previously unsuccessfully auditioned for The X Factor and for P. Diddy's Starmaker in New York.^[44] She sang "A Change Is Gonna Come" during her initial audition.^[44] At her bootcamp audition, she sang "Like a Star" to which Cowell said "That is a recording voice!" and Nicole Scherzinger said, "I told you first time I saw her, she is our generation’s songbird."^[46] She sang "Fireflies" at judges' houses and made it through to the live shows.^[9] For the first live show she sang "Teardrops" and in the second live show she sang "Feeling Good". In the third live show she sang "Why Don't You Do Right?" and in the fourth live show she sang "Wicked Game". In the fifth live show, Rebecca received a standing ovation from Simon and Cheryl following her performance of "Make You Feel My Love". In the following episode, Rebecca received another standing ovation from Dannii and Cheryl after singing "Candle in the Wind". In the final, she performed a duet with Christina Aguilera, performing Aguilera's hit single "Beautiful". Ferguson's winner's song was a cover of Duffy's "Distant Dreamer", but she finished as the runner-up to Matt Cardle, making her the first female runner-up on The X Factor. After the final, it was announced that Ferguson had been signed by Syco Music,.^[47]

In January 2011 it was reported by The Sun that she had signed a joint record deal between Syco and Epic Records.^[48] Her debut album has been confirmed and is due for release on 5 December 2011.^[49]

She is the new face of Walkers crisps Sunbites as of June 2011.^[50]

On 20 November 2011, Ferguson released her debut single "Nothing's Real But Love", which debuted at number 10 on the UK Singles Chart. Nothing's Real But Love also charted at no.23 in Ireland, 16 in Italy and 24 in Switzerland. On December 5, 2011, Rebecca released her debut album, "Heaven", and it debuted at number 3 in the UK, Heaven also charted at number 7 in Switzerland, 9 in Ireland and 30 in Italy. Her second single, "Too Good to Lose", was released on March 4, 2012, and charted at 186 on the UK Singles Chart. Glitter & Gold was released as her third single on the 29th April 2012 and charted at 152 on the UK Singles Chart.

Cher Lloyd[link]

Cher Lloyd (born 28 July 1993)^{[citation needed]} from Malvern, where she lives with her parents Darren and Dina, brother Josh, 15, and sisters Sophie, 14, and Rosie, 12.^[51] She is of Romani ancestry through her mother.^[52][53] She studied performing arts at Dyson Perrins CE Sports College in 2009.^[51] She auditioned for The X Factor twice before (when the minimum age was lower) singing ballads but it was only after reinventing herself that she got through to see the judges.^[54] Lloyd has previously performed at holiday camps and said she gets a mixed reaction to her music.^[55] She auditioned singing the Keri Hilson version of "Turn My Swag On".^[55] At bootcamp she sang a rap version of "Viva La Vida"^[8] adding lyrics she claimed to have written herself, but this was disputed and reports claimed she took them from a song by rapper Swizz Beatz.^[56][57] At the judges' houses she performed "Cooler Than Me"^[24] but was unable to complete her song after suffering from tonsillitis.^[9] She was given a second chance, but broke down sobbing and could not complete the song. She was still picked, however, as one of the final three girls by mentor Cole.

Lloyd sang a rendition of "Just Be Good to Me" in the first of the live shows. In the second live show she sang "Hard Knock Life (Ghetto Anthem)". In the third live show, she sang a mashup of "No Diggity" and "Shout", and in the fourth live show she sang "Stay". This was the first time Lloyd did not rap in her performance, and Cowell even called it "the performance of the series". In week five, Lloyd sang "Empire State of Mind", but judges stated that after last week they were disappointed, and Cowell said it was copycat. In the seventh week, Lloyd was in the final showdown for the first time after her rendition of "Imagine", but Cowell, Cole and Walsh all saved her in the final showdown, resulting in Paije Richardson being eliminated. In the following week Cher performed "Girlfriend", followed by "Walk This Way". Both performances were credited by the judges and she was saved by the public vote the following night, securing her position in the semi final.^[31] In the semi-final, Lloyd performed "Nothin' on You" and "Love the Way You Lie" and was in the final showdown with Mary Byrne. She was saved by the judges again and put through to the final, even though voting statistics after the show revealed that Lloyd had the fewest votes.^[58] In the final she performed a mash-up of "The Clapping Song" and "Get Ur Freak On", followed by a duet with will.i.am, which was a mashup of "Where Is the Love?" and "I Gotta Feeling". Lloyd was then eliminated in fourth place, having received the fewest public votes.^[59]

After the final, it was announced that Lloyd had been signed by Syco Music.^[47] Her debut album has been confirmed and was released 7 November 2011.^[60] She has worked with songwriter Autumn Rowe and producer RedOne.^[61][62] Her debut single "Swagger Jagger" received its first airplay on 20 June. The single was leaked onto the internet on 15 June^[63] but this version was later confirmed as only the demo of the track on Lloyd's Twitter account.^[64] Lloyd performed at T4 on the Beach in 2011.^[65]

Katie Waissel[link]

Katie Waissel, 24 during the series, is a recording artist from Harefield, Middlesex.^[66] She is a former receptionist, waitress and once worked at OK! magazine.^[67] She has previously performed under the name Katie Vogel on the series Green Eyed World^[68] and had a recording contract under the name Lola Fontaine.^[69][70] Waissel has been on tour with East 17 and Andy Abraham^[71] and has performed in Hollywood for Paris Hilton together with friend and fellow contestant Storm Lee.^[72]

At her audition, she sang Queen's "We Are the Champions", but forgot some of the words.^[70] She then sang "At Last" by Etta James, and was put through to bootcamp by Cole and Cowell. At bootcamp, she sang "Make You Feel My Love" and reacted with shock when she was the first girl put through to the judges houses.^[16] She fell to pieces when performing the song "Smile" in front of Cole and will.i.am at the judges' houses stage,^[70] and had to stop to compose herself after crying, stating that "it's an emotional song for me".^[73] Despite stumbling through her song for the second time, Cole picked her as one of her final three girls. This was a very controversial decision as many viewers did not feel that Waissel deserved her place in the live shows over girls who did not forget their words or break down during their performance.

In week 1 of the live shows, she sang "We Are the Champions". She was put in the bottom three as a result of the public vote, but because Nicolo Festa polled the fewest votes, Waissel had to sing in the final showdown with F.Y.D., the other bottom three act from week 1. She sang "Don't Let Me Down" in the final showdown and was saved from elimination by the judges by a majority vote, with only Cowell (F.Y.D.'s mentor) voting against her. In the second week of the live shows, Katie sang "I'd Rather Go Blind". She managed to avoid the bottom three this week and secured a place to the third week, where she sang "I Wan'na Be Like You (The Monkey Song)". In the fourth week she sang "Bewitched" and was in the final showdown with Belle Amie. Judges votes were deadlocked this time, but Waissel was announced as safe when it was revealed that Belle Amie had the fewest votes.^[74] She was in the final showdown again the next week after performing "Don't Speak" but was saved by a majority vote over Treyc Cohen when Cole refused to vote for either of her acts, while Cowell and Walsh voted to eliminate Cohen.^[40]

Waissel was in the final showdown for a third successive week when she sang "Saturday Night's Alright for Fighting", and this had been the fourth time Waissel had been in the final showdown. The judges votes were tied once again, but the public vote showed Aiden Grimshaw received fewer votes and he was eliminated.^[25] This was the first time in the show's history that a contestant was put into the final showdown four times and survived. In week 8, she performed "Sex on Fire" and "Everybody Hurts". In the series' third double elimination, Waissel received the fewest number of votes and was automatically eliminated.^[75]

Over 28s[link]

The Over 28s category was mentored by Louis Walsh. Contestants in this category are of either sex, aged 29 or over. There is no upper age limit. The eight candidates were John Adeleye, Mary Byrne, Wagner, Stephen Hunter, Justin Vanderhyde, Storm Lee, Yuli Minguel, and Elesha Moses. Walsh chose:

John Adeleye[link]

John Adeleye (as J'Nay) performing at Dalston Festival Vortex in 2008

John Adeleye, 30-31, is a care worker and award winning songwriter from Harlesden, London^[76] and had an artistic career under the stage name J'Nay. He was a founding member of the band Nu-park, with fellow musician Shad. The band was further expanded to become a 5-piece band by including Mylz (Tyga), Raheem and Anton Gordon.^[77] and a solo artist after the breakup of the group. In 2008, J'Nay released his EP Merry Go Round containing 5 tracks.^[77] Adeleye was working as an activity co-ordinator at Forrester Court care home in Westminster helping people with early onset dementia for two years,^[76] where he says he was inspired to enter The X Factor by the residents and his distinctive singing means they always remember him. He said one resident said to him "'When you sing, I feel a lot better'. That's when I decided to enter."^[78] Adeleye, under the name J'Nay, won the MOBO Unsung Award at the 2003 MOBO Awards.^[79] He has appeared on GMTV^[80] and Newsround,^[81] was signed by record company Universal-Island and has toured with Jamelia.^[82] At his first audition he sang "You Are So Beautiful",^[82] at the judges house stage he sang "Billionaire".^[24] In week 1 of the live shows, Adeleye sang "One Sweet Day" and in week 2, "A Song for You". He was safe in both weeks. After performing "Zoom" in week 3, he was voted in final showdown by the public and had a showdown with Treyc Cohen.. In the final showdown the, he sang "Because of You", but was voted out by Simon Cowell, who could have gone to deadlock. He finished 12th overall. However, at the end of the series, it was revealed that John had received more votes than Treyc, so if Cowell had gone to deadlock Adeleye would have remained in the competition.

Mary Byrne[link]

Mary Byrne (born November 1959) is a supermarket worker from Ballyfermot, Dublin, Ireland.^[83] She says she wanted to be a singer all her life but never had the confidence until now. She is a single mother to her 23-year-old daughter Deborah.^[84][85] She has performed with her brother Tommy under the name The Tommy Lee Experience.^[86] In 2008 she won the Irish television singing contest Nollaig No. 1 using the name Mary Lee, releasing the single "Siúil Leat".^[83][87]

At her first audition, she sang "I (Who Have Nothing)"^[9] and at the judges houses she performed "Fix You",^[9] though she forgot the words to the song.^[88] Her employer, Tesco Ireland, started a campaign for all staff to support her.^[89] Before the first live show, Byrne suffered a throat infection and the show's bosses asked her to quit smoking to avoid making the infection worse.^[88] She performed "It's a Man's Man's Man's World" in the first week, "You Don't Have to Say You Love Me" in week 2 and "I Who Have Nothing" in week 3, the song she had sung in her audition. In week 8 she performed "All I Want Is You" and "Brass in Pocket" and was in the bottom three. She was saved by the judges after Katie Waissel and Wagner were eliminated.^[75] In week 9 she performed "Never Can Say Goodbye" and "The Way We Were" and was in the bottom two with Cher Lloyd. She sang "It's a Man's Man's Man's World" as her sing-off song but was eliminated by the judges.^[58] She finished in fifth place.

Byrne's debut album, Mine & Yours, was released on March 28, 2011.^[90] The album charted at number-one on the Irish Album Charts^[91] The album also charted at no.6 in the UK Album Chart^[92]

Storm Lee[link]

Storm Lee (born Lee Gardner), 37, is a singer songwriter born in Edinburgh,^[93] who later moved to Los Angeles when he was 17.^[94] Lee's mother, Arras Robertson, left the family home when he was four years old after her marriage with his father Robert Gardner collapsed in 1977. Lee and his mother only speak through Facebook, he has a half-sister Susan and older brother Gary, 40.^[95] He came out as gay on the Edinburgh Evening News in 2007, saying it's not a "huge part of my life, it's just part of my personality".^[96] His previous jobs include make-up artist, furniture re-upholster and a cleaner at KFC.^[97] Lee has performed in Hollywood for Paris Hilton together with friend and fellow contestant Katie Waissel^[72] and at the 2009 NewNowNext Awards.^[98] Lee appeared on US show The Singing Bee as part of the house band.^[93][99] and has a credit for original music on Bridezillas.^[100]

Lee has released his own album Soulfillapopkilla in January 2007.^[99][101] He collaborated with Andy Caldwell on the track "Black Diamond Sky" on Caldwell's Obsession album ^[102] and with Ricky Martin on four tracks for Martin's album.^[103] Lee has recorded backing vocals for many artists including Meat Loaf, Ozzy Osbourne and Carrie Underwood.^[93][97] He can be heard on backing vocals on Meat Loaf albums Bat Out of Hell III: The Monster Is Loose^[104] and It's All Coming Back to Me Now,^[105] Glee: The Music, Volume 3 Showstoppers,^[106] and Naked Truth^[107] by Sarah Hudson. Lee co-wrote a track on the album Confessional^[108] by Bryan Rice and he was nominated for a Latin Grammy for the composition on the Lipstick album by Alejandra Guzmán.^[99][109] He also played tenor saxophone on Utopía^[110] by Belinda.

Lee's first audition was with "Every Breath You Take" and "I Still Haven't Found What I'm Looking For".^[93] At the bootcamp he sang "Sweet Child O'Mine"^[8] and at the judges houses stage sang "Without You".^[9] He was eliminated from the live shows on October 17, after singing "Born to Run" and receiving the fewest public votes for week 2. He finished 14th overall. His album Soulfillapopkilla was re-released on 1 March 2011.^[111]

Wagner[link]

Wagner Fiuza-Carrilho ( /ˈvɑːɡnə/; born 1956), the oldest of the finalists, is a Brazilian born retired PE teacher and karate enthusiast now living in Pensnett, Dudley.^[112] Fiuza-Carrilho previously owned a martial arts and dance gym in Cradley Heath before becoming a P.E. teacher for a year at Leasowes Community College; he has also worked for a conservatory fitting company.^[113][114] He said he entered the contest to get a "new start" and to be able to provide for his son.^[112] At the judges houses Fiuza-Carrilho performed "You Got The Love"^[24] to which Walsh joked to his assistant judge Sharon Osbourne that "He fancies you".^[115] Although he was not picked to be one the initial top 12 acts,^[9] mentor Walsh chose him as his wildcard for the live finals.^[1] Wagner moved out of The X Factor house after the second live show after complaining about his fellow contestants' behaviour.^[116]