In physics, a wave is a disturbance that travels through space and time, usually accompanied by the transfer of energy.

Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass transport. They consist, instead, of oscillations or vibrations around almost fixed locations. For example, a cork on rippling water will bob up and down, staying in about the same place while the wave itself moves onwards.

One type of wave is a mechanical wave, which propagates through a medium in which 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 bumping into their neighbors. This transfers some energy to these neighbors, which will cause a cascade of collisions between neighbouring molecules. When air molecules collide with their neighbors, they also bounce away from them (restoring force). This keeps the molecules from continuing to travel in the direction of the wave.

Another type of wave can travel through a vacuum, e.g. electromagnetic radiation (including visible light, ultraviolet radiation, infrared radiation, gamma rays, X-rays, and radio waves). This type of wave consists of periodic oscillations in electrical and magnetic fields.

A main distinction can be made between transverse and longitudinal waves. Transverse waves occur when a disturbance sends waves perpendicular (at right angles) to the original wave. Longitudinal waves occur when a disturbance sends waves in the same direction as the original wave.

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.

General features

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 . 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 / light waves in a vacuum, where the concept of medium does not apply. There are water waves on the ocean surface; light waves emitted by the Sun; microwaves used in microwave ovens; radio waves broadcast by radio stations; and sound waves generated by radio receivers, telephone handsets and living creatures (as voices).

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 etc.; and so on.

Other properties, however, although they are usually described in an origin-specific manner, 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 from their physical origin. 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 ). 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

Wave equation

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

in the $x$ direction in space. E.g., let the positive $x$ direction be to the right, and the negative $x$ direction be to the left.

with constant amplitude $u$

with constant velocity $v$, where $v$ is

*independent of wavelength (no dispersion)

independent of amplitude (linear media, not nonlinear).

with constant waveform, or shape

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

: $u(x,\; \backslash \; t)\; =\; F(x\; -\; v\; \backslash \; t)$ (waveform $F$ traveling to the right) : $u(x,\; \backslash \; t)\; =\; G(x\; +\; v\; \backslash \; t)$ (waveform $G$ traveling to the left)

or, more generally, by d'Alembert's formula:

:$u(x,t)=F(x-vt)+G(x+vt).\; \backslash ,$

representing two component waveforms $F$ and $G$ traveling through the medium in opposite directions. This wave can also be represented by the partial differential equation

:$\backslash frac\{1\}\{v^2\}\backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; t^2\}=\backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; x^2\}.\; \backslash ,$

General solutions are based upon Duhamel's principle.

Wave forms

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).

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''.

Amplitude and modulation

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:

:$u(x,\; \backslash \; t)\; =\; A(x,\; \backslash \; t)\backslash sin\; (kx\; -\; \backslash omega\; t\; +\; \backslash phi)\; \backslash \; ,$

where $A(x,\backslash \; t)$ is the amplitude envelope of the wave, $k$ is the ''wavenumber'' and $\backslash phi$ is the ''phase''. If the group velocity $v\_g$ (see below) is wavelength-independent, this equation can be simplified as:

:$u(x,\; \backslash \; t)\; =\; A(x\; -\; v\_g\; \backslash \; t)\backslash sin\; (kx\; -\; \backslash omega\; t\; +\; \backslash phi)\; \backslash \; ,$

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''.

Phase velocity and 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.

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

:$\backslash psi\; (x,\; \backslash \; t)\; =\; A\; e^\{i\; \backslash left(\; kx\; -\; \backslash omega\; t\; \backslash right)\}\; \backslash \; ,$

which can be related to the usual sine and cosine forms using Euler's formula. Rewriting the argument, $kx-\backslash omega\; t\; =\; (\backslash frac\{2\backslash pi\}\{\backslash lambda\})(x\; -\; vt)$, makes clear that this expression describes a vibration of wavelength $\backslash lambda\; =\; \backslash frac\{2\backslash pi\}\{k\}$ traveling in the ''x''-direction with a constant ''phase velocity'' $v\_p\; =\; \backslash frac\{\backslash omega\}\{k\}\backslash ,$.

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:

:$\backslash psi\; (x,\; \backslash \; t)\; =\; \backslash int\_\{-\backslash infty\}\; ^\{\backslash infty\}\backslash \; dk\_1\; \backslash \; A(k\_1)\backslash \; e^\{i\backslash left(k\_1x\; -\; \backslash omega\; t\; \backslash right)\}\; \backslash \; ,$

where now ''A(k''₁'')'' (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peak in a region of wave vectors Δ''k'' surrounding the point ''k''₁ = ''k''. In exponential form:

:$A\; =\; A\_o\; (k\_1)\; e^\; \{i\; \backslash alpha\; (k\_1)\}\; \backslash \; ,$

with ''A''_o the magnitude of ''A''. For example, a common choice for ''A''_o is a Gaussian wave packet:

:$A\_o\; (k\_1)\; =\; N\backslash \; e^\{-\backslash sigma^2\; (k\_1-k)^2\; /\; 2\}\; \backslash \; ,$

where σ determines the spread of ''k''₁-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(''k''₁), and where it varies rapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ. 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.) The condition for φ to vary slowly is that its rate of change with ''k''₁ be small; this rate of variation is:

:$\backslash left\; .\; \backslash frac\{d\; \backslash varphi\; \}\{d\; k\_1\}\; \backslash right\; |\; \_\{k\_1\; =\; k\; \}\; =\; x\; -\; t\; \backslash left\; .\; \backslash frac\{d\; \backslash omega\}\{dk\_1\}\backslash right\; |\; \_\{k\_1\; =\; k\; \}\; +\backslash left\; .\; \backslash frac\{d\; \backslash alpha\}\{d\; k\_1\}\backslash right\; |\; \_\{k\_1\; =\; k\; \}\; \backslash \; ,$

where the evaluation is made at ''k''₁ = ''k'' because ''A(k''₁'')'' 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'':

:$v\_g\; =\; \backslash frac\{d\; \backslash omega\}\{dk\}\; \backslash \; .$

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 = (ħ''k'')²/(2''m''). Consequently, for that wave situation, the group velocity is

:$v\_g\; =\; \backslash frac\; \{\backslash hbar\; k\}\{m\}\; \backslash \; ,$

showing that the velocity of a localized particle in quantum mechanics is its group velocity. Because the group velocity varies with ''k'', the shape of the wave packet broadens with time, and the particle becomes less localized. 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

Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (or ''harmonic wave'' or ''sinusoid'') with an amplitude $u$ described by the equation:

:$u(x,\; \backslash \; t)=\; A\; \backslash sin\; (kx\; -\; \backslash omega\; t\; +\; \backslash phi)\; \backslash \; ,$

where

$A$ 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.

$x$ is the space coordinate

$t$ is the time coordinate

$k$ is the wavenumber

$\backslash omega$ is the angular frequency

$\backslash phi$ 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 $\backslash lambda$ is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber $k$, the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation

:$k\; =\; \backslash frac\{2\; \backslash pi\}\{\backslash lambda\}.\; \backslash ,$

The period $T$ is the time for one complete cycle of an oscillation of a wave. The frequency $f$ is the number of periods per unit time (per second) and is typically measured in hertz. These are related by:

:$f=\backslash frac\{1\}\{T\}.\; \backslash ,$

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

The angular frequency $\backslash omega$ represents the frequency in radians per second. It is related to the frequency or period by

:$\backslash omega\; =\; 2\; \backslash pi\; f\; =\; \backslash frac\{2\; \backslash pi\}\{T\}.\; \backslash ,$

The wavelength $\backslash lambda$ of a sinusoidal waveform traveling at constant speed $v$ is given by: :$\backslash lambda\; =\; \backslash frac\{v\}\{f\},$

where $v$ is called the phase speed (magnitude of the phase velocity) of the wave and $f$ 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.

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. 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. The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.

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. 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. When a medium is nonlinear, the response to complex waves cannot be determined from a sine-wave decomposition.

Plane waves

Standing waves

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.

Physical properties

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

Transmission and media

Waves normally move in 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

Reflection

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

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

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

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

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

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.

Mechanical waves

Waves on strings

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 ''):

:$v=\backslash sqrt\{\backslash frac\{T\}\{\backslash mu\}\},\; \backslash ,$

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

Acoustic waves

Acoustic or sound waves travel at speed given by

:$v=\backslash sqrt\{\backslash frac\{B\}\{\backslash rho\_0\}\},\; \backslash ,$

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

Water waves

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

Shock waves

Other

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

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

Electromagnetic waves

(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

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. Quantum mechanics also describes particle properties that other waves, such as light and sound, have on the atomic scale and below.

thumb|A propagating wave packet; in general, the ''envelope'' of the wave packet moves at a different speed than the constituent waves.

de Broglie waves

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

:$\backslash lambda\; =\; \backslash frac\{h\}\{p\},$

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:

:$\backslash psi\; (\backslash mathbf\{r\},\; \backslash \; t=0)\; =A\backslash \; e^\{i\backslash mathbf\{k\; \backslash cdot\; r\}\}\; \backslash \; ,$

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

:$\backslash lambda\; =\; \backslash frac\; \{2\; \backslash pi\}\{k\}\; \backslash \; ,$

and the momentum by:

:$\backslash mathbf\; p\; =\; \backslash hbar\; \backslash mathbf\{k\}\; \backslash \; .$

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, 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''. Gaussian wave packets also are used to analyze water waves.

For example, a Gaussian wavefunction ψ might take the form:

:$\backslash psi(x,\backslash \; t=0)\; =\; A\backslash \; \backslash exp\; \backslash left(\; -\backslash frac\{x^2\}\{2\backslash sigma^2\}\; +\; i\; k\_0\; x\; \backslash right)\; \backslash \; ,$

at some initial time ''t'' = 0, where the central wavelength is related to the central wave vector ''k''₀ as λ₀ = 2π / ''k''₀. It is well known from the theory of Fourier analysis, 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. Given the Gaussian:

:$f(x)\; =\; e^\{-x^2\; /\; (2\backslash sigma^2)\}\; \backslash \; ,$

the Fourier transform is:

:$\backslash tilde\{\; f\}\; (k)\; =\; \backslash sigma\; e^\{-\backslash sigma^2\; k^2\; /\; 2\}\; \backslash \; .$

The Gaussian in space therefore is made up of waves:

:$f(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash \; \backslash tilde\{f\}\; (k)\; e^\{ikx\}\; \backslash \; dk\; \backslash \; ;$

that is, a number of waves of wavelengths λ such that ''k''λ = 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''.

Gravitational waves

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

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

See also

Sources

Campbell, M. and Greated, C. (1987). ''The Musician’s Guide to Acoustics''. New York: Schirmer Books.

. | ref=harv | postscript=. }}. .

Vassilakis, P.N. (2001). ''Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance''. Doctoral Dissertation. University of California, Los Angeles.

External links

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.

Category:Fundamental physics concepts Category:Partial differential equations

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Patrick Shane Dorian (born July 19, 1972), or "Shane", is an American surfer from Kailua-Kona, Hawaii. He spent 11 years touring on the World Championship Tour surfing.

Early life

Dorian was born to Patrick, a former Hollywood actor and stunt double for Elvis Presley, and Susan, who competed in female bodybuilding events. When Shane was three, his parents opened a restaurant called "Dorian's" on the beach. Being a child, Dorian was too young to wait tables, and long hours at the restaurant resulted in him turning to the ocean for amusement.

Dorian attended Konawaena High School.

Surfing career

He started out on a bodyboard alongside friend, and future bodyboarder champion, Mike Stewart. Dorian received his first surfboard, from his dad, on his fifth birthday. He soon abandoned the bodyboard to focus on surfing.

Surfing and surfing education became a priority for Dorian. Dorian's mother, who had divorced Patrick when Shane was 12, worked out a system where Shane would spend the second and third quarters of the school year on the North Shore of Oahu (the peak winter season) and the first and fourth quarters back home on the Big Island.

In 1987, Shane Dorian had his first major success a the Gotcha Pro competition at Sandy Beach. He was able to beat some big names, made it through four rounds, and received some media exposure. He became friends with a couple of the up-and-coming surfers on Oahu, Brock Little and Todd Chesser who played a role in the development of his surfing.

World Championship Tour

Dorian joined the ASP World Tour in 1993 where he hovered outside of the top 10 for a number of years before cracking the barrier with a fourth-place world finish in 2000. He's also won a number of events in his career, including the Rip Curl Pro in 1999 and the 2000 Billabong Pro at Mundaka. In 2004, Dorian retired from the WTC tour, but not from a being a professional; choosing to focus on becoming a better surfer. Shane Dorian took the top prize at the 2008 Global Big Wave awards in California.

He has been a long-time member of the Billabong team, his main sponsor.

In the documentary ''Solid: The Two Days That Teahupoo Blew Minds'', fellow surfers gave him a standing ovation for his wave conquering rides.

Filmography

''In God's Hands'' (1998), with Surfers Matt George and Matty Liu

''Thicker Than Water'' (2000) a surf film by Jack Johnson and the Malloys.

''The Blueprint'' (2005)

''Solid: The Two Days That Teahupoo Blew Minds'' (2005)

''Campaign 2'' (2005)

''A Fly in the Champagne'' (2009)

References

External links

Billabong Rider Profile

iMDB: In God's Hands

Category:American surfers Category:1972 births Category:Living people Category:Sportspeople from Hawaii

Mark Robert Healey (born January 13, 1971 in Ipswich) is a British computer games developer. Healey started his career making games for the Commodore 64 home computer - his first published game was KGB Super Spy for Codemasters which led to developing the Educational Fun School series of games for Europress Software. Healey later Joined Bullfrog Productions to work with Peter Molyneux on titles such as Theme Park, Magic Carpet and Dungeon Keeper. When Molyneux left Bullfrog to form Lionhead Studios, Healey joined him, and worked as a senior artist/designer on the BAFTA award winning Black & White. Whilst still at Lionhead, he developed ''Rag Doll Kung Fu'' independently in his spare time, which was the first third party game to be distributed over Steam - Valve Corporation's online distribution system. He is a co-founder and Creative Director of Media Molecule, a computer games studio which published its first videogame, a cooperative platformer for the PlayStation 3 entitled ''LittleBigPlanet'', which has been pioneering in the area of user created content.

References

External links

Media Molecule

}}

Category:British video game designers Category:Living people

Coordinates	18°55′23″N48°17′19″N
birth name	Bethany Meilani Hamilton
birth date	February 08, 1990
birth place	Kauai, Hawaii, United States
death date
residence	Hawaii
nickname	B-Ham, Bethy
height
weight
sponsors	Rip Curl
major achievements	ESPY Award (2004)NSSA National Helping Kids Champion (2005)2nd place, ASP World Junior Championships (2008)
favorite waves
favorite maneuvers
website	BethanyHamilton.com }}

Bethany Meilani Hamilton (born February 8, 1990), daughter of Thomas and Cherilyn Hamilton, is an American professional surfer. She is known for surviving a shark attack in which she lost her left arm, and for overcoming the serious and debilitating injury to ultimately return to professional surfing. She wrote about her experience in the 2004 autobiography ''Soul Surfer: A True Story of Faith, Family, and Fighting to Get Back on the Board''. In April 2011, the feature film ''Soul Surfer'' was released, based on the book and additional interviews.

Shark attack and recovery

On October 31, 2003, at the age of 13, Hamilton went for a morning surf along Tunnels Beach, Kauai with one of her friends, Alana Blanchard, and Blanchard's father and brother. Around 7:30 a.m., with numerous turtles in the area, she was lying on her surfboard with her left arm dangling in the water, when a tiger shark attacked her, ripping her left arm off just below the shoulder. The Blanchards helped paddle her back to shore, then Alana's father fashioned a tourniquet out of a surfboard leash, and wrapped it around the short piece left of her arm, before rushing her to Wilcox Memorial Hospital. Her father was supposed to have knee surgery that morning, but she took his place in the operating room. She then spent seven more days in recovery at the hospital.

Despite the trauma of the incident, Hamilton was determined to return to surfing. Less than one month after the incident, she returned to her board and went surfing again. Initially, she adopted a custom-made board that was longer and slightly thicker, making it easier to paddle. She has observed that she has to kick a lot more to make up for the loss of her left arm. After teaching herself to surf with one arm, she again began surfing competitively. She is now back to using competitive performance short-boards.

In July 2004, Hamilton won the Best Comeback Athlete ESPY Award. She was also presented with a special "courage award" at the 2004 Teen Choice Awards.

In 2005, Hamilton took 1st place in the National Scholastic Surfing Association (NSSA) National Championships, a goal she had been trying to achieve since before the shark attack. In 2008, she began competing full-time on the Association of Surfing Professionals (ASP) World Qualifying Series (WQS). In her first competition against many of the world's best women surfers, she finished 2nd.

Media

Since the attack, she has appeared on ''20/20'', ''Good Morning America'', ''Inside Edition'', ''The Oprah Winfrey Show'', ''The Ellen DeGeneres Show'', ''The Today Show'' and ''The Tonight Show'', as well as in magazines ''People'', ''Time'' and ''American Girl''. Additionally, she was the cover story in the premier issue of ''niNe'' magazine.

In 2004, MTV Books published Hamilton's book, ''Soul Surfer: A True Story of Faith, Family, and Fighting to Get Back on the Board'' (ISBN 0-7434-9922-0), which describes her ordeal. Her story is also told in the 2007 short subject documentary film, ''Heart of a Soul Surfer'', directed by Becky Baumgartner. Described as a "faith-based documentary", the film addresses Hamilton's courage and faith in Jesus Christ in the aftermath of the shark attack, and follows her quest for spiritual meaning.

On August 7, 2009, Hamilton won $25,000 on ''Are You Smarter Than a 5th Grader?''. She was stuck on a third grade grammar question involving articles.

On May 16, 2010, she appeared on an episode of ABC's ''Extreme Makeover: Home Edition''.

In March 2011, Bethany did a video for the Christian organization, I Am Second, telling about her struggle after the shark attack and how she trusted in God to get her through it.

On April 8, 2011, a docudrama feature film ''Soul Surfer'', based on her 2004 book, was released. Hamilton is portrayed by AnnaSophia Robb and Dennis Quaid, Helen Hunt, Craig T. Nelson, Carrie Underwood and Kevin Sorbo costar in the film. Hamilton performs all the one-armed surfing stunts in the film.

On the Weekend of August 20, 2011, Bethany was a guest speaker at Fellowship Church in Grapevine, Texas. A video can be seen at its website showing Pastor Ed Young interviewing Bethany. Fellowship Church saw over 25,000 people in attendance across the three services that is offered at five different campuses around the Dallas/Fort Worth metroplex and in Miami, Florida.

References

External links

CBS News interview

Category:1990 births Category:American amputees Category:Animal attack victims Category:Female surfers Category:American surfers Category:American Christians Category:Living people Category:Sportspeople from Hawaii

an:Bethany Hamilton cs:Bethany Hamilton de:Bethany Hamilton es:Bethany Hamilton fr:Bethany Hamilton no:Bethany Hamilton pl:Bethany Hamilton sv:Bethany Hamilton

Coordinates	18°55′23″N48°17′19″N
name	Brian Conley
birth name	Brian Paul Conley
birth date	August 07, 1961
birth place	Paddington, London
nationality	British
occupation	Comedian, singer, actor
height	5' 11"
spouse	Anne-Marie(1996 - present)
children	2 daughters
website
footnotes	}}

Brian Conley (born 7 August 1961) is an English comedian, television presenter, singer and actor. At the peak of his television career, he was the highest-paid male television personality in the UK. Outside of television, he is best known for his frequent portrayals of Buttons in pantomime versions of ''Cinderella''.

Early life

Brian Paul Conley was born in Paddington, London. His father was a taxi driver. He was brought up in West London and studied Performing Arts at the Barbara Speake Stage School. As a teenager, Conley had a few minor television appearances, including an advert for hot dogs and a small role in a 1977 episode of the science fiction series ''Survivors''. At the age of 16, by lying about his age, Conley started work as a Pontin's Bluecoat.

Career

Conley's first major show business success was fronting a comedy showband called ''Tomfoolery'', who performed in pubs and clubs across England and Wales, sometimes as a support act for artists such as Johnny Mathis and The Nolans. Although the group broke up due to internal disputes when Conley was 19, his work with the band led to him being talent spotted by agent Bob Voice. As a result of this, Conley started working as a warm-up man for television personalities such as The Krankies, Kenny Everett and Terry Wogan.

Conley's career was then advanced by TVS casting director Bill Hatterley, who secured him onscreen appearances on comedy shows such as ''Make Me Laugh'' (1982), ''The Laughter Show'' (1984-1985), ''Live from Her Majesty's'' (1984-87) and ''Five Alive'' (1986). In 1989, with the support of London Weekend Television (LWT)'s light entertainment controller, Conley starred in his own comedy sketch show, ''Brian Conley: This Way Up''. This programme was popular, but not hugely successful.

After two series of the show, Conley made his first appearance in the West End, playing the lead role of Bill Snibson in a production of ''Me & My Girl''.

In 1992, LWT offered him another opportunity to star in his own comedy programme, with ''The Brian Conley Show''. At the suggestion of producer and director Nigel Lythgoe, this new show had a variety format rather than being purely sketches. This different format proved popular, and the show became Britain's most-watched light entertainment programme. Conley's next success was a sitcom entitled ''Time After Time'', in which he played the lead role. The show was named 'Best ITV Sitcom' at the 1994 British Comedy Awards.

He then went on to play the hugely popular Doug 'Dynamo' Digby, starring opposite Amanda Holden, Nigel Planer and Noddy Holder in The Grimleys. He was also given two "An Audience with..." shows.

Conley recorded a live show in 1996 titled "Brian Conley: Alive and Dangerous", which was televised and featured stand-up plus special Nick Frisby/Larry the Loafer and Dangerous Brian sketches. The show was put on VHS and later on DVD.

Around 2000, Conley was given a new show for ITV consisting of chat and music called, once again, The Brian Conley Show. The show lasted for three series. The show's guests included some major Hollywood names like Kathleen Turner and Leslie Nielsen.

Conley also worked at the BBC for a couple of shows. His first being "Got Your Number", a game show part of a long series of National Lottery game shows. Conley would later return to the BBC to present Let Me Entertain You in 2006.

He has done much stage work, including the parts of Buttons in ''Cinderella'' and Caractacus Potts in ''Chitty Chitty Bang Bang''.

In 2003 he starred in the movie Equilibrium.

In 2006, Brian Conley returned to television, hosting a daytime variety show called Let Me Entertain You, a Lion TV production for BBC Two where 13-year-old Operatic Baritone Matthew Crane was the First Series Champion. On 1 December of the same year, he appeared as a guest presenter on The New Paul O'Grady Show on Channel 4, where Larry The Loafer made an appearance in the beginning, and Brian referred himself to "Dangerous Brian" as he was about to enter the "Tank of Doom" as part of a Bushtucker trial.

Later in 2007, Conley presented a second series of Let Me Entertain You on BBC Two, and presented a gameshow, ''Dirty Rotten Cheater'' for the BBC. Twenty shows were recorded in May at the Maidstone Studios in Kent.

Conley presented the Midweek Lottery show on BBC 1 in 2008. On 24 April 2008, Brian took over from ill presenter Paul O'Grady on The Paul O'Grady Show on Channel 4, where he starred once again as 'Dangerous Brian' and he contested against 'Nearly Dangerous Joyce'. Larry the loafer also made an appearance. On 22 June 2008, Brian starred as a cameo on ''Last of the Summer Wine'' as "Boothroyd", Barry's fitness mad neighbour.

Brian appeared as part of Team Ant on Ant and Dec's Saturday Night Takeaway, and took over from Michael Ball in the musical ''Hairspray'' playing the role of Edna Turnblad at London's Shaftesbury Theatre. He will then play the same role in the first UK Tour of Hairspray, sharing the role with Michael Ball and actor Michael Starke, at selected UK venues. He has also partaken in his own live show, "The Best of Brian Conley," which showcases 'the best' of Brian's material from his past shows, interspersed with occasional showreels of either Larry the Loafer, Dangerous Brian (depending on who Brian is about to come on stage as) or a generic showreel of his work. Like The Brian Conley Show, the stage show invites guests to perform on stage alongside Brian. In 2010, the show went on tour again, covering more areas of the country.

Personal life

Conley is married to Anne-Marie, who became his wife on 4 August 1996, in a ceremony at Stoke Poges chapel, Buckinghamshire. Conley had proposed eight months earlier, while wearing a suit of armour. The wedding day itself also had comic elements, causing celebrity guest Linda Lusardi to remark that it was 'the funniest wedding I've been to' — Conley made several memorable quips during the event, and the couple left the church in the ''Chitty Chitty Bang Bang'' car. The couple have two children, both daughters.

Television, film and theatre credits

Television

As title star

''Brian Conley: This Way Up'' (1989–1990)

''The Brian Conley Show'' (1992-1995)

''Brian Conley: Alive and Dangerous'' (1996)

''Brian Conley's Crazy Christmas'' (1997)

''The Brian Conley Show'' (2000-2002)

''An Audience with...'' (2002)

As presenter

''Simply the Best! LWT'S Most Memorable Moments'' (1998)

''The National Lottery: We've Got Your Number'' (1999)

''The Royal Variety Performance'' (1999)

''Judgement Day'' (2003)

''Let Me Entertain You'' (2006-7)

''The New Paul O'Grady Show'' (2006 & 2009 — 2 episodes as guest presenter)

''Dirty Rotten Cheater'' (2007)

As actor

''Survivors'' (1977) — as Michael, episode 'The Peacemakers'

''Outside Chance'' (1993, pilot for ''Time After Time'') — as Kenny Conway

''Time After Time'' (1994-1995) — as Kenny Conway

''Privates'' (1999)

''The Grimleys'' (1999-2000) — as Doug 'Dynamo' Digby

''The Life and Times of Vivienne Vyle'' (2007) — as Chris Connor

''Last of the Summer Wine'' (2008) — as Boothroyd, episode 'Enter the Finger'

As guest/performer

''Make Me Laugh'' (1982)

''Punchlines!'' (1983-84, 2 episodes)

''Knees-Up'' (1983-1984)

''The Laughter Show'' (1984-1985)

''Live from Her Majesty's'' (1984-87, 4 episodes)

''The Keith Harris Show'' (1985, 1 episode)

''Five Alive'' (1986)

''Laughs from the Palladium'' (1987)

''You Bet!'' (1991, 1 episode)

''Telethon Night Out'' (1992)

''The Royal Variety Performance'' (1993)

''Surprise, Surprise'' (1993-94, 2 episodes)

''Children in Need'' (1996)

''The Royal Variety Performance'' (1996)

''All Star Family Fortunes'' (2001, 1 episode)

''SM:TV Gold'' (2003, 1 episode)

''The Wright Stuff'' (2005, 1 episode)

''Comic Relief in da Bungalow'' (2005)

''Dick and Dom in da Bungalow'' (2006, 1 episode)

''The Best of the Royal Variety'' (2006)

''Grumpy Old New Year'' (2006)

''The Wright Stuff'' (2007-2009, 16 episodes)

''The Comedy Map of Britain'' (2007, 1 episode)

''The Grumpy Guide to... Youth Culture'' (2007)

''Cash in the Celebrity Attic'' (2008, 1 episode)

''Ant & Dec's Saturday Night Takeaway'' (2009, 6 episodes)

Conley's appearances on chat shows include ''Aspel & Company'' (1993), ''This Morning'' (1993 & 2005), ''Open House with Gloria Hunniford'' (1998), ''The New Paul O'Grady Show'' (2005 & 2009), ''GMTV'' (2005), ''Breakfast'' (2008), ''The Alan Titchmarsh Show'' (2009), ''The ONE Show'' (2009), ''The 5 O'Clock Show'' (2010), ''The Michael Ball Show'' (2010) and ''Loose Women'' (2007-11, various episodes).

Additionally, Conley has appeared as a talking head on many compilation shows and documentaries including: ''The 100 Greatest Musicals'' (2003), ''Bob Monkhouse's Comedy Heroes'' (2004), ''Who Killed Saturday Night TV?'' (2004), ''The Story of Light Entertainment'' (2006), ''100 Greatest Stand-Ups'' (2007), ''50 Greatest Comedy Catchphrases'' (2008) and ''The Nolans: In the Mood for Dancing'' (2009).

Film

''Cinderella: The Show Must Go On'' (1986, TV) — as Dandini

''West is West'' (1987) — as Sue's friend

''Circus'' (2000) — as Bruno

''Hotel!'' (2001, TV) — as Inspector Cochrane

''Dream'' (2001) — as Charlie Allen

''Arthur's Dyke'' (2001) — as Dave

''Cruise of the Gods'' (2002) — as self

''Equilibrium'' (2002) — as Reading Room Proprietor

''Marple: By the Pricking of my Thumbs'' (2006, TV) — as Eric Johnson

''I Am Bob'' (2007) — as The Compere

Theatre

Pantomime appearances

(unknown year) — ''Dick Whittington'' (as Idle Jack), The Hexagon, Reading

1979-80 — ''Robinson Crusoe'' (as part of Tomfoolery), Swansea Grand Theatre

1984-5 — ''Mother Goose'', Churchill Theatre, Bromley

1989-90 — ''Dick Whittington'' (as a villager), Nottingham Theatre Royal

1992-3 — ''Aladdin'', Birmingham Hippodrome

1993-4 — ''Cinderella'' (as Buttons), The Mayflower Theatre, Southampton

1997-8 — ''Cinderella'' (as Buttons), The Mayflower Theatre, Southampton

1998-9 — ''Cinderella'' (as Buttons), Birmingham Hippodrome

1999-2000 — ''Cinderella'' (as Buttons), Theatre Royal, Plymouth

2001-2 — ''Dick Whittington'', Birmingham Hippodrome

2002-3 — ''Cinderella'' (as Buttons), The Mayflower Theatre, Southampton

2003-4 — ''Cinderella'' (as Buttons), Manchester Opera House

2004-5 — ''Cinderella'' (as Buttons), Theatre Royal, Plymouth

2005-6 - ''Aladdin'' (as Aladdin), The Orchard Theatre, Dartford

2006-7 — ''Cinderella'' (as Buttons), Birmingham Hippodrome

2007-8 — ''Cinderella'' (as Buttons), Wycombe Swan Theatre, High Wycombe

2008-9 — ''Cinderella'' (as Buttons), Nottingham Theatre Royal

2009-10 — ''Cinderella'' (as Buttons), Cliffs Pavilion, Southend-on-Sea

2010-11 — ''Cinderella'' (as Buttons), The New Theatre, Cardiff

2011-12 — ''Cinderella'' (as Buttons), Birmingham Hippodrome

Other roles

''Me and My Girl'' (as Bill Snibson), Adelphi Theatre, 1991.

''Jolson'' (as Al Jolson), Victoria Palace Theatre, 1995-98.

''Elton John's Glasses'' (as Bill), ''Queen's Theatre'', 1998.

''Chitty Chitty Bang Bang'' (as Caractacus Potts), London Palladium, 2005.

''The Music Man'' (as Professor Harold Hill), Chichester Festival Theatre, 2008.

''Hairspray'' (as Edna Turnblad), Shaftesbury Theatre, 2009-10.

''Brother Love's Travelling Salvation Show'' (as Brother Love), UK tour, 2011.

Discography

''Stage to Stage'' (Telstar, 1996)

''Songs from the Shows'' (2002)

''Brian Conley Sings'' (Sony, 2002)

''Let the Good Times Roll'' (Universal, 2002)

Videography

''Brian Conley: Alive + Extra Dangerous'' (Universal, 1996)

Awards and nominations

;Awards

1995 National Television Awards, Most Popular Comedy Performer

2004 Manchester Evening News Theatre Awards, Most Popular Performer

2008 TMA Best Performance in a Musical Award (for ''The Music Man'')

;Nominations

1996 Olivier Award for Best Musical Actor (for ''Jolson'')

2000 National Television Awards, Most Popular Comedy Performer

References

External links

Brian Conley Official Site

BBC Comedy Guide to Brian Conley: This Way Up

BBC Comedy Guide to The Brian Conley Show

Category:1961 births Category:Living people Category:English comedians Category:English male singers Category:English television presenters Category:People from Paddington

fr:Brian Conley