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.

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2011)

Antônio Carlos Brasileiro de Almeida Jobim (January 25, 1927 – December 8, 1994), also known as Tom Jobim (Portuguese pronunciation: [ˈtõ ʒoˈbĩ]), was a Brazilian songwriter, composer, arranger, singer, and pianist/guitarist. He was a primary force behind the creation of the bossa nova style, and his songs have been performed by many singers and instrumentalists within Brazil and internationally.

Widely known as the composer of "The Girl from Ipanema" (Garota de Ipanema), one of the most recorded songs of all time, Jobim has left a large number of songs that are today included in the standard Jazz and Pop repertoires.

Early life[link]

Antonio Carlos Jobim was born in the middle-class district of Tijuca, in Rio de Janeiro. His father Jorge de Oliveira Jobim (São Gabriel, Rio Grande do Sul, April 23, 1889 - July 19, 1935) was a writer, diplomat, professor and journalist. He came from a prominent family, being the great-grand nephew of José Martins da Cruz Jobim,^[1] senator, privy councillor and physician of Emperor Dom Pedro II. While studying medicine in Europe, José Martins added Jobim to his last name, paying homage to the village where his family came from in Portugal, the parish of Santa Cruz de Jovim, Porto.^[2][3]

When Antonio was still an infant, his parents separated and his mother, Nilza Brasileiro de Almeida (c. 1910 - November 17, 1989), moved with her children (Antonio Carlos and his sister Helena Isaura, born February 23, 1931) to Ipanema, the beachside neighborhood the composer would later celebrate in his songs. In 1935, when the elder Jobim died, Nilza married Celso da Frota Pessoa (died February 2, 1979), who would encourage his stepson's career. He was the one that gave Jobim his first piano. As a young man of limited means, Jobim earned his living by playing in nightclubs and bars and later as an arranger for a recording label, before starting to achieve success as a composer.

Jobim's musical roots were planted firmly in the work of Pixinguinha, the legendary musician and composer who began modern Brazilian music in the 1930s. Among his teachers were Lúcia Branco, and, from 1941 on, Hans-Joachim Koellreutter. Jobim was also influenced by the French composers Claude Debussy and Maurice Ravel, by the Brazilian composer Heitor Villa-Lobos, and by jazz. Among many themes, his lyrics talked about love, self discovery, betrayal, joy and especially about the birds and natural wonders of Brazil, like the "Mata Atlântica" forest, characters of Brazilian folklore like Matita Pereira (Saci Pererê), and his home city of Rio de Janeiro.

Career[link]

Jobim became prominent in Brazil when he teamed up with poet and diplomat Vinícius de Moraes to write the music for the play Orfeu de Conceição (1956). The most popular song from the show was "Se Todos Fossem Iguais a Você" ("If everyone were like you"). Later, when the play was turned into a film, producer Sacha Gordine did not want to use any of the existing music from the play. Gordine asked de Moraes and Jobim for a new score for the film Black Orpheus (1959). Moraes was at the time away in Montevideo, Uruguay, working for the Itamaraty (the Brazilian Ministry of Foreign Affairs) and so he and Jobim were only able to write three songs, primarily over the telephone ("A Felicidade", "Frevo",and "O Nosso Amor"). This collaboration proved successful, and Vinicius went on to pen the lyrics to some of Jobim's most popular songs.

A key event in making Jobim's music known in the English speaking world was his collaboration with the American jazz saxophonist Stan Getz, João Gilberto and Gilberto's wife at the time, Astrud Gilberto, which resulted in two albums, Getz/Gilberto (1963) and Getz/Gilberto Vol. 2 (1964). The release of Getz/Gilberto created a bossa nova craze in the United States, and subsequently internationally. Getz had previously recorded Jazz Samba with Charlie Byrd (1962), and Jazz Samba Encore! with Luiz Bonfá (1964). Jobim wrote many of the songs on Getz/Gilberto, which became one of the best-selling jazz albums of all time, and turned Astrud Gilberto, who sang on "The Girl from Ipanema" and "Corcovado", into an international sensation. At the Grammy Awards of 1965 Getz/Gilberto won the Grammy Award for Album of the Year, Grammy Award for Best Jazz Instrumental Album, Individual or Group and the Grammy Award for Best Engineered Album, Non-Classical. "The Girl from Ipanema" won the award for Grammy Award for Record of the Year.

Personal life[link]

Jobim was married to Thereza Otero Hermanny on October 15, 1949 and had two children with her: Paulo Jobim (born 1950), an architect and musician, married and father of Daniel Jobim and Dora Jobim (born 1976); and Elizabeth "Beth" Jobim (born 1957), a painter. Jobim and Otero divorced in 1978. On April 30, 1986 he married 29-year old photographer Ana Beatriz Lontra, with whom he had two more children: João Francisco Jobim (1979–1998) and Maria Luiza Helena Jobim (born 1987). His grandson Daniel Jobim (born 1973), Paulo's son, followed his grandfather's steps, becoming a pianist and composer.

Death[link]

In early 1994, after finishing his album Antonio Brasileiro, Jobim complained to his doctor, Roberto Hugo Costa Lima, of urinary problems, which included urinating blood. A bladder tumor was detected, but Jobim postponed the recommended immediate surgery for several months, while he tried spiritual treatment with a Brazilian medium and started working with his album Tom Jobim. After receiving a message allegedly coming from Frederik von Stein, a dead German doctor, that recommended not having the surgery, Jobim decided to stop listening to the spiritual guidance and have the surgery instead. He was operated in Mount Sinai Hospital, in New York, on December 2, 1994. On December 8, while recovering from surgery, he had a cardiac arrest caused by a pulmonary embolism and two hours later another cardiac arrest, from which he died.^[4] His last words were: "Remember me".^{[citation needed]} He was survived by his children and grandchildren. His last album, Antonio Brasileiro, was released three days after his death.^[5]

Jobim's body was flown back to Brazil on December 9, 1994 and was given a private funeral on December 13, 1994 in Rio de Janeiro. His family, his friends Miúcha, Edu Lobo, João Gilberto, Astrud Gilberto and his close friends came to his funeral.^{[citation needed]} His body lay in state until given a proper burial on December 20, 1994. He is buried in the Cemitério São João Batista in Rio de Janeiro.^[6]

Legacy[link]

Jobim is one of the most important songwriters of the 20th century. Many of Jobim's songs are jazz standards. American jazz singers Ella Fitzgerald and Frank Sinatra prominently featured Jobim's songs on their albums Ella Abraça Jobim (1981), and Francis Albert Sinatra & Antonio Carlos Jobim (1967), respectively. The 1996 CD Wave: The Antonio Carlos Jobim Songbook included performances of Jobim tunes by Oscar Peterson, Herbie Hancock, Chick Corea, and Toots Thielemans. Jobim was an innovator in the use of sophisticated harmonic structures in popular song. Some of his melodic twists, like the melody insisting on the major seventh of the chord, became common use in Jazz and easy listening music after him.^[7] The Brazilian collaborators and interpreters of Jobim's music include João Gilberto (often credited as a co-creator of bossa nova), Chico Buarque, Gal Costa, Elis Regina, Sérgio Mendes, Astrud Gilberto, and Flora Purim. Eumir Deodato and the conductor/composer Claus Ogerman arranged many recordings of Jobim tunes.^[8] He won a Lifetime Achievement award at the 54th Grammy Awards.^[9]

As a well-deserved posthumous homage, on January 5, 1999 the Municipality of Rio de Janeiro changed the name of Rio's Galeão International Airport , located on Governador Island, to bear the composer's name. Galeão Airport is explicitly mentioned in his composition "Samba do avião."

Discography[link]

Studio albums[link]

• 1963: The Composer of Desafinado, Plays
• 1965: The Wonderful World of Antonio Carlos Jobim
• 1967: Wave
• 1967: A Certain Mr. Jobim
• 1970: Stone Flower
• 1970: Tide
• 1972: Look to the Sky
• 1973: Matita Perê
• 1973: Jobim
• 1974: Elis & Tom (with Elis Regina)
• 1976: Urubu
• 1977: Miúcha & Antonio Carlos Jobim - Vol. 1 (with Miúcha)
• 1979: Miúcha & Antonio Carlos Jobim - Vol. 2 (with Miúcha)
• 1980: Terra Brasilis
• 1981: Edu & Tom (with Edu Lobo)
• 1987: Passarim
• 1987: Inédito
• 1989: Echoes of Rio
• 1994: Miúcha e Tom Jobim (with Miúcha)
• 1994: Antonio Brasileiro
• 1998: Tom Jobim (Brazil only)

Compilations[link]

• 1999: The Best of Tom Jobim
• 2000: The Tom Jobim Sessions (outtakes with various artists)
• 2002: The Outtakes (outtakes as a solo artist)
• 2006: Sinatra-Jobim (outtakes with Frank Sinatra)
• 2009: The Complete Tom Jobim (boxset)

Live albums[link]

• 1977: Gravado ao vivo no Canecão (with Vinicius, Toquinho, Miúcha)
• 1986: Jazzvisions: Rio Revisited (with Gal Costa)
• 1996: Antonio Carlos Jobim and Friends (with Joe Henderson, Shirley Horn, Jon Hendricks, Herbie Hancock, Gonzalo Rubalcaba)
• 2001: Tom Canta Vinicius: Ao Vivo (recorded in 1990)
• 2004: Em Minas Ao Vivo: Piano e Voz (recorded in 1981)

Soundtracks[link]

• 1959: Black Orpheus (Soundtrack)
• 1970: The Adventurers (Soundtrack)
• 1983: Gabriela, Cravo e Canela (Soundtrack)^[10]
• 1986: Moments of Play (Soundtrack)^[11]

As contributor[link]

• 1958: Canção do Amor Demais – Elizete Cardoso
• 1959: Amor de gente moça – Silvia Telles
• 1959: Chega de Saudade – João Gilberto
• 1959: Por tôda a minha vida – Lenita Bruno
• 1960: O Amor, o Sorriso e a Flor – João Gilberto
• 1965: The Swinger from Rio – Sérgio Mendes
• 1966: Love, Strings and Jobim
• 1995: Abandoned Garden – Michael Franks

With Stan Getz[link]

• 1962: Jazz Samba
• 1963: Getz/Gilberto
• 1963: Jazz Samba Encore!
• 1964: Getz/Gilberto Vol. 2
• 2007: Greatest Hits

With Frank Sinatra[link]

• 1967: Francis Albert Sinatra & Antonio Carlos Jobim
• 1971: Sinatra & Company
• 1979: Sinatra-Jobim Sessions
• 1994: Duets II

Compositions[link]

• "A Felicidade"
• "Água de Beber"
• "Anos Dourados" (Looks Like December)
• "As Praias Desertas"
• "Águas de Março" (Waters of March)
• "Bonita"
• "Brigas Nuca Mais"
• "Chega de Saudade" (No More Blues)
• "Chovendo na Roseira"
• "Corcovado" (Quiet Nights of Quiet Stars)
• "Desafinado" (Slightly Out of Tune)
• "Dindi"
• "Favela"
• "Fotografia"
• "Garota de Ipanema" (The Girl From Ipanema)
• "Insensatez (How Insensitive)"
• "Inútil Paisagem (If You Never Come to Me)"
• "Look to the Sky"
• "Meditação" (Meditation)
• "O Amor Em Paz" (Once I Loved)
• "Passarim"
• "Sabiá"
• "Samba de Uma Nota Só" (One Note Samba)
• "Samba do Avião"
• "So Danço Samba"
• "Só Tinha De Ser Com Você
• "Se Todos Fossem Iguais A Você (Someone to Light Up My Life)"
• "Tema de Amor de Gabriela"
• "Triste"
• "Vivo Sonhando" (Dreamer)
• "Vou te Contar" (Wave)

Concert films[link]

• 2002: Antonio Carlos Jobim: An All-Star Tribute (with Herbie Hancock, Joe Henderson, Shirley Horn, Jon Hendricks, Gonzalo Rubalcaba)^[12]
• 2007: Antonio Carlos Jobim In Concert
• 2007: Live at Montreal Jazz Festival
• 2009: Tom Jobim: The Waters of March

Notes[link]

References[link]

• McGowan, Chris; Pessanha, Ricardo (1998). The Brazilian Sound: Samba, Bossa Nova and the Popular Music of Brazil (2nd edition ed.). Philadelphia: Temple University Press. ISBN 1-56639-545-3, ISBN 1-56639-544-5. 
• Castro, Ruy (2000). Bossa Nova: The Story of the Brazilian Music That Seduced the World (1st English-Language Edition ed.). Chicago, IL: A Capella Books. ISBN 1-55652-409-9. 
• Cabral, Sergio (2008). Antonio Carlos Jobim - Uma Biografia (1st Edition ed.). São Pulo, Brazil: IBEP Nacional. ISBN 85-7865-011-5. 

External links[link]

• Nos Bastidores do legendario concerto da bossa nova no Carnegie Hall in 1962
• Jobim Guitar Tab Archives & Song Lyrics Page
• Antônio Carlos Jobim page at The Brazilian Sound
• Antônio Carlos Jobim at Find-A-Grave
• Interview, 1988
• Antônio Carlos Jobim: Clube do Tom
• Remembering Antonio Carlos Jobim
• Tribute to Tom Jobim Website

Herbie Hancock
Hancock in concert at the Nice Jazz Festival 2010
Background information
Birth name	Herbert Jeffrey Hancock
Also known as	Herbie Hancock
Born	(1940-04-12) April 12, 1940 (age 72) Chicago, Illinois, United States
Genres	Jazz, bebop, post bop, jazz fusion, hard bop, jazz-funk, funk, R&B, electro funk, classical
Occupations	Musician, composer, bandleader
Instruments	Piano, synthesizer, organ, clavinet, keytar, vocoder
Years active	1961–present
Labels	Columbia, Blue Note, Verve, Warner Bros.
Associated acts	Miles Davis Quintet, Jaco Pastorius, Joni Mitchell, Stevie Wonder
Website	www.herbiehancock.com

Herbert Jeffrey "Herbie" Hancock (born April 12, 1940) is an American pianist, keyboardist, bandleader and composer.^[1] As part of Miles Davis's Second Great Quintet, Hancock helped to redefine the role of a jazz rhythm section and was one of the primary architects of the "post-bop" sound. He was one of the first jazz musicians to embrace music synthesizers and funk music (characterized by syncopated drum beats). Hancock's music is often melodic and accessible; he has had many songs "cross over" and achieved success among pop audiences. His music embraces elements of funk and soul while adopting freer stylistic elements from jazz. In his jazz improvisation, he possesses a unique creative blend of jazz, blues, and modern classical music, with harmonic stylings much like the styles of Claude Debussy and Maurice Ravel.

Hancock's best-known solo works include "Cantaloupe Island", "Watermelon Man" (later performed by dozens of musicians, including bandleader Mongo Santamaría), "Maiden Voyage", "Chameleon", and the singles "I Thought It Was You" and "Rockit". His 2007 tribute album River: The Joni Letters won the 2008 Grammy Award for Album of the Year, only the second jazz album ever to win the award after Getz/Gilberto in 1965.

Hancock is a member of Sōka Gakkai International.^[2][3]

On July 22, 2011 at a ceremony in Paris, Hancock was named UNESCO Goodwill Ambassador for the promotion of Intercultural Dialogue.

Early life and career[link]

Hancock was born in Chicago, Illinois. Like many jazz pianists, Hancock started with a classical music education. He studied from age seven, and his talent was recognized early. Considered a child prodigy,^[4] he played the first movement of Mozart's Piano Concerto No. 5 at a young people's concert with the Chicago Symphony at age eleven.^[5]

Through his teens, Hancock never had a jazz teacher, but developed his ear and sense of harmony. He was also influenced by records of the vocal group the Hi-Lo's. He reported that:

the time I actually heard the Hi-Lo's, I started picking that stuff out; my ear was happening. I could hear stuff and that's when I really learned some much farther-out voicings -like the harmonies I used on 'Speak Like a Child' -just being able to do that. I really got that from Clare Fischer's arrangements for the Hi-Lo's. Clare Fischer was a major influence on my harmonic concept... He and Bill Evans, and Ravel and Gil Evans, finally. You know, that's where it came from.^[6]

In 1960, he heard Chris Anderson play just once, and begged him to accept him as a student.^[7] Hancock often mentions Anderson as his harmonic guru. Hancock left Grinnell College, moved to Chicago and began working with Donald Byrd and Coleman Hawkins, during which period he also took courses at Roosevelt University. (He later graduated from Grinnell with degrees in electrical engineering and music. Grinnell also awarded him an honorary Doctor of Fine Arts degree in 1972).^[5][8] Donald Byrd was attending the Manhattan School of Music in New York at the time and suggested that Hancock study composition with Vittorio Giannini, which he did for a short time in 1960. The pianist quickly earned a reputation, and played subsequent sessions with Oliver Nelson and Phil Woods. He recorded his first solo album Takin' Off for Blue Note Records in 1962. "Watermelon Man" (from Takin' Off) was to provide Mongo Santamaría with a hit single, but more importantly for Hancock, Takin' Off caught the attention of Miles Davis, who was at that time assembling a new band. Hancock was introduced to Davis by the young drummer Tony Williams, a member of the new band.

Miles Davis quintet and Blue Note[link]

Hancock received considerable attention when, in May 1963,^[5] he joined Miles Davis's Second Great Quintet. Davis personally sought out Hancock, whom he saw as one of the most promising talents in jazz. The rhythm section Davis organized was young but effective, comprising bassist Ron Carter, 17-year-old drummer Tony Williams, and Hancock on piano. After George Coleman and Sam Rivers each took a turn at the saxophone spot, the quintet would gel with Wayne Shorter on tenor saxophone. This quintet is often regarded as one of the finest jazz ensembles, and the rhythm section has been especially praised for its innovation and flexibility.

The second great quintet was where Hancock found his own voice as a pianist. Not only did he find new ways to use common chords, but he also popularized chords that had not previously been used in jazz. Hancock also developed a unique taste for "orchestral" accompaniment – using quartal harmony and Debussy-like harmonies, with stark contrasts then unheard of in jazz. With Williams and Carter he wove a labyrinth of rhythmic intricacy on, around and over existing melodic and chordal schemes. In the later half of the sixties their approach became so sophisticated and unorthodox that conventional chord changes would hardly be discernible; hence their improvisational concept would become known as "Time, No Changes". 

While in Davis's band, Hancock also found time to record dozens of sessions for the Blue Note label, both under his own name and as a sideman with other musicians such as Wayne Shorter, Tony Williams, Grant Green, Bobby Hutcherson, Sam Rivers, Donald Byrd, Kenny Dorham, Hank Mobley, Lee Morgan and Freddie Hubbard.

His albums Empyrean Isles (1964) and Maiden Voyage (1965) were to be two of the most famous and influential jazz LPs of the sixties, winning praise for both their innovation and accessibility (the latter demonstrated by the subsequent enormous popularity of the Maiden Voyage title track as a jazz standard, and by the jazz rap group US3 having a hit single with "Cantaloop" (derived from "Cantaloupe Island" on Empyrean Isles) some twenty five years later). Empyrean Isles featured the Davis rhythm section of Hancock, Carter and Williams with the addition of Freddie Hubbard on cornet, while Maiden Voyage also added former Davis saxophonist George Coleman (with Hubbard remaining on trumpet). Both albums are regarded as among the principal foundations of the post-bop style. Hancock also recorded several less-well-known but still critically acclaimed albums with larger ensembles – My Point of View (1963), Speak Like a Child (1968) and The Prisoner (1969) featured flugelhorn, alto flute and bass trombone. 1963's Inventions and Dimensions was an album of almost entirely improvised music, teaming Hancock with bassist Paul Chambers and two Latin percussionists, Willie Bobo and Osvaldo "Chihuahua" Martinez.

During this period, Hancock also composed the score to Michelangelo Antonioni's film Blowup, the first of many soundtracks he recorded in his career.

Davis had begun incorporating elements of rock and popular music into his recordings by the end of Hancock's tenure with the band. Despite some initial reluctance, Hancock began doubling on electric keyboards including the Fender Rhodes electric piano at Davis's insistence. Hancock adapted quickly to the new instruments, which proved to be instrumental in his future artistic endeavors.

Under the pretext that he had returned late from a honeymoon in Brazil, Hancock was dismissed from Davis's band. In the summer of 1968 Hancock formed his own sextet. However, although Davis soon disbanded his quintet to search for a new sound, Hancock, despite his departure from the working band, continued to appear on Miles Davis records for the next few years. Noteworthy appearances include In a Silent Way, A Tribute to Jack Johnson and On the Corner.

[edit] Fat Albert and Mwandishi

Hancock playing a Roland AX-7 keytar, at The Roundhouse, Camden, London, 2006

Hancock left Blue Note in 1969, signing up with Warner Bros. Records. In 1969, Hancock composed the soundtrack for the Bill Cosby animated children's television show Fat Albert and the Cosby Kids. Titled Fat Albert Rotunda, the album was mainly an R&B-influenced album with strong jazz overtones. One of the jazzier songs on the record, "Tell Me A Bedtime Story", was later re-worked as a more electronic sounding song for the Quincy Jones album, Sounds...and Stuff Like That.

Hancock became fascinated with accumulating musical gadgets and toys. Together with the profound influence of Davis's Bitches Brew, this fascination would culminate in a series of albums in which electronic instruments are coupled with acoustic instruments.

Hancock's first ventures into electronic music started with a sextet comprising Hancock, bassist Buster Williams and drummer Billy Hart, and a trio of horn players: Eddie Henderson (trumpet), Julian Priester (trombone), and multireedist Bennie Maupin. Dr. Patrick Gleeson was eventually added to the mix to play and program the synthesizers. In fact, Hancock was one of the first jazz pianists to completely embrace electronic keyboards.^{[citation needed]}

The sextet, later a septet with the addition of Gleeson, made three experimental albums under Hancock's name: Mwandishi (1971), Crossings (1972) (both on Warner Bros. Records), and Sextant (1973) (released on Columbia Records); two more, Realization and Inside Out, were recorded under Henderson's name with essentially the same personnel. The music often had very free improvisations and showed influence from the electronic music of some contemporary classical composers.

Synthesizer player Patrick Gleeson, one of the first musicians to play synthesizer on any jazz recording, introduced the instrument on Crossings, released in 1972, one of a handful of influential electronic jazz/fusion recordings to feature synthesizer that same year. On Crossings (as well as on I Sing the Body Electric), the synthesizer is used more as an improvisatory global orchestration device than as a strictly melodic instrument. This reflected Gleeson's (and Powell's) interest in contemporary European electronic music techniques and in the West Coast synthesis techniques of Morton Subotnick and other contemporaries, several of whom were resident at one time or another, as was Gleeson, at The Mills College Tape Music Center. An early review of Crossings in Downbeat magazine complained about the synthesizer, but a few years later the magazine noted in a cover story on Gleeson that he was "a pioneer" in the field of electronics in jazz. Gleeson used a modular Moog III for the recording of the album, but used an ARP 2600 synthesizer, and occasionally an ARP Soloist for the group's live performances. On Sextant Gleeson used the more compact ARP synthesizers instead of the larger Moog III for both studio and live performances. In the albums following The Crossings, Hancock started to play synth himself and unlike Gleeson, he plays it as a melodical and rhythm instrument just like electric pianos.

Hancock's three records released in 1971–1973, became later known as the "Mwandishi" albums, so-called after a Swahili name Hancock sometimes used during this era (Mwandishi is Swahili for writer). The first two, including Fat Albert Rotunda were made available on the 2-CD set Mwandishi: the Complete Warner Bros. Recordings, released in 1994, but are now sold as individual CD editions. Of the three electronic albums, Sextant is probably the most experimental since the Arp synthesizers are used extensively, and some advanced improvisation ("post-modal free impressionism") is found on the tracks "Hornets" and "Hidden Shadows" (which is in the meter 19/4). "Hornets" was later revised on the 2001 album Future2Future as "Virtual Hornets".

Among the instruments Hancock and Gleeson used were Fender Rhodes piano, ARP Odyssey, ARP 2600, ARP Pro Soloist Synthesizer, a Mellotron and the Moog synthesizer III.

All three Warner Bros. albums Fat Albert Rotunda, Mwandishi, and Crossings, were remastered in 2001 and released in Europe but were not released in the U.S.A. as of June 2005. In the Winter of 2006–7 a remastered edition of Crossings was announced and scheduled for release in the Spring.

[edit] Headhunters and Death Wish

Hancock performing in concert, 2006

After the sometimes "airy" and decidedly experimental "Mwandishi" albums, Hancock was eager to perform more "earthy" and "funky" music. The Mwandishi albums – though these days seen as respected early fusion recordings – had seen mixed reviews and poor sales, so it is probable that Hancock was motivated by financial concerns as well as artistic restlessness.^{[citation needed]} Hancock was also bothered by the fact that many people did not understand avant-garde music. He explained that he loved funk music, especially Sly Stone's music, so he wanted to try to make funk himself.

He gathered a new band, which he called The Headhunters, keeping only Maupin from the sextet and adding bassist Paul Jackson, percussionist Bill Summers, and drummer Harvey Mason. The album Head Hunters, released in 1973, was a major hit and crossed over to pop audiences, though it prompted criticism from some jazz fans. Head Hunters was recorded at Different Fur studios.

Despite charges of "selling out", Stephen Erlewine of Allmusic positively reviewed the album among other friendly critics, saying, "Head Hunters still sounds fresh and vital three decades after its initial release, and its genre-bending proved vastly influential on not only jazz, but funk, soul, and hip-hop."^[9]

Mason was replaced by Mike Clark, and the band released a second album, Thrust, the following year. (A live album from a Japan performance, consisting of compositions from those first two Head Hunters releases was released in 1975 as Flood. The record has since been released on CD in Japan.) This was almost as well received as its predecessor, if not attaining the same level of commercial success. The Headhunters made another successful album (called Survival of the Fittest) without Hancock, while Hancock himself started to make even more commercial albums, often featuring members of the band, but no longer billed as The Headhunters. The Headhunters reunited with Hancock in 1998 for Return of the Headhunters, and a version of the band (featuring Jackson and Clark) continues to play live and record.

In 1973, Hancock composed his second masterful soundtrack to the controversial film The Spook Who Sat By The Door. Then in 1974, he also composed the soundtrack to the first Death Wish film. One of his memorable songs, "Joanna's Theme", would later be re-recorded in 1997 on his duet album with Wayne Shorter 1 + 1.

Hancock's next jazz-funk albums of the 1970s were Man-Child (1975), and Secrets (1976), which point toward the more commercial direction Hancock would take over the next decade. These albums feature the members of the 'Headhunters' band, but also a variety of other musicians in important roles.

[edit] Back to the basics: VSOP and Future Shock

During late 1970s and early 1980s, Hancock toured with his "V.S.O.P." quintet, which featured all the members of the 1960s Miles Davis quintet except Davis, who was replaced by trumpet giant Freddie Hubbard. There was constant speculation that one day Davis would reunite with his classic band, but he never did so. VSOP recorded several live albums in the late 1970s, including VSOP (1976), and VSOP: The Quintet (1977).

In 1978, Hancock recorded a duet with Chick Corea, who had replaced him in the Miles Davis band a decade earlier. He also released a solo acoustic piano album titled The Piano (1978), which, like so many Hancock albums at the time, was initially released only in Japan. (It was finally released in the US in 2004.) Several other Japan-only releases have yet to surface in the US, such as Dedication (1974), VSOP: Tempest in the Colosseum (1977), and Direct Step (1978). Live Under the Sky was a VSOP album remastered for the US in 2004, and included an entire second concert from the July 1979 tour.

From 1978–1982, Hancock recorded many albums consisting of jazz-inflected disco and pop music, beginning with Sunlight (featuring guest musicians including Tony Williams and Jaco Pastorius on the last track) (1978). Singing through a vocoder, he earned a British hit,^[10] "I Thought It Was You", although critics were unimpressed.^[11] This led to more vocoder on the 1979 follow-up, Feets, Don't Fail Me Now, which gave him another UK hit in "You Bet Your Love".^[10] Albums such as Monster (1980), Magic Windows (1981), and Lite Me Up (1982) were some of Hancock's most criticized and unwelcomed albums, the market at the time being somewhat saturated with similar pop-jazz hybrids from the likes of former bandmate Freddie Hubbard. Hancock himself had quite a limited role in some of those albums, leaving singing, composing and even producing to others. Mr. Hands (1980) is perhaps the one album during this period that was critically acclaimed. To the delight of many fans, there were no vocals on the album, and one track featured Jaco Pastorius on bass. The album contained a wide variety of different styles, including a disco instrumental song, a Latin-jazz number and an electronic piece in which Hancock plays alone with the help of computers.

Hancock also found time to record more traditional jazz while creating more commercially oriented music. He toured with Tony Williams and Ron Carter in 1981, recording Herbie Hancock Trio, a five-track live album released only in Japan. A month later, he recorded Quartet with Wynton Marsalis, released in the US the following year. Hancock, Williams and Carter toured internationally with Wynton and his brother Branford Marsalis in what was affectionately known as "VSOP II". This quintet can be heard on Marsalis' debut album on Columbia (1981). In 1982 he contributed to the Simple Minds album New Gold Dream (81,82,83,84), playing a synthesizer solo on the track "Hunter and The Hunted".

In 1983, Hancock had a mainstream hit with the Grammy-award winning instrumental single "Rockit" from the album Future Shock. It was perhaps the first mainstream single to feature scratching, and also featured an innovative animated music video which was directed by Godley and Creme and showed several robot-like artworks by Jim Whiting. The video was a hit on MTV and reached No.8 in the UK.^[12] The video won five different categories at the inaugural MTV Video Music Awards. This single ushered in a collaboration with noted bassist and producer Bill Laswell. Hancock experimented with electronic music on a string of three LPs produced by Laswell: Future Shock (1983), Sound-System (1984) and Perfect Machine (1988). Despite the success of "Rockit", Hancock's trio of Laswell-produced albums (particularly the latter two) are among the most critically derided of his entire career, perhaps even more so than his erstwhile pop-jazz experiments. Hancock's level of actual contribution to these albums was also questioned, with some critics contending that the Laswell albums should have been labelled "Bill Laswell featuring Herbie Hancock".

During this period, he appeared onstage at the Grammy awards with Stevie Wonder, Howard Jones, and Thomas Dolby, in a famous synthesizer jam (The video on Youtube can be found here.). Lesser known works from the 1980s are the live album Jazz Africa and the studio album Village Life (1984), which were recorded with Gambian kora player Foday Musa Suso.^[13] Also, in 1985 Hancock performed as a guest on the album So Red The Rose by the Duran Duran offshoot group Arcadia. He also provided introductory and closing comments for the PBS rebroadcast in the United States of the BBC educational series from the mid-1980s, Rockschool (not to be confused with the most recent Gene Simmons' Rock School series).

In 1986, Hancock performed and acted in the film 'Round Midnight. He also wrote the score/soundtrack, for which he won an Academy Award for Original Music Score. Often he would write music for TV commercials. "Maiden Voyage", in fact, started out as a cologne advertisement. At the end of the Perfect Machine tour, Hancock decided to leave Columbia Records after a 15-plus-year relationship.

As of June 2005, almost half of his Columbia recordings have been remastered. The first three US releases, Sextant, Head Hunters and Thrust as well as the last four releases Future Shock, Sound-System, the soundtrack to Round Midnight and Perfect Machine. Everything released in America from Man-Child to Quartet has yet to be remastered. Some albums, made and initially released in the US, were remastered between 1999 and 2001 in other countries such as Magic Windows and Monster. Hancock also re-released some of his Japan-only releases in the West, such as The Piano.

1990s and later[link]

Hancock live in concert

After leaving Columbia, Hancock took a break. Then, with friends Ron Carter, Tony Williams, Wayne Shorter, and Davis admirer Wallace Roney, they recorded A Tribute to Miles, which was released in 1994. The album contained two live recordings and studio recording classics, with Roney playing Davis's part as trumpet player. The album won a Grammy for best group album. He also toured with Jack DeJohnette, Dave Holland and Pat Metheny in 1990 on their Parallel Realities tour, which included a memorable performance at the Montreux Jazz Festival in July 1990.

Hancock's next album, Dis Is Da Drum, released in 1994, saw him return to Acid Jazz. Also in 1994, he appeared on the Red Hot Organization's compilation album Stolen Moments: Red Hot + Cool. The album, meant to raise awareness and funds in support of the AIDS epidemic in relation to the African-American community, was heralded as "Album of the Year" by Time Magazine.

1995's The New Standard found Hancock and an all-star band including John Scofield, Jack DeJohnette and Michael Brecker interpreting pop songs by Nirvana, Stevie Wonder, The Beatles, Prince, Peter Gabriel and others. A 1997 duet album with Wayne Shorter entitled 1 + 1 was successful, the song "Aung San Suu Kyi" winning the Grammy Award for Best Instrumental Composition. Hancock also achieved great success in 1998 with his album Gershwin's World, which featured inventive readings of George and Ira Gershwin standards by Hancock and a plethora of guest stars, including Stevie Wonder, Joni Mitchell and Shorter. Hancock toured the world in the support of Gershwin's World with a sextet that featured Cyro Baptista, Terri Lynne Carrington, Ira Coleman, Eli Degibri and Eddie Henderson.

In 2001, Hancock recorded Future2Future, which reunited Hancock with Bill Laswell and featured doses of electronica as well as turntablist Rob Swift of The X-Ecutioners. Hancock later toured with the band, and released a live concert DVD with a different lineup which also included the "Rockit" music video. Also in 2001, Hancock partnered with Michael Brecker and Roy Hargrove to record a live concert album saluting Davis and John Coltrane entitled Directions in Music: Live at Massey Hall, recorded live in Toronto. The threesome toured to support the album, and toured on and off through 2005.

The year 2005 saw the release of a duet album called Possibilities. It features duets with Carlos Santana, Paul Simon, Annie Lennox, John Mayer, Christina Aguilera, Sting and others. In 2006, Possibilities was nominated for Grammy awards in two categories: "A Song For You", featuring Christina Aguilera was nominated for a Grammy Award for Best Pop Instrumental Performance, and "Gelo No Montanha", featuring Trey Anastasio on guitar, was nominated for a Grammy Award for Best Instrumental Performance. Neither nomination resulted in an award.

Also in 2005, Hancock toured Europe with a new quartet that included Beninese guitarist Lionel Loueke, and explored textures ranging from ambient to straight jazz to African music. Plus, during the summer of 2005, Hancock re-staffed the famous Headhunters and went on tour with them, including a performance at The Bonnaroo Music & Arts Festival.

However, this lineup did not consist of any of the original Headhunters musicians. The group included Marcus Miller, Terri Lyne Carrington, Lionel Loueke and John Mayer. Hancock also served as the first artist in residence for Bonnaroo that summer.

Also in 2006, Sony BMG Music Entertainment (which bought out Hancock's old label, Columbia Records) released the two-disc retrospective The Essential Herbie Hancock. This two-disc set is the first compilation of Herbie's work at Warner Bros. Records, Blue Note Records, Columbia and at Verve/Polygram. This became Hancock's second major compilation of work since the 2002 Columbia-onlyThe Herbie Hancock Box, which was released at first in a plastic 4x4 cube then re-released in 2004 in a long box set. Hancock also in 2006, recorded a new song with Josh Groban and Eric Mouquet (co-founder of Deep Forest) entitled "Machine". It is featured on Josh Groban's CD Awake. Hancock also recorded and improvised with guitarist Lionel Loueke on Loueke's debut album Virgin Forest on the ObliqSound label in 2006, resulting in two improvisational tracks "Le Réveil des Agneaux (The Awakening of the Lambs)" and "La Poursuite du lion (The Lion's Pursuit)".

Hancock, a longtime associate and friend of Joni Mitchell released a 2007 album, River: The Joni Letters, that paid tribute to her work. Norah Jones and Tina Turner recorded vocals,^[14] as did Corinne Bailey Rae, and Leonard Cohen contributed a spoken piece set to Hancock's piano. Mitchell herself also made an appearance. The album was released on September 25, simultaneously with the release of Mitchell's album Shine.^[15] "River" was nominated for and won the 2008 Album of the Year Grammy Award, only the second jazz album ever to receive either honor. The album also won a Grammy for Best Contemporary Jazz Album, and the song "Both Sides Now" was nominated for Best Instrumental Jazz Solo.

Recently Hancock performed at the Shriner's Children's Hospital Charity Fundraiser with Sheila E, Jim Brickman, Kirk Whalum and Wendy Alane Wright.

His latest work includes assisting the production of the Kanye West track "RoboCop", found on 808s & Heartbreak.

On June 14, 2008, Hancock performed at Rhythm on the Vine at the South Coast Winery in Temecula, California, for Shriners Hospital for Children. Other performers at the event, that raised $515,000 for Shriners Hospital, were contemporary music artist Jim Brickman, and Sheila E. & the E. Family Band.^[16]

On January 18, 2009, Hancock performed at the We Are One concert, marking the start of inaugural celebrations for American President Barack Obama.^[17] Hancock also performed the Rhapsody in Blue at the 2009 Classical BRIT Awards with classical pianist Lang Lang. Hancock was named as the Los Angeles Philharmonic's creative chair for jazz for 2010–12.^[18] In June 2010, Hancock released his newest album, The Imagine Project.

On June 5, 2010, Hancock received an Alumni Award from his alma mater, Grinnell College.^[19]

Discography[link]

Filmography[link]

As a Leader[link]

• 2000: Dejohnette, Hancock, Holland and Metheny – Live in Concert
• 2002: Herbie Hancock Trio: Hurricane! with Ron Carter and Billy Cobham^[20]
• 2002: The Jazz Channel Presents Herbie Hancock (BET on Jazz) with Cyro Baptista, Terri Lynne Carrington, Ira Coleman, Eli Degibri and Eddie Henderson (recorded in 2000)
• 2004: Herbie Hancock – Future2Future Live
• 2006: Herbie Hancock – Possibilities with John Mayer, Christina Aguilera, Joss Stone, and more

Awards[link]

Academy Awards[link]

• 1986, Original Soundtrack, for Round Midnight

Grammy Awards[link]

1. 1984, Best R&B Instrumental Performance, for Rockit
2. 1985, Best R&B Instrumental Performance, for Sound-System
3. 1988, Best Instrumental Composition, for Call Sheet Blues
4. 1995, Best Jazz Instrumental Performance, Individual Or Group, for A Tribute to Miles
5. 1997, Best Instrumental Composition, for Manhattan (Island Of Lights And Love)
6. 1999, Best Instrumental Arrangement Accompanying Vocal(s), for St. Louis Blues
7. 1999, Best Jazz Instrumental Performance, Individual Or Group, for Gershwin's World
8. 2003, Best Jazz Instrumental Album, Individual or Group, for Directions in Music: Live at Massey Hall
9. 2003, Best Jazz Instrumental Solo, for My Ship
10. 2005, Best Jazz Instrumental Solo, for Speak Like a Child
11. 2008, Album of the Year, for River: The Joni Letters
12. 2008, Best Contemporary Jazz Album, for River: The Joni Letters
13. 2011, Best Improvised Jazz Solo, for A Change Is Gonna Come
14. 2011, Best Pop Collaboration with Vocals, for Imagine

Playboy Music Poll[link]

• Best Jazz Group, 1985
• Best Jazz Keyboards, 1985
• Best Jazz Album – Rockit, 1985
• Best Jazz Keyboards, 1986
• Best R&B Instrumentalist, 1987
• Best Jazz Instrumentalist, 1988

Keyboard Magazine's Readers Poll[link]

• Best Jazz & Pop Keyboardist, 1983
• Best Jazz Pianist, 1987
• Best Jazz Keyboardist, 1987
• Best Jazz Pianist, 1988

Other notable awards[link]

• MTV Awards (5 awards in total) – Best Concept Video – Rockit, 1983–84
• Gold Note Jazz Awards – NY Chapter of the National Black MBA Association, 1985
• French Award Officer of the Order of Arts & Letters - Paris, 1985
• BMI Film Music Award Round Midnight, 1986
• U.S. Radio Award "Best Original Music Scoring – Thom McAnn Shoes", 1986
• Los Angeles Film Critics Association "Best Score – Round Midnight", 1986
• BMI Film Music Award Colors, 1989
• Miles Davis Award, granted by the Montreal International Jazz Festival, 1997
• Soul Train Music Award "Best Jazz Album – The New Standard", 1997
• Festival International Jazz de Montreal Prix Miles Davis, 1997
• VH1's 100 Greatest Videos Rockit is "10th Greatest Video", 2001
• NEA Jazz Masters Award, 2004
• Downbeat Magazine Readers Poll Hall of Fame, 2005

^[21]

References[link]

External links[link]

• Official website of Herbie Hancock
• Official Herbie Hancock MySpace page
• Herbie Hancock at Verve Records
• River:The Joni Letters at Verve Records
• Possibilities Herbie Hancock
• Discography Herbie Hancock Discography
• Interview with Herbie Hancock on music and technology from AppleMatters
• Interview with Herbie Hancock on the "Possibilities" album release from LiveDaily
• Herbie Hancock: Outside The Comfort Zone Herbie Hancock interview from JamBase
• "Herbie Hancock: Essential Recordings" by Ted Gioia (www.jazz.com)
• Herbie Hancock's Grinnell College Alumni Award citation from Grinnell College Alumni Assembly on June 5, 2010.