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.

Guthrie Govan
Background information
Birth name	Guthrie Govan
Born	(1971-12-27) 27 December 1971 (age 40) Chelmsford, Essex
Genres	Jazz fusion, rock, blues, progressive rock, funk, country, bluegrass
Instruments	Guitar
Years active	1991–present
Labels	Cornford Records
Associated acts	Erotic Cakes, Young Punx, The Fellowship, GPS, Asia, Dizzee Rascal, Docker's Guild, The Aristocrats
Website	Official website
Notable instruments
Suhr Standards Classic Classic T

Guthrie Govan (born 27 December 1971 in Chelmsford, Essex, England) is a virtuoso guitarist known for his work with the bands Asia (2001–2006), GPS, The Young Punx and The Fellowship as well as Erotic Cakes (a vehicle for his own music). He is a noted guitar teacher through his work with the UK magazine Guitar Techniques, Guildford’s Academy of Contemporary Music and currently the Brighton Institute of Modern Music. He is the 1993 winner of Guitarist magazine's "Guitarist of the Year" competition.

Biography and early playing career[link]

Govan began playing guitar aged three, encouraged by his father but initially learning mainly by ear. At the age of nine he and his brother Seth Govan played guitar on a Thames Television programme called Ace Reports. At secondary school he was exposed, via older classmates, to "shred" guitarists of the time.^[1]

After leaving school, Govan read English at the University of Oxford, though he left after a year to pursue a career in music. Around this time (by Govan’s own estimation, 1991^[1]) he sent demos of his work to Mike Varney of Shrapnel Records. Varney was impressed and offered him a record deal; ultimately however, Govan declined. Regarding his reasons he has explained: "it was as though all I really wanted to know was that I was good enough […] I found I was getting a bit wary of the shred movement."^[2]

In 1993 he won Guitarist magazine’s "Guitarist of the Year" competition with his instrumental piece Wonderful Slippery Thing (a version of which would eventually appear on his debut solo album); the demo of the track earned him a place amongst several other entrants in the live final, which he then won. Subsequently, he submitted a sample transcription (of a Shawn Lane piece) to Guitar Techniques magazine; this earned him a job as a contributor to the magazine, ending a spell working in fast food.^[1]

Guitar tuition[link]

Govan has since worked with Guitar Techniques,^[1] including an article providing a transcription of his track Wonderful Slippery Thing.^[3] Since the late 1990s, Govan has taught at the Guitar Institute in Acton, Thames Valley University, and the Academy of Contemporary Music; he currently teaches at Brighton Institute of Modern Music. In this context, he is known for his ability to teach a wide range of styles.^[1] Govan has published two books on guitar playing: Creative Guitar Volume 1: Cutting Edge Techniques and Creative Guitar Volume 2: Advanced Techniques.^[4][5]

Playing career in the 2000s[link]

Asia, GPS and The Young Punx[link]

Govan began his involvement with Asia playing on the album Aura. With his work on the album complete, Govan was added to the Asia line-up for the tour to support the new album. Bad Asteroid, an instrumental in the live shows, was a Govan original composition; it dates back to his early 1990s demo. He went on to play on the band’s 2004 album Silent Nation.

In 2006, Asia keyboardist Geoff Downes left to reform an earlier Asia line-up; Govan and the other two band members, John Payne and Jay Schellen, along with and Erik Norlander continued as Asia featuring John Payne. Govan left in mid-2009 and was replaced by Mitch Perry. Payne, Govan and Schellen also formed GPS (named after the members' initials) and after the addition of Ryo Okumoto on keyboards the band released the album Window to the Soul (2006).

Govan performs on a number of tracks on Your Music Is Killing Me, the debut album of UK electronic dance music act The Young Punx and is part of their live act. He also appears in the music video for their track Rockall.

Govan has played live guitar for Dizzee Rascal, including a performance on Later With Jools Holland.^[6]

Docker's Guild[link]

In December 2010, Govan performed on all tracks on Docker's Guild album The Mystic Technocracy - Season 1: The Age of Ignorance, due for release in 2012 on Lion Music. The project also features some other Asia artists, notably John Payne on vocals and Simon Hanhart, who mixed the album. Other special guests include Gregg Bissonette (drums), Tony Franklin (bass), Jeff Watson (guitars), Douglas R. Docker (keyboards), Goran Edman, Tony Mills and Amanda Somerville (vocals).^[7]

Erotic Cakes and The Fellowship[link]

Govan’s debut solo album, Erotic Cakes, was released through Cornford Records in August 2006. In the album's sleeve notes, Govan states the album is named after an episode of The Simpsons; he also explains that when playing live, the band is billed as Erotic Cakes rather than simply Guthrie Govan.^[8] In addition to bass by Govan’s brother Seth and drumming by Pete Riley, the album features guitar solos by Richie Kotzen (whose Los Angeles studio was used to record the guitar tracks on the album) on Ner Ner and Bumblefoot on Rhode Island Shred respectively. Seth Govan and Pete Riley also complete the Erotic Cakes live band. Previous versions of Waves and Rhode Island Shred appeared on the compilation Guitar on the Edge, Vol 1. no.4 (Legato Records, 1993).^[8] A version of his Guitarist of the Year winning piece, Wonderful Slippery Thing appears on the album. While this is his first album, Govan had been collaborating with drummer Pete Riley on the music since the mid-1990s.^[1]

The Erotic Cakes band line up, with the addition of saxophonist Zak Barrett, also forms jazz-fusion band The Fellowship. The band plays at the Bassment club in Chelmsford, Essex, every Thursday night and has done so for several years.^[1][9]

Playing career in the 2010s[link]

In late 2011, Govan announced a new tour with new band The Aristocrats, featuring Bryan Beller on bass and Marco Minnemann on the drums. Their self-titled album was released worldwide in September 2011.

In 2012, TBS began running a TV ad for the popular show "House of Payne" that features part of his solo from his jam track "Les is More" available through www.JamTrackCentral.com

Influences and technique[link]

Govan’s earliest influences were Jimi Hendrix and Cream-era Eric Clapton; as such he describes himself as coming from a "blues rock background".^[1] While he is "wary" of 1980s technique driven guitar music ("shred") he cites the imagination of Steve Vai (as well as Frank Zappa, with whom Vai played) and the passion of Yngwie Malmsteen as influences. Jazz and fusion elements are an important part of his style: he cites Joe Pass as a pivotal influence in this respect.^[2]

Govan is known for his virtuoso command of the electric guitar, due to both his technical ability and proficiency in various styles. Govan’s MySpace profile lists quotes from various guitarists to this effect;^[10] elsewhere, fellow virtuosos Joe Satriani and Paul Gilbert have praised Govan's playing.^[11][12]

Equipment[link]

Govan mainly uses various Suhr guitars: three different Guthrie Govan Signature Model guitars, three custom Standards, a Modern 24-fret model, a Classic and Classic T.^[13] He plays Vigier fretless guitar. Govan also favours Cornford amps: the sleeve notes of Erotic Cakes state he has used an RK100, MK50 and Hellcat.^[1] In the last European tour of 2010 and 2011 he played Brunetti amplifiers (CustomWork Mercury 50) like his friend Dave Kilminster, and he recorded live his last CD/DVD during the Lapsus (Torino, Italy) show on the December 8th 2011.^{[citation needed]}

On December 17, 2011 Guthrie and the Erotic Cakes band had their gear stolen after a show in Rome. Among the equipment stolen were Brunetti Mercury heads, a 1989 Warwick Streamer bass, yet-to-be-edited footage from the show played at Lapsus, Torino IT and more.^[14]

References[link]

External links[link]

• Official Guthrie Govan Web Site
• Cornford Records on Myspace Samples of the debut album
• A YouTube clip of Govan improvising Sample of his fusion-jazz style
• Another YouTube clip of Govan improvising Bluesy style showcasing Govan's string overbends to create melody
• Official Aristocrats Band Site