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# wave

[weyv]
wave, in oceanography, an oscillating movement up and down, of a body of water caused by the frictional drag of the wind, or on a larger scale, by submarine earthquakes, volcanoes, and landslides. In seismology, waves moving though the earth are caused by the propagation of a disturbance generated by an earthquake or explosion. In atmospheric science, waves are periodic disturbances in the air flow.

## Oceanographic Waves

In a body of water, waves consist of a series of crests and troughs, where wavelength is the distance between two successive crests (or successive troughs). As waves are generated, the water particles are set in motion, following vertical circular orbits. Water particles momentarily move forward as the wave crest passes and backward as the trough passes. Thus, except for a slight forward drag, the water particles remain in essentially the same place as successive waves pass. The orbital motion of the water particles decreases in size at depths below the surface, so that at a depth equal to about one half of the wave's length, the water particles are barely oscillating back and forth. Thus, for even the largest waves, their effect is negligible below a depth of 980 ft (300 m).

The height and period of water waves in the deep ocean are determined by wind velocity, the duration of the wind, and the fetch (the distance the wind has blown across the water). In stormy areas, the waves are not uniform but form a confusing pattern of many waves of different periods and heights. Storms also produce white caps at wind speeds c.8 mi per hr (13 km per hr). Major storm waves can be over a half mile long and travel close to c.25 mi per hr (40 km per hour). A wave in the Gulf of Mexico associated with Hurricane Ivan (2004) measured 91 ft (27.7 m) high, and scientists believe that other waves produced by Ivan may have reached as much as 132 ft (40 m) high. Waves of similar heights, sometimes called rogue waves, most commonly occur in regions of strong ocean currents, which can amplify wind-driven waves when they flow in opposing directions; sandbanks may also act to focus wave energy and give rise to rogue waves.

When waves approach a shore, the orbital motion of the water particles becomes influenced by the bottom of the body of water and the wavelength decreases as the wave slows. As the water becomes shallower the wave steepens further until it "breaks" in a breaker, or surf, carrying the water forward and onto the beach in a turbulent fashion. Because waves usually approach the shore at an angle, a longshore (littoral) current is generated parallel to the shoreline. These currents can be effective in eroding and transporting sediment along the shore (see coast protection; beach).

In many enclosed or partly enclosed bodies of water such as lakes or bays, a wave form called a standing wave, or seiche, commonly develops as a result of storms or rapid changes in air pressure. These waves do not move forward, but the water surface moves up and down at antinodal points, while it remains stationary at nodal points.

Internal waves can form within waters that are density stratified and are similar to wind-driven waves. They usually cannot be seen on the surface, although oil slicks, plankton, and sediment tend to collect on the surface above troughs of internal waves. Any condition that causes waters of different density to come into contact with one another can lead to internal waves. They tend to have lower velocities but greater heights than surface waves. Very little is known about internal waves, which may move sediment on deeper parts of continental shelves.

Just as a rock dropped into water produces waves, sudden displacements such as landslides and earthquakes can produce high energy waves of short duration that can devastate coastal regions (see tsunami). Hurricanes traveling over shallow coastal waters can generate storm surges that in turn can cause devastating coastal flooding (see under storm).

## Seismic and Atmospheric Waves

Seismic waves are generated in the earth by the movements of earthquakes or explosions. Depending on the material traveled through, surface and internal waves move at variable velocities. Layers of the earth, including the core, mantle, and crust, have been discerned using seismic wave profiles. Seismic waves from explosions have been used to understand the subsurface structure of the crust and upper mantle and in the exploration for oil and gas deposits. Atmospheric waves are caused by differences in temperature, the Coriolis effect, and the influence of highlands.

wave, in physics, the transfer of energy by the regular vibration, or oscillatory motion, either of some material medium or by the variation in magnitude of the field vectors of an electromagnetic field (see electromagnetic radiation). Many familiar phenomena are associated with energy transfer in the form of waves. Sound is a longitudinal wave that travels through material media by alternatively forcing the molecules of the medium closer together, then spreading them apart. Light and other forms of electromagnetic radiation travel through space as transverse waves; the displacements at right angles to the direction of the waves are the field intensity vectors rather than motions of the material particles of some medium. With the development of the quantum theory, it was found that particles in motion also have certain wave properties, including an associated wavelength and frequency related to their momentum and energy. Thus, the study of waves and wave motion has applications throughout the entire range of physical phenomena.

## Classification of Waves

Waves may be classified according to the direction of vibration relative to that of the energy transfer. In longitudinal, or compressional, waves the vibration is in the same direction as the transfer of energy; in transverse waves the vibration is at right angles to the transfer of energy; in torsional waves the vibration consists of a twisting motion as the medium rotates back and forth around the direction of energy transfer. The three types of waves are illustrated by an example in which a coil spring is held stretched out by two persons. If the person holding one end pulls a few coils toward himself and releases them, a longitudinal wave will travel along the spring, with coils alternately being pressed closer together, then stretched apart, as the wave passes. If the first person then shakes his end up and down or from side to side, a transverse wave will travel along the spring. Finally, if he grabs several coils and twists them around the axis of the spring, a torsional wave will travel along the spring.

A wave may be a combination of types. Water waves in deep water are mainly transverse. However, as they approach a shore they interact with the bottom and acquire a longitudinal component. When the longitudinal component becomes very large compared to the transverse component, the wave breaks.

## Parameters of Waves

The maximum displacement of the medium in either direction is the amplitude of the wave. The distance between successive crests or successive troughs (corresponding to maximum displacements in the same direction) is the wavelength of the wave. The frequency of the wave is equal to the number of crests (or troughs) that pass a given fixed point per unit of time. Closely related to the frequency is the period of the wave, which is the time lapse between the passage of successive crests (or troughs). The frequency of a wave is the inverse of the period.

One full wavelength of a wave represents one complete cycle, that is, one complete vibration in each direction. The various parts of a cycle are described by the phase of the wave; all waves are referenced to an imaginary synchronous motion in a circle; thus the phase is measured in angular degrees, one complete cycle being 360°. Two waves whose corresponding parts occur at the same time are said to be in phase. If the two waves are at different parts of their cycles, they are out of phase. Waves out of phase by 180° are in phase opposition. The various phase relationships between combining waves determines the type of interference that takes place.

The speed of a wave is determined by its wavelength λ and its frequency ν, according to the equation v=λν, where v is the speed, or velocity. Since frequency is inversely related to the period T, this equation also takes the form v=λ/T. The speed of a wave tells how quickly the energy it carries is being transferred. It is important to note that the speed is that of the wave itself and not of the medium through which it is traveling. The medium itself does not move except to oscillate as the wave passes.

## Wave Fronts and Rays

In the graphic representation and analysis of wave behavior, two concepts are widely used—wave fronts and rays. A wave front is a line representing all parts of a wave that are in phase and an equal number of wavelengths from the source of the wave. The shape of the wave front depends upon the nature of the source; a point source will emit waves having circular or spherical wave fronts, while a large, extended source will emit waves whose wave fronts are effectively flat, or plane. A ray is a line extending outward from the source and representing the direction of propagation of the wave at any point along it. Rays are perpendicular to wave fronts.

Principle that subatomic particles possess some wavelike characteristics, and that electromagnetic waves, such as light, possess some particlelike characteristics. In 1905, by demonstrating the photoelectric effect, Albert Einstein showed that light, which until then had been thought of as a form of electromagnetic wave (see electromagnetic radiation), must also be thought of as localized in packets of discrete energy (see photon). In 1924 Louis-Victor Broglie proposed that electrons have wave properties such as wavelength and frequency; their wavelike nature was experimentally established in 1927 by the demonstration of their diffraction. The theory of quantum electrodynamics combines the wave theory and the particle theory of electromagnetic radiation.

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or abrasion platform

Gently sloping rock ledge that extends from the high-tide level at a steep cliff base to below the low-tide level. It develops as a result of wave abrasion; beaches protect the shore from abrasion and therefore prevent the formation of platforms. A platform is broadened as waves erode a notch at the base of the sea cliff, causing overhanging rock to fall. As the sea cliffs are attacked, weak rocks are quickly eroded, leaving the more resistant rocks as protrusions.

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Device that constrains the path of electromagnetic waves (see electromagnetic radiation). It can be used to transmit power or signals in the form of waves while minimizing power loss. Common examples are metallic tubes, coaxial cables, and optical fibres (see fibre optics). Waveguides transmit energy by propagating transmitted electromagnetic waves through the inside of a tube to a receiver at the other end. Metal waveguides are used in such technologies as microwave ovens, radar systems, radio relay systems, and radio telescopes.

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Variable quantity that mathematically describes the wave characteristics of a particle. It is related to the likelihood of the particle being at a given point in space at a given time, and may be thought of as an expression for the amplitude of the particle wave, though this is strictly not physically meaningful. The square of the wave function is the significant quantity, as it gives the probability for finding the particle at a given point in space and time. Seealso wave-particle duality.

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Imaginary surface that represents corresponding points of waves vibrating in unison. As identical waves from the same source travel through a homogeneous medium, corresponding crests and troughs are in phase at any instant; that is, they have completed the same fraction of their periodic motion. Any surface drawn through all points of the same phase constitutes a wave front.

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In oceanography, a ridge or swell on the surface of a body of water, normally having a forward motion distinct from the motions of the particles that compose it. Ocean waves are fairly regular, with an identifiable wavelength between adjacent crests and with a definite frequency of oscillation. Waves result when a generating force (usually the wind) displaces surface water and a restoring force returns it to its undisturbed position. Surface tension alone is the restoring force for small waves. For large waves, gravity is more important.

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Vibrational or stress waves in elastic media that have a frequency above 20 kilohertz, the highest frequency of sound waves that can be detected by the human ear. They can be generated or detected by piezoelectric transducers (see piezoelectricity). High-power ultrasonics produce distortion in a medium; applications include ultrasonic welding, drilling, irradiation of fluid suspensions (as in wine clarification), cleaning of surfaces (such as jewelry), and disruption of biological structures. Low-power ultrasonic waves do not cause distortions; uses include sonar, structure testing, and medical imaging and diagnosis. Some animals, including bats, employ ultrasonic echolocation for navigation.

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or seismic sea wave or tidal wave

Catastrophic ocean wave, usually caused by a submarine earthquake. Underwater or coastal landslides or volcanic eruptions also may cause tsunamis. The term tsunami is Japanese for “harbour wave.” The term tidal wave is a misnomer, because the wave has no connection with the tides. Perhaps the most destructive tsunami ever occurred in 2004 in the Indian Ocean, after an earthquake struck the seafloor off the Indonesian island of Sumatra. More than 200,000 people were killed in Indonesia, Thailand, India, Sri Lanka and other countries as far away as Somalia on the Horn of Africa.

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Artificial imitation of sound to accompany action and supply realism in a dramatic production. Sound effects were first used in the theatre, where they can represent a range of action too vast or difficult to present onstage, from battles and gunshots to trotting horses and rainstorms. Various methods were devised by backstage technicians to reproduce sounds (e.g., rattling sheet metal to create thunder); today most sound effects are reproduced by recordings. An important part of old-fashioned radio dramas, sound effects are still painstakingly added to television and movie soundtracks.

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Sharp rise in aerodynamic drag that occurs as an aircraft approaches the speed of sound. At sea level the speed of sound is about 750 miles (1,200 km) per hour, and at 36,000 feet (11,000 metres) it is about 650 miles (1,050 km) per hour. The sound barrier was formerly an obstacle to supersonic flight. If an aircraft flies at somewhat less than sonic speed, the pressure waves (sound waves) it creates outspeed their sources and spread out ahead of it. Once the aircraft reaches sonic speed the waves are unable to get out of its way. Strong local shock waves form on the wings and body; airflow around the craft becomes unsteady, and severe buffeting may result, with serious stability difficulties and loss of control over flight characteristics. Generally, aircraft properly designed for supersonic flight have little difficulty in passing through the sound barrier, but the effect on those designed for efficient operation at subsonic speeds may become extremely dangerous. The first pilot to break the sound barrier was Chuck Yeager (1947), in the experimental X-1 aircraft.

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Mechanical disturbance that propagates as a longitudinal wave through a solid, liquid, or gas. A sound wave is generated by a vibrating object. The vibrations cause alternating compressions (regions of crowding) and rarefactions (regions of scarcity) in the particles of the medium. The particles move back and forth in the direction of propagation of the wave. The speed of sound through a medium depends on the medium's elasticity, density, and temperature. In dry air at 32 °F (0 °C), the speed of sound is 1,086 feet (331 metres) per second. The frequency of a sound wave, perceived as pitch, is the number of compressions (or rarefactions) that pass a fixed point per unit time. The frequencies audible to the human ear range from approximately 20 hertz to 20 kilohertz. Intensity is the average flow of energy per unit time through a given area of the medium and is related to loudness. Seealso acoustics; ear; hearing; ultrasonics.

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formerly Melville Sound

Body of water, northern Canada. Located in the Arctic Archipelago, between Melville and Victoria islands, the sound is 250 mi (400 km) long and 100 mi (160 km) wide. Its discovery, when reached from the east (1819–20) by William E. Parry and from the west (1850–54) by Robert McClure, proved the existence of the Northwest Passage. The sound is navigable only under favourable weather conditions.

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Deep inlet, Norwegian Sea, eastern central coast of Greenland. It runs inland for 70 mi (110 km) and has numerous fjords (the longest is 280 mi, or 451 km) and two large islands. It was charted by William Scoresby in 1822.

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Arm of the Pacific Ocean indenting northwestern Washington, U.S. It was explored by the British navigator George Vancouver in 1792 and named by him for Peter Puget, a second lieutenant in his expedition, who probed the main channel. It has many deepwater harbours, including Seattle, Tacoma, Everett, and Port Townsend, which are shipping ports for the rich farmlands along the river estuaries. It provides a sheltered area for recreational boating and salmon fishing.

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Inlet of the Gulf of Alaska, southern Alaska, U.S. It lies east of the Kenai Peninsula and spans 90–100 mi (145–160 km). It was named by the British captain George Vancouver in 1778 to honour a son of George III. In 1989 one of the largest oil spills in history occurred when the tanker Exxon Valdez ran aground on Bligh Reef and lost 10.9 million gallons of crude oil into the sound, with disastrous effects on its ecology.

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Shallow body of water, eastern shore of North Carolina, U.S. It is separated from the Atlantic Ocean by the Outer Banks. It extends 80 mi (130 km) south from Roanoke Island and is 8–30 mi (13–48 km) wide. Numerous waterfowl nest along the coastal waters; there is some commercial fishing, especially for oysters.

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Inlet of the Tasman Sea, southwestern coast of South Island, New Zealand. About 2 mi (3 km) wide, the sound extends inland for 12 mi (19 km). It was named by a whaler in the 1820s for its resemblance to Milford Haven in Wales. It is the northernmost fjord in Fiordland National Park and is the site of Milford Sound town, one of the region's few permanently inhabited places.

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Bay, western extension of the Ross Sea, Antarctica. Lying at the edge of the Ross Ice Shelf, the channel is 92 mi (148 km) long and up to 46 mi (74 km) wide; it has been a major centre for Antarctic explorations. First discovered in 1841 by Scottish explorer James C. Ross, it served as one of the main access routes to the Antarctic continent. Ross Island, on the shores of the sound, was the site of headquarters for British explorers Robert Falcon Scott and Ernest Shackleton.

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Body of water between the southern shore of Connecticut and the northern shore of Long Island, New York, U.S. It connects with the East River and with Block Island Sound. Covering 1,180 sq mi (3,056 sq km), it is 90 mi (145 km) long and 3–20 mi (5–32 km) wide. Its shores have many residential communities and summer resorts.

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Coastal inlet, northeastern North Carolina, U.S. Protected from the Atlantic Ocean by the Outer Banks, it is about 50 mi (80 km) long and 5–14 mi (8–23 km) wide. It is connected with Chesapeake Bay by the Dismal Swamp Canal and the Albemarle and Chesapeake Canal. Elizabeth City is its chief port. Explored by Ralph Lane in 1586, it was later named for George Monck, duke of Albemarle.

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Vibration generated by an earthquake, explosion, or similar phenomenon and propagated within the Earth or along its surface. Earthquakes generate two principal types of waves: body waves, which travel within the Earth, and surface waves, which travel along the surface. Seismograms (recorded traces of the amplitude and frequency of seismic waves) yield information about the Earth and its subsurface structure; artificially generated seismic waves are used in oil and gas prospecting.

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Energy propagated through free space or through a material medium in the form of electromagnetic waves. Examples include radio waves, infrared radiation, visible light, ultraviolet radiation, X rays, and gamma rays. Electromagnetic radiation exhibits wavelike properties such as reflection, refraction, diffraction, and interference, but also exhibits particlelike properties in that its energy occurs in discrete packets, or quanta. Though all types of electromagnetic radiation travel at the same speed, they vary in frequency and wavelength, and interact with matter differently. A vacuum is the only perfectly transparent medium; all others absorb some frequencies of electromagnetic radiation.

A wave is a disturbance that propagates through space and time, usually with transference of energy. While a mechanical wave exists in a medium (which on deformation is capable of producing elastic restoring forces), waves of electromagnetic radiation (and probably gravitational radiation) can travel through vacuum, that is, without a medium. Waves travel and transfer energy from one point to another, often with little or no permanent displacement of the particles of the medium (that is, with little or no associated mass transport); instead there are oscillations around almost fixed locations.

## Definitions

Agreeing on a single, all-encompassing definition for the term wave is non-trivial. A vibration can be defined as a back-and-forth motion around a point m around a reference value. However, defining the necessary and sufficient characteristics that qualify a phenomenon to be called a wave is, at least, flexible. The term is often understood intuitively as the transport of disturbances in space, not associated with 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: 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 / light waves in a vacuum, where the concept of medium does not apply.

For such reasons, wave theory represents a peculiar branch of physics that is concerned with the properties of wave processes independently from their physical origin (Ostrovsky and Potapov, 1999). The peculiarity lies in the fact that this independence from physical origin is accompanied by a heavy reliance on origin when describing any 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-like transfer / transformation of vibratory energy. Concepts such as mass, momentum, inertia, or elasticity, become therefore crucial in describing acoustic (as opposed to optic) wave processes. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved (for example, in the case of air: vortices, radiation pressure, shock waves, etc., in the case of solids: Rayleigh waves, dispersion, etc., and so on).

Other properties, however, although they are usually described in an origin-specific manner, may be generalized to all waves. For example, based on the mechanical origin of acoustic waves there can be a moving disturbance in space-time 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 (or rather infinitely fast wave motion). 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 (or rather infinitely slow 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 wave phenomena with origins different from that of sound waves can be equally significant to the understanding of sound phenomena. A relevant example is Young's principle of interference (Young, 1802, in Hunt, 1978: 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.

## Characteristics

Periodic waves are characterized by crests (highs) and troughs (lows), and may usually be categorized as either longitudinal or transverse. Transverse waves are those with vibrations perpendicular to the direction of the propagation of the wave; examples include waves on a string and electromagnetic waves. Longitudinal waves are those with vibrations parallel to the direction of the propagation of the wave; examples include most sound waves.

When an object bobs up and down on a ripple in a pond, it experiences an orbital trajectory because ripples are not simple transverse sinusoidal waves .

Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths.

All waves have common behavior under a number of standard situations. All waves can experience the following:

### Polarization

A wave is polarized, if it can only oscillate in one direction. 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, because for these waves the direction of oscillation is along the direction of travel. A wave can be polarized by using a polarizing filter.

### Examples

Examples of waves include:

## Mathematical description

From a mathematical point of view, the most primitive or fundamental wave is harmonic (sinusoidal) wave which is described by the equation $f\left(x,t\right) = Asin\left(omega t-kx\right)\right),$ where $A$ is the amplitude of a wave - a measure of the maximum disturbance in the medium during one wave cycle (the maximum distance from the highest point of the crest to the equilibrium). In the illustration to the right, this is the maximum vertical distance between the baseline and the wave. The units of the amplitude depend on the type of wave — waves on a string have an amplitude expressed as a distance (meters), sound waves as pressure (pascals) and electromagnetic waves as the amplitude of the electric field (volts/meter). The amplitude may be constant (in which case the wave is a c.w. or continuous wave), or may vary with time and/or position. The form of the variation of amplitude is called the envelope of the wave.

The wavelength (denoted as $lambda$) is the distance between two sequential crests (or troughs). This generally is measured in meters; it is also commonly measured in nanometers for the optical part of the electromagnetic spectrum.

A wavenumber $k$ can be associated with the wavelength by the relation


k = frac{2 pi}{lambda}. ,

The period $T$ is the time for one complete cycle for an oscillation of a wave. The frequency $f$ (also frequently denoted as $nu$) is how many periods per unit time (for example one second) and is measured in hertz. These are related by:


f=frac{1}{T}. ,

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

The angular frequency $omega$ represents the frequency in terms of radians per second. It is related to the frequency by


omega = 2 pi f = frac{2 pi}{T}. ,

There are two velocities that are associated with waves. The first is the phase velocity, which gives the rate at which the wave propagates, is given by

$v_p = frac\left\{omega\right\}\left\{k\right\} = \left\{lambda\right\}f.$

The second is the group velocity, which gives the velocity at which variations in the shape of the wave's amplitude propagate through space. This is the rate at which information can be transmitted by the wave. It is given by


v_g = frac{partial omega}{partial k}. ,

### The wave equation

The wave equation is a differential equation that describes the evolution of a harmonic wave over time. The equation has slightly different forms depending on how the wave is transmitted, and the medium it is traveling through. Considering a one-dimensional wave that is traveling down a rope along the x-axis with velocity $v$ and amplitude $u$ (which generally depends on both x and t), the wave equation is


frac{1}{v^2}frac{partial^2 u}{partial t^2}=frac{partial^2 u}{partial x^2}. ,

In three dimensions, this becomes


frac{1}{v^2}frac{partial^2 u}{partial t^2} = nabla^2 u. ,

where $nabla^2$ is the Laplacian.

The velocity v will depend on both the type of wave and the medium through which it is being transmitted.

A general solution for the wave equation in one dimension was given by d'Alembert. It is


u(x,t)=F(x-vt)+G(x+vt). ,

This can be viewed as two pulses traveling down the rope in opposite directions; F in the +x direction, and G in the −x direction. If we substitute for x above, replacing it with directions x, y, z, we then can describe a wave propagating in three dimensions.

The Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of this equation are wave functions which can be used to describe the probability density of a particle. Quantum mechanics also describes particle properties that other waves, such as light and sound, have on the atomic scale and below.

### Traveling waves

Simple wave or a traveling wave, also sometimes called a progressive wave is a disturbance that varies both with time $t$ and distance $z$ in the following way:


y(z,t) = A(z, t)sin (kz - omega t + phi), ,

where $A\left(z,t\right)$ is the amplitude envelope of the wave, $k$ is the wave number and $phi$ is the phase. The phase velocity vp of this wave is given by


v_p = frac{omega}{k}= lambda f, ,

where $lambda$ is the wavelength of the wave.

### Standing wave

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, longitudinal waves propagate out to where the string is held in place at the bridge and the "nut", where upon 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 on average no net propagation of energy.

Also see: Acoustic resonance, Helmholtz resonator, and organ pipe

### Propagation through strings

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


v=sqrt{frac{T}{mu}}. ,

## Transmission medium

The medium that carries a wave is called a transmission medium. It 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 if its physical properties are unchanged at different locations in space.
• An isotropic medium if its physical properties are the same in different directions.

## Sources

• 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.
• Hall, D. E. (1980). Musical Acoustics: An Introduction. Belmont, California: Wadsworth Publishing Company.
• Hunt, F. V. (1978). Origins in Acoustics. New York: Acoustical Society of America Press, (1992).
• Ostrovsky, L. A. and Potapov, A. S. (1999). Modulated Waves, Theory and Applications. Baltimore: The Johns Hopkins University Press.
• Vassilakis, P.N. (2001) Perceptual and Physical Properties of Amplitude Fluctuation and their Musical Significance. Doctoral Dissertation. University of California, Los Angeles.

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