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.

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.


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.


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:


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 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(x,t) = Asin(omega t-kx)), 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{omega}{k} = {lambda}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(z,t) 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.

See also


  • 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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