Wave

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Different types of wave with varying rectifications Rectified waves.png
Different types of wave with varying rectifications

In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport (Mass transfer). Waves consist of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. From the perspective of mathematics, waves, as functions of time and space, are a class of signals. [1]

Physics Study of the fundamental properties of matter and energy

Physics is the natural science that studies matter and its motion and behavior through space and time and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Energy quantitative physical property transferred to objects to perform heating or work on them

In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object. Energy is a conserved quantity; the law of conservation of energy states that energy can be converted in form, but not created or destroyed. The SI unit of energy is the joule, which is the energy transferred to an object by the work of moving it a distance of 1 metre against a force of 1 newton.

Matter substance that has rest mass and volume, or several other definitions

In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic particles, and in everyday as well as scientific usage, "matter" generally includes atoms and anything made up of them, and any particles that act as if they have both rest mass and volume. However it does not include massless particles such as photons, or other energy phenomena or waves such as light or sound. Matter exists in various states. These include classical everyday phases such as solid, liquid, and gas – for example water exists as ice, liquid water, and gaseous steam – but other states are possible, including plasma, Bose–Einstein condensates, fermionic condensates, and quark–gluon plasma.

Contents

There are two main types of waves: mechanical and electromagnetic. Mechanical waves propagate through a physical matter, whose substance is being deformed. Restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbours. When the 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. Electromagnetic waves do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields originally generated by charged particles, and can therefore travel through a vacuum. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays.

Mechanical wave

A mechanical wave is a wave that is an oscillation of matter, and therefore transfers energy through a medium. While waves can move over long distances, the movement of the medium of transmission—the material—is limited. Therefore, the oscillating material does not move far from its initial equilibrium position. Mechanical waves transport energy. This energy propagates in the same direction as the wave. Any kind of wave has a certain energy. Mechanical waves can be produced only in media which possess elasticity and inertia.

Restoring force, in a physics context, is a force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium. The restoring force is a function only of position of the mass or particle. It is always directed back toward the equilibrium position of the system. The restoring force is often referred to in simple harmonic motion. The force which is responsible to restore original size and shape is called restoring force.

Sound mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing; pressure wave, generated by vibrating structure

In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.

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. Further, the behavior of particles in quantum mechanics are described by waves. In addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields.

Wave equation Second-order linear differential equation important in physics

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

Quantum mechanics branch of physics dealing with phenomena at scales of the order of the Planck constant

Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

In physics, a gravitational field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenomena, and is measured in newtons per kilogram (N/kg). In its original concept, gravity was a force between point masses. Following Isaac Newton, Pierre-Simon Laplace attempted to model gravity as some kind of radiation field or fluid, and since the 19th century explanations for gravity have usually been taught in terms of a field model, rather than a point attraction.

A wave can be transverse, where a disturbance creates oscillations that are perpendicular to the propagation of energy transfer, or longitudinal: the oscillations are parallel to the direction of energy propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse in free space.

Transverse wave moving wave that consists of oscillations occurring perpendicular (right angled) to the direction of energy transfer (or the propagation of the wave)

A transverse wave is a moving wave that consists of oscillations occurring perpendicular to the direction of energy transfer.

Longitudinal wave waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the direction of propagation of the wave

Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the direction of propagation of the wave. Mechanical longitudinal waves are also called compressional or compression waves, because they produce compression and rarefaction when traveling through a medium, and pressure waves, because they produce increases and decreases in pressure.

In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet.

General features

Surface waves in water showing water ripples 2006-01-14 Surface waves.jpg
Surface waves in water showing water ripples

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 as a wave results in a blurred line.

Vibration mechanical phenomenon whereby oscillations occur about an equilibrium point; precisely used to describe mechanical oscillation

Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem. The oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road.

Phenomenon philosophical concept

A phenomenon is any thing which manifests itself. Phenomena are often, but not always, understood as "things that appear" or "experiences" for a sentient being, or in principle may be so.

The term wave is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of 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 1982 , p. 8). However, this motion 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.

Standing wave wave that remains in a constant position

In physics, a standing wave, also known as a stationary wave, is a wave which oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with time, and the oscillations at different points throughout the wave are in phase. The locations at which the amplitude is minimum are called nodes, and the locations where the amplitude is maximum are called antinodes.

Vacuum Space that is empty of matter

Vacuum is space devoid of matter. The word stems from the Latin adjective vacuus for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss ideal test results that would occur in a perfect vacuum, which they sometimes simply call "vacuum" or free space, and use the term partial vacuum to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is lower than atmospheric pressure. The Latin term in vacuo is used to describe an object that is surrounded by a vacuum.

Radar object detection system based on radio waves

Radar is a detection system that uses radio waves to determine the range, angle, or velocity of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. A radar system consists of a transmitter producing electromagnetic waves in the radio or microwaves domain, a transmitting antenna, a receiving antenna and a receiver and processor to determine properties of the object(s). Radio waves from the transmitter reflect off the object and return to the receiver, giving information about the object's location and speed.

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 independent of their physical origin. [2] 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. 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 repeatedly .

Mathematical description of one-dimensional waves

Wave equation

Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have a single spatial dimension. Consider this wave as traveling

Wavelength l, can be measured between any two corresponding points on a waveform Nonsinusoidal wavelength.svg
Wavelength λ, can be measured between any two corresponding points on a waveform
Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that f(x,t) + g(x,t) = u(x,t) Superpositionprinciple.gif
Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that f(x,t) + g(x,t) = u(x,t)

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

(waveform traveling to the right)
(waveform traveling to the left)

or, more generally, by d'Alembert's formula: [4]

representing two component waveforms and traveling through the medium in opposite directions. A generalized representation of this wave can be obtained [5] as the partial differential equation

General solutions are based upon Duhamel's principle. [6]

Wave forms

Sine, square, triangle and sawtooth waveforms. Waveforms.svg
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). [7]

In the case of a periodic function F with period λ, that is, F(x + λvt) = 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(xv(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. [8]

Amplitude and modulation

Amplitude modulation can be achieved through f(x,t) = 1.00*sin(2*pi/0.10*(x-1.00*t)) and g(x,t) = 1.00*sin(2*pi/0.11*(x-1.00*t))only the resultant is visible to improve clarity of waveform. Amplitudemodulation.gif
Amplitude modulation can be achieved through f(x,t) = 1.00*sin(2*pi/0.10*(x-1.00*t)) and g(x,t) = 1.00*sin(2*pi/0.11*(x-1.00*t))only the resultant is visible to improve clarity of waveform.
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. Wave packet.svg
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: [9] [10] [11]

where is the amplitude envelope of the wave, is the wavenumber and is the phase . If the group velocity (see below) is wavelength-independent, this equation can be simplified as: [12]

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. [12] [13]

Phase velocity and group velocity

The red square moves with the phase velocity, while the green circles propagate with the group velocity Wave group.gif
The red square moves with the phase velocity, while the green circles propagate with the group velocity

There are two velocities that are associated with waves, the phase velocity and the group velocity.

Phase velocity is the rate at which the phase of the wave propagates in space: any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and period T as

A wave with the group and phase velocities going in different directions Wave opposite-group-phase-velocity.gif
A wave with the group and phase velocities going in different directions

Group velocity is a property of waves that have a defined envelope, measuring propagation through space (i.e. phase velocity) of the overall shape of the waves' amplitudes – modulation or envelope of the wave.

Sine waves

Sinusoidal waves correspond to simple harmonic motion. Simple harmonic motion animation.gif
Sinusoidal waves correspond to simple harmonic motion.

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

where

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

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

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

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

The wavelength of a sinusoidal waveform traveling at constant speed is given by: [14]

where is called the phase speed (magnitude of the phase velocity) of the wave and 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. [15]

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. [16] 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. [17] The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time. [18] [19]

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. 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. [20] [21] 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. [22] When a medium is nonlinear, then the response to complex waves cannot be determined from a sine-wave decomposition.

Plane waves

Standing waves

Standing wave in stationary medium. The red dots represent the wave nodes Standing wave.gif
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.


Physical properties

Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism Light dispersion of a mercury-vapor lamp with a flint glass prism IPNrdeg0125.jpg
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

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:

Absorption

Absorption of waves means, if a kind of wave strikes a matter, it will be absorbed by the matter. When a wave with that same natural frequency impinges upon an atom, then the electrons of that atom will be set into vibrational motion. If a wave of a given frequency strikes a material with electrons having the same vibrational frequencies, then those electrons will absorb the energy of the wave and transform it into vibrational motion.

Reflection

When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and line normal to the surface equals the angle made by the reflected wave and the same normal line.

Refraction

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. Wave refraction.gif
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

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.

Interference

Waves that encounter each other combine through superposition to create a new wave called an interference pattern. Important interference patterns occur for waves that are in phase.

Polarization

Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View.svg

The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions. Transverse waves can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of linear polarization. A transverse wave is linearly polarized if it oscillates in only one direction or plane. In the case of linear polarization, it is often useful to add the relative orientation of that plane, perpendicular to the direction of travel, in which the oscillation occurs, such as "horizontal" for instance, if the plane of polarization is parallel to the ground. Electromagnetic waves propagating in free space, for instance, are transverse; they can be polarized by the use of a polarizing filter.

Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel.

Dispersion

Schematic of light being dispersed by a prism. Click to see animation. Light dispersion conceptual waves.gif
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. [23]

Mechanical waves

Waves on strings

The speed of a transverse 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 ):

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

Acoustic waves

Acoustic or sound waves travel at speed given by

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

Water waves

Shallow water wave.gif

Seismic waves

Seismic waves are waves of energy that travel through the Earth's layers, and are a result of earthquakes, volcanic eruptions, magma movement, large landslides and large man-made explosions that give out low-frequency acoustic energy.

Shock waves

Formation of a shock wave by a plane. Transonico-en.svg
Formation of a shock wave by a plane.

A shock wave is a type of propagating disturbance. When a wave moves faster than the local speed of sound in a fluid, it is a shock wave. Like an ordinary wave, a shock wave carries energy and can propagate through a medium; however, it is characterized by an abrupt, nearly discontinuous change in pressure, temperature and density of the medium. [24]

Other

Electromagnetic waves

Onde electromagnetique.png

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, X-rays, and Gamma rays.

Quantum mechanical waves

Schrödinger equation

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.

Dirac equation

The Dirac equation is a relativistic wave equation detailing electromagnetic interactions. Dirac waves accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The wave equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-½ particles.

A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves. Wave packet (dispersion).gif
A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.

de Broglie waves

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

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 as follows:

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

and the momentum by:

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, [27] 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. [28] Gaussian wave packets also are used to analyze water waves. [29]

For example, a Gaussian wavefunction ψ might take the form: [30]

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, [31] 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. [32] Given the Gaussian:

the Fourier transform is:

The Gaussian in space therefore is made up of waves:

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 GravitationalWave CrossPolarization.gif
Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles

Gravity waves

Gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. A ripple on a pond is one example.

Gravitational waves

Gravitational waves also travel through space. The first observation of gravitational waves was announced on 11 February 2016. [33] Gravitational waves are disturbances in the curvature of spacetime, predicted by Einstein's theory of general relativity.

See also

Waves in general

Parameters

Waveforms

Electromagnetic waves

In fluids

In quantum mechanics

In relativity

Other specific types of waves

Related Research Articles

Electromagnetic radiation form of energy emitted and absorbed by charged particles, which exhibits wave-like behavior as it travels through space

In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.

Group velocity physical quantity

The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.

Phase velocity rate at which the phase of the wave propagates in space

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as

Wavelength spatial period of the wave—the distance over which the waves shape repeats, and thus the inverse of the spatial frequency

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

Plane wave Type of wave propagating in 3 dimensions

In the physics of wave propagation, a plane wave is a wave whose wavefronts are infinite parallel planes. Mathematically a plane wave takes the form

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

Wavenumber spatial frequency of a wave

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter can exhibit wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave. The concept that matter behaves like a wave was proposed by Louis de Broglie in 1924. It is also referred to as the de Broglie hypothesis. Matter waves are referred to as de Broglie waves.

Normal mode pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation

A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at the fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. When relating to music, normal modes of vibrating instruments are called "harmonics" or "overtones".

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In physics, Landau damping, named after its discoverer, the eminent Soviet physicist Lev Davidovich Landau (1908–68), is the effect of damping of longitudinal space charge waves in plasma or a similar environment. This phenomenon prevents an instability from developing, and creates a region of stability in the parameter space. It was later argued by Donald Lynden-Bell that a similar phenomenon was occurring in galactic dynamics, where the gas of electrons interacting by electrostatic forces is replaced by a "gas of stars" interacting by gravitation forces. Landau damping can be manipulated exactly in numerical simulations such as particle-in-cell simulation. It was proved to exist experimentally by Malmberg and Wharton in 1964, almost two decades after its prediction by Landau in 1946.

Dispersion relation Relation of wavelength/wavenumber as a function of a waves frequency

In physical sciences and electrical engineering, dispersion relations describe the effect of dispersion in a medium on the properties of a wave traveling within that medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. From this relation the phase velocity and group velocity of the wave have convenient expressions which then determine the refractive index of the medium. More general than the geometry-dependent and material-dependent dispersion relations, there are the overarching Kramers–Kronig relations that describe the frequency dependence of wave propagation and attenuation.

An optical medium is material through which electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it. The medium has an intrinsic impedance, given by

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

The word electricity refers generally to the movement of electrons through a conductor in the presence of potential and an electric field. The speed of this flow has multiple meanings. In everyday electrical and electronic devices, the signals or energy travel as electromagnetic waves typically on the order of 50%–99% of the speed of light, while the electrons themselves move (drift) much more slowly.

Lamb waves

Lamb waves propagate in solid plates. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the plate normal. In 1917, the English mathematician Horace Lamb published his classic analysis and description of acoustic waves of this type. Their properties turned out to be quite complex. An infinite medium supports just two wave modes traveling at unique velocities; but plates support two infinite sets of Lamb wave modes, whose velocities depend on the relationship between wavelength and plate thickness.

In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.

Envelope (waves) function describing the extremes of an oscillating signal

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

References

  1. Pragnan Chakravorty, "What Is a Signal? [Lecture Notes]," IEEE Signal Processing Magazine, vol. 35, no. 5, pp. 175-177, Sept. 2018. Template:Doi.org
  2. Lev A. Ostrovsky & Alexander I. Potapov (2001). Modulated waves: theory and application. Johns Hopkins University Press. ISBN   978-0-8018-7325-6.
  3. Michael A. Slawinski (2003). "Wave equations". Seismic waves and rays in elastic media. Elsevier. pp. 131 ff. ISBN   978-0-08-043930-3.
  4. Karl F Graaf (1991). Wave motion in elastic solids (Reprint of Oxford 1975 ed.). Dover. pp. 13–14. ISBN   978-0-486-66745-4.
  5. For an example derivation, see the steps leading up to eq. (17) in Francis Redfern. "Kinematic Derivation of the Wave Equation". Physics Journal.
  6. Jalal M. Ihsan Shatah; Michael Struwe (2000). "The linear wave equation". Geometric wave equations. American Mathematical Society Bookstore. pp. 37ff. ISBN   978-0-8218-2749-9.
  7. Louis Lyons (1998). All you wanted to know about mathematics but were afraid to ask. Cambridge University Press. pp. 128 ff. ISBN   978-0-521-43601-4.
  8. Alexander McPherson (2009). "Waves and their properties". Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN   978-0-470-18590-2.
  9. Christian Jirauschek (2005). FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection. Cuvillier Verlag. p. 9. ISBN   978-3-86537-419-6.
  10. Fritz Kurt Kneubühl (1997). Oscillations and waves. Springer. p. 365. ISBN   978-3-540-62001-3.
  11. Mark Lundstrom (2000). Fundamentals of carrier transport. Cambridge University Press. p. 33. ISBN   978-0-521-63134-1.
  12. 1 2 Chin-Lin Chen (2006). "§13.7.3 Pulse envelope in nondispersive media". Foundations for guided-wave optics. Wiley. p. 363. ISBN   978-0-471-75687-3.
  13. Stefano Longhi; Davide Janner (2008). "Localization and Wannier wave packets in photonic crystals". In Hugo E. Hernández-Figueroa; Michel Zamboni-Rached; Erasmo Recami. Localized Waves. Wiley-Interscience. p. 329. ISBN   978-0-470-10885-7.
  14. David C. Cassidy; Gerald James Holton; Floyd James Rutherford (2002). Understanding physics. Birkhäuser. pp. 339ff. ISBN   978-0-387-98756-9.
  15. Paul R Pinet (2009). op. cit. p. 242. ISBN   978-0-7637-5993-3.
  16. Mischa Schwartz; William R. Bennett & Seymour Stein (1995). Communication Systems and Techniques. John Wiley and Sons. p. 208. ISBN   978-0-7803-4715-1.
  17. See Eq. 5.10 and discussion in A.G.G.M. Tielens (2005). The physics and chemistry of the interstellar medium. Cambridge University Press. pp. 119 ff. ISBN   978-0-521-82634-1.; Eq. 6.36 and associated discussion in Otfried Madelung (1996). Introduction to solid-state theory (3rd ed.). Springer. pp. 261 ff. ISBN   978-3-540-60443-3.; and Eq. 3.5 in F Mainardi (1996). "Transient waves in linear viscoelastic media". In Ardéshir Guran; A. Bostrom; Herbert Überall; O. Leroy. Acoustic Interactions with Submerged Elastic Structures: Nondestructive testing, acoustic wave propagation and scattering. World Scientific. p. 134. ISBN   978-981-02-4271-8.
  18. Aleksandr Tikhonovich Filippov (2000). The versatile soliton. Springer. p. 106. ISBN   978-0-8176-3635-7.
  19. Seth Stein, Michael E. Wysession (2003). An introduction to seismology, earthquakes, and earth structure. Wiley-Blackwell. p. 31. ISBN   978-0-86542-078-6.
  20. Seth Stein, Michael E. Wysession (2003). op. cit.. p. 32. ISBN   978-0-86542-078-6.
  21. Kimball A. Milton; Julian Seymour Schwinger (2006). Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators. Springer. p. 16. ISBN   978-3-540-29304-0. Thus, an arbitrary function f(r, t) can be synthesized by a proper superposition of the functions exp[i (k·r−ωt)]...
  22. Raymond A. Serway & John W. Jewett (2005). "§14.1 The Principle of Superposition". Principles of physics (4th ed.). Cengage Learning. p. 433. ISBN   978-0-534-49143-7.
  23. Newton, Isaac (1704). "Prop VII Theor V". Opticks: Or, A treatise of the Reflections, Refractions, Inflexions and Colours of Light. Also Two treatises of the Species and Magnitude of Curvilinear Figures. 1. London. p. 118. All the Colours in the Universe which are made by Light... are either the Colours of homogeneal Lights, or compounded of these...
  24. Anderson, John D. Jr. (January 2001) [1984], Fundamentals of Aerodynamics (3rd ed.), McGraw-Hill Science/Engineering/Math, ISBN   978-0-07-237335-6
  25. M.J. Lighthill; G.B. Whitham (1955). "On kinematic waves. II. A theory of traffic flow on long crowded roads". Proceedings of the Royal Society of London. Series A. 229 (1178): 281–345. Bibcode:1955RSPSA.229..281L. CiteSeerX   10.1.1.205.4573 . doi:10.1098/rspa.1955.0088. And: P.I. Richards (1956). "Shockwaves on the highway". Operations Research. 4 (1): 42–51. doi:10.1287/opre.4.1.42.
  26. A.T. Fromhold (1991). "Wave packet solutions". Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59 ff. ISBN   978-0-486-66741-6. (p. 61) ...the individual waves move more slowly than the packet and therefore pass back through the packet as it advances
  27. Ming Chiang Li (1980). "Electron Interference". In L. Marton; Claire Marton. Advances in Electronics and Electron Physics. 53. Academic Press. p. 271. ISBN   978-0-12-014653-6.
  28. See for example Walter Greiner; D. Allan Bromley (2007). Quantum Mechanics (2 ed.). Springer. p. 60. ISBN   978-3-540-67458-0. and John Joseph Gilman (2003). Electronic basis of the strength of materials. Cambridge University Press. p. 57. ISBN   978-0-521-62005-5.,Donald D. Fitts (1999). Principles of quantum mechanics. Cambridge University Press. p. 17. ISBN   978-0-521-65841-6..
  29. Chiang C. Mei (1989). The applied dynamics of ocean surface waves (2nd ed.). World Scientific. p. 47. ISBN   978-9971-5-0789-3.
  30. Walter Greiner; D. Allan Bromley (2007). Quantum Mechanics (2nd ed.). Springer. p. 60. ISBN   978-3-540-67458-0.
  31. Siegmund Brandt; Hans Dieter Dahmen (2001). The picture book of quantum mechanics (3rd ed.). Springer. p. 23. ISBN   978-0-387-95141-6.
  32. Cyrus D. Cantrell (2000). Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 677. ISBN   978-0-521-59827-9.
  33. "Gravitational waves detected for 1st time, 'opens a brand new window on the universe'". CBC. 11 February 2016.

Sources