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Dispersion Relation

Dispersion relation

The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagates through space. The group velocity is defined by the equation: :

v_g \equiv \frac

where: :v_g is the group velocity :ω is the wave's angular frequency :k is the wave number The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform. However, if the wave is travelling through an absorptive medium, this does not always hold. It is possible to design experiments where the group velocity of laser light pulses sent through specially prepared materials significantly exceeds the signal velocity. (It is also possible to stop the laser pulse.) The function ω(k), which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity. Otherwise, the envelope of the wave will become distorted as it propagates. This "group velocity dispersion" is an important effect in the propagation of signals through optical fibers and in the design of short pulse lasers. The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.

References

- Brillouin, Léon. Wave Propagation and Group Velocity. Academic Press Inc., New York (1960).

External links

- Greg Egan has an excellent Java applet on [http://gregegan.customer.netspace.net.au/APPLETS/20/20.html his web site] that illustrates the apparent difference in group velocity from phase velocity. Category:Radio frequency propagation Category:Optics Category:Wave mechanics Category:Physical quantity ja:群速度

Wave

:This article is about waves in the most general scientific sense; a separate article focuses on ocean waves. For other meanings see wave (disambiguation). Soundwave redirects here. for the Transformers character, see Soundwave (Transformers) A wave is a disturbance that propagates in a periodically repeating fashion, often transferring energy. A mechanical wave exists in a medium (which on deformation is capable of producing elastic restoring forces) through which they travel and can transfer energy from one place to another without any of the particles of the medium being displaced permanently; there is no associated mass transport. Instead, any particular point oscillates around a fixed position. However, electromagnetic radiation, and probably gravitational radiation are not mechanical waves, and can travel through a vacuum, without a medium. Waves are characterised by crests (highs) and troughs (lows), either perpendicular (in the case of transverse waves) or parallel (in the case of longitudinal waves) to wave motion.
The medium which carries a wave
A medium that can carry a wave is classified by one or more of the following properties:
- A linear medium if the amplitudes of different waves at any particular point in the medium can be added.
- A bounded medium if it is finite in extent, otherwise unbounded.
- 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.
Examples of waves
troughs
- Ocean surface waves, which are perturbations that propagate through water (see also surfing and tsunami).
- Visible light, radio waves, x-rays, gamma rays, infrared rays, and ultraviolet rays make up electromagnetic radiation. In this case propagation is possible without a medium, through vacuum. These electromagnetic waves travel at about 300,000 km/s.
- Sound - a mechanical wave that propagates through air, liquid or solids, and is of a frequency detected by the auditory system. Similar are seismic waves in earthquakes, of which there are the S, P and L kinds.
- Gravitational waves, which are fluctuations in the gravitational field predicted by General relativity. These waves are nonlinear.
Characteristic properties
All waves have common behaviour under a number of standard situations. All waves can experience the following:
- Reflection – the change of direction of waves, due to hitting a reflective surface.
- Refraction – the change of direction of a wave due to them entering a new medium.
- Diffraction – the spreading out of waves, for example when they travel through a small slit.
- Interference – the superposition of two waves that come into contact with each other.
- Dispersion – the splitting up of waves by frequency.
- Rectilinear propagation – the movement of waves in straight lines.
Transverse and longitudinal waves
Rectilinear propagation 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. Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow elliptical paths.
Polarization
Transverse waves can be polarized. Unpolarised waves can oscillate in any direction in the plane perpendicular to the direction of travel, while polarized waves oscillate in only one direction perpendicular to the line of travel.
Physical description of a wave
Image:wave.png Waves can be described using a number of standard variables including: frequency, wavelength, amplitude and period. The amplitude of a wave is the measure of the magnitude of the maximum disturbance in the medium during one wave cycle, and is measured in units depending on the type of wave. For examples, 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 period (T) is the time for one complete cycle for an oscillation of a wave. The frequency (F) is how many periods per unit time (for example one second) and is measured in hertz. These are related by: : $f=\frac$ When waves are expressed mathematically, the angular frequency (ω, radians/second) is often used; it is related to the frequency f by: : $f=\frac$ .
Travelling waves
Waves that remain in one place are called standing waves - e.g. vibrations on a violin string. Waves that are moving are called travelling waves, and have a disturbance that varies both with time t and distance z. This can be expressed mathematically as: : $y=A(z,t) \cos (\omega t - kz + \phi),\,f$ where A(z, t) is the amplitude envelope of the wave, k is the wave number and φ is the phase. The velocity v of this wave is given by: : $v=\frac= \lambda f,$ where λ is the wavelength of the wave.
Propagation through strings
The speed of a wave travelling along a string (v) is directly proportional to the square root of the tension (T) over the linear density (ρ): : $v=\sqrt.$ This equation can be found using dimensional analysis
The wave equation
The wave equation is a differential equation which describes a harmonic wave passing through a medium, discussed above. The equation has different forms depending on how the wave is transmitted, and on what medium. Not all waves are sinusoidal. One example of a non-sinusoidal wave is a pulse that travels down a rope resting on the ground, extending in direction x, travelling at velocity c. The height of the pulse above the ground is φ. The distance the pulse travels between some time t and time 0 is ct. In one dimension the wave equation has the form : $\frac\frac=\frac. \$ A general solution, given by d'Alembert is : $\phi(x,t)=F(x-ct)+G(x+ct). \$ considered to be the shapes of two pulses travelling down the rope, F in the +x direction, and G in the -x direction. If we substitute for x above, instead directions x, y, z, we then can describe a wave propagating in three dimensions. A non-linear wave-equation can cause mass transport. The Schrödinger equation describes the wave-like behaviour 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.
External links

- [http://www.lightandmatter.com/area1book3.html Vibrations and Waves] - an online textbook
- [http://kestrel.nmt.edu/~raymond/classes/ph13xbook/node1.html A Radically Modern Approach to Introductory Physics] - an online physics textbook that starts with waves rather than mechanics
See also

- List of wave topics
- Capillary waves
- Doppler effect
- Group velocity
- Phase velocity
- Ripple tank
- Standing wave
- Audience wave
- Ocean surface wave
- Waving Category:Partial differential equations ko:파동 ms:Gelombang ja:波動 simple:Wave

Velocity

This article is about velocity in physics. For other meanings, see velocity (disambiguation). The velocity of an object is simply its speed in a particular direction. Note that both speed and direction are required to define a velocity.

Explanation

The velocity (v) is an physical quantity of the motion. A change in an object's velocity can therefore arise from either a change in its speed or in its direction. For example an aeroplane that is circling at a constant speed of 200km/h is changing its velocity because it is continously changing its direction. A aeroplane that is taking-off may go from zero to 200km/h in a straight line and so would also be changing its velocity. A change in velocity is called an acceleration. Objects are only accelerated if a force is applied to them. (The amount of acceleration depends the size of the force and the mass of the object being shifted, see Newton's Second Law of Motion.) In the case of the circling aeroplane, the pilot banks to use the force of lift from the wings to change direction. In another example the Space Shuttle orbits the earth at a constant speed but is constantly changing its velocity because of the circular orbit. In this case the force causing the acceleration is provided by the earth's gravity acting on the shuttle. The average speed v of an object moving a distance d during a time interval t is described by the formula: :

v = \frac

Acceleration is the rate of change of an object's velocity over time. The average acceleration of a of an object whose speed changes from v_i to v_f during a time interval t is given by: :

a = \frac

Where

v_i

= an object's initial velocity and

v_f

= the object's final velocity over a period of time t

Formal description

Velocity (symbol: v) is a vector measurement of the rate and direction of motion. The scalar absolute value (magnitude) of velocity is speed. Velocity can also be defined as rate of change of displacement or just as the rate of displacement, depending on how the term displacement is used. It is thus a vector quantity with dimension length/time. In the SI (metric) system it is measured in metre per second The instantaneous velocity vector v of an object that has position at time t is given by x(t) can be computed as the derivative :

v= = \lim_

The instantaneous acceleration vector a of an object that has position at time t is given by x(t) is :

\mathbf = \frac   = \frac

The equation for an object's velocity can be obtained mathematically by taking the integral of the equation for its acceleration beginning from some initial period time $t_0$ to some point in time later $t_n$ . The final velocity v_f of an object which starts with velocity v_i and then accelerates at constant acceleration a for a period of time t is: :

v_f = v_i + a t

The average velocity of an object undergoing constant acceleration is (v_i + v_f)/2. To find the displacement d of such an accelerating object during a time interval t, substitute this expression into the first formula to get: :

d = t \times \frac

When only the object's initial velocity is known, the expression :

d = v_i t + \frac

can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time, also known as Torricelli's Equation: :

v_f^2 = v_i^2 + 2 a d

The above equations are valid for both classical mechanics and special relativity. Where classical mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in classical mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. The kinetic energy (energy of motion) of a moving object is linear with both its mass and the square of its velocity: :

E_ = \begin \frac \end mv^2

The kinetic energy is a scalar quantity.

Polar coordinates

In polar coordinates, a two-dimensional velocity can be decomposed into a radial velocity, defined as the component of velocity away from or toward the origin, and transverse velocity, the component of velocity along a circle centred at the origin, and equal to the distance to the origin times the angular velocity. Angular momentum in scalar form is the distance to the origin times the transverse speed, or equivalently, the distance squared times the angular speed, with a plus or minus to distinguish clockwise and anti-clockwise direction. If forces are in the radial direction only, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion.

Wave number

Wavenumber in most physical sciences is a wave property inversely related to wavelength, having units of inverse length (radians per meter). Wavenumber is the spatial analogue of angular frequency. Application of a Fourier transformation on data in the time domain yields a frequency spectrum; applied on data in the spatial domain (data as a function of position) yields a spectrum as a function of wavenumber. The exact definition is dependent on the field.

Circular wavenumber

The circular wavenumber, k, often misleadingly abbreviated as "wavenumber", is defined as :

k \equiv \frac = \frac=\frac,

where λ is the wavelength in the medium, ν is the frequency, v_p is the phase velocity of wave, and ω is the angular frequency. The wavenumber is the Scalar of the wave vector.

Wavenumber

In spectroscopy, the wavenumber

\tilde

of electromagnetic radiation is defined as :

\tilde = 1/\lambda,

where

\lambda

as a length in the SI-system is measured in meters (m) and commonly quantified in centimetres (cm=10^-2m) refers to the wavelength in vacuum. The unit of this quantity is cm^-1, pronounced as reciprocal centimeter, or "inverse centimeter". The historical reason for using this quantity is that this quantity is proportional to energy, but not dependent on the speed of light or Planck's constant, which were not known with sufficient accuracy (or rather not at all known). For example, the wavenumbers of the emissions lines of hydrogen atoms are given by :

\tilde = R\left( - \right)

where λ is the wavelength, R is Rydberg constant,

n_1

and

n_2

are the orbit numbers, and

n_2

is greater than

n_1

. Spectroscopists often express various quantities, such as frequency and energy in cm^-1. In colloquial usage, the unit cm^-1 is sometimes referred to as a "wavenumber", which confuses the role of a unit with that of a quantity. An incorrect phrase such as "The energy is 300 wavenumbers" should be read as "The energy corresponds to a wavenumber of 300 reciprocal centimeters", or as "The energy corresponds to a wavenumber of 300 inverse centimeters".

Atmospheric science

Wavenumber in atmospheric science is defined as length of the spatial domain divided by the wavelength, or equivalently the number of times a wave has the same phase over the spatial domain. The domain might be 2π for the non-dimensional case, or :

2\pi R \cos\left(\phi\right)

for an atmospheric wave, where R is Earth's radius and φ is latitude. Wavenumber-frequency diagrams are a common way of visualizing atmospheric waves. ja:波数 Category:Wave mechanics

Energy

Energy is a measure of being able to do work. This is a fundamental concept pertaining to the ability for action. In physics, it is a quantity that every physical system possesses. This quantity is not absolute but relative to a state of the system known as its reference state or reference level. The energy of a physical system is defined as the amount of mechanical work that the system can produce if it changes its state to its reference state; for example if a liter of water cools down to 0°C or if a car hits a tree and decelerates from 120 km/h to 0 km/h. Energy of an object can be in several forms, potential—due to the position of the object relative to other objects; kinetic—energy because of its motion; chemical—due to chemical bonds between atoms that make up the substance; electrical—due to its charge; thermal—due to its heat; and nuclear—due to the instability of the nuclei of its atoms. In the case where the "object" is an electromagnetic wave or light, then radiant energy can also be defined. One form of energy can be readily transformed into another; for instance, a battery converts chemical energy into electrical energy, which can be converted into thermal energy. Similarly, potential energy is converted into kinetic energy of moving water and turbine in a dam, which in turn transforms into electric energy by generator. The law of conservation of energy states that in a closed system the total amount of energy, corresponding to the sum of a system's constituent energy components, remains constant. This law follows from translational symmetry of time (that is, independence of any physical process on the moment it started). Some works (thus some forms of energy) are not easily measured by the unaided observer. The term 'energy' is also used in a spiritual or non-scientific way that cannot be quantified, to make certain prepositions look like they are more plausible, by imitating the scientific terminology. Usually this has something to do with mystical and/or healing type references such as acupuncture and reiki.

Forms of Energy

Below is a list of different energy forms. Lotka (1956, p. 5) asked an interesting question about what defines an energy form. :"We are equipped with two separate and distinct senses, the one responding to electromagnetic waves ranging from about 4×10^-4 to 8×10^-4 mm., light waves; the other to somewhat longer waves otherwise of the same character, heat waves. Accordingly we have two separate terms in our language light and heat, to denote two phenomena which, objectively considered, are not separated by any line of division, but merge into one another by gradual transition. Here the question might be raised whether an electromagnetic wave of a length of 9×10^-4 mm. is a light wave or a heat wave." That is to ask, if all forms of energy are defined in terms of infinitesimal increments of the wave spectrum, what makes one form of energy different to another?
- Kinetic energy: the energy of moving objects
- Thermal energy: the energy associated with heat
- Sound energy: the energy of compression waves
- Electrical energy: the energy of moving charged particles
- Potential Energy: the energy that an object has due to position; also known as stored energy
- Chemical energy: the stored energy of chemical substances
- Nuclear energy: the stored energy of the atomic nucleus
- Radiant energy: the energy of electromagnetic waves, including light

Units

SI

The SI unit for both energy and work is the joule (J), named in honour of James Prescott Joule and his experiments on the mechanical equivalent of heat. In slightly more fundamental terms, 1 joule is equal to 1 newton-metre and, in terms of SI base units:

1\ \mathrm = 1\ \mathrm \left( \frac \right ) ^ 2 = 1\ \frac

An energy unit that is used in particle physics is the electronvolt (eV). One eV is equivalent to 1.60217653×10⁻¹⁹ J. In spectroscopy the unit cm^-1 = 0.0001239 eV is used to represent energy since energy is inversely proportional to wavelength from the equation

E = h \nu = h c/\lambda

. (Note that torque, which is typically expressed in newton-metres, has the same dimension and this is not a simple coincidence: a torque of 1 newton-metre applied on 1 radian requires exactly 1 newton-metre=joule of energy.)

Other units of energy

In cgs units, one erg is 1 g cm² s⁻², equal to 1.0×10⁻⁷ J. Another obsolete metric unit is the litre-atmosphere (101.325 J). The imperial/US units for both energy and work include the foot-pound force (1.3558 J), the British thermal unit (Btu) which has various values in the region of 1055 J, and the horsepower-hour (2.6845 MJ). The energy unit used for everyday electricity, particularly for utility bills, is the kilowatt-hour (kW h), and one kW h is equivalent to 3.6×10⁶ J (3600 kJ or 3.6 MJ; the metric units usually are self-consistent, and this particular one may seem arbitrary; it's not, the metric measurement for time is the second, and there are 3,600 seconds in an hour -- in other words, 1 kW second = 1 kJ, but the kW h is a more convenient unit for everyday use). The calorie is mainly used in nutrition and equals the amount of heat necessary to raise the temperature of one kilogram of water by 1 degree Celsius, at a pressure of 1 atm. This amount of heat depends somewhat on the initial temperature of the water, which results in various different units sharing the name of "calorie" but having slightly different energy values. It is equal to 4.1868 kJ. The calories used for food energy in nutrition are the large calories based on the kilogram rather than the gram, often identified as food calories. These are sometimes called kilocalories with that calorie being the small calorie based on the gram, and as a result the prefixes are generally avoided for the large calories (i.e., 1 kcal is 4.184 kJ, never 4.184 MJ, even if "calories" are also used for the other, larger unit in the same document or the same nutrition label). Food calories are sometimes noted as Calories (1000 calories) or simply abbreviated Cal with the capital C, but that convention is more often found in chemistry or physics textbooks—which do not use these large calories—than it is in real-world applications by those who do use these calories. (This convention is also, of course, useless when the word calorie appears in a location where it would ordinarily be capitalized, as at the beginning of a sentence or in the first column of a nutrition label as a substitute for the quantity being measured, which is energy, when all the other quantities such as "Iron" and "Sugars" are also capitalized.)

Transfer of energy

Work

Main article: mechanical work. Work is a defined as a path integral of force F over distance s: :

W = \int \mathbf \cdot \mathrm\mathbf

The equation above says that the work (

W

) is equal to the integral of the dot product of the force (

\mathbf

) on a body and the infinitesimal of the body's position (

\mathbf

Heat

Main article: Heat. Heat is the common name for thermal energy of an object that is due to the motion of the atoms and molecules that constitute the object. This motion can be translational (motion of molecules or atoms as a whole; vibrational - relative motion of atoms within molecules or rotational motion. It is the form of energy which is usually linked with a change in temperature or in a change in phase of matter. In chemistry, heat is the amount of energy which is absorbed or released when atoms are rearranged between various molecules by a chemical reaction. The relationship between heat and energy is similar to that between work and energy. Heat flows from areas of high temperature to areas of low temperature. All objects (matter) have a certain amount of internal energy that is related to the random motion of their atoms or molecules. This internal energy is directly proportional to the temperature of the object. When two bodies of different temperature come in to thermal contact, they will exchange internal energy until the temperature is equalised. The amount of energy transferred is the amount of heat exchanged. It is a common misconception to confuse heat with internal energy, but there is a difference: the change of the internal energy is the heat that flows from the surroundings into the system plus the work performed by the surroundings on the system. Heat Energy is transferred in three different ways: conduction, convection and/or radiation.

Conservation of energy

The first law of thermodynamics says that the total inflow of energy into a system must equal the total outflow of energy from the system, plus the change in the energy contained within the system. This law is used in all branches of physics, but frequently violated by quantum mechanics (see off shell). Noether's theorem relates the conservation of energy to the time invariance of physical laws. An example of the conversion and conservation of energy is a pendulum. At its highest points the kinetic energy is zero and the potential gravitational energy is at its maximum. At its lowest point the kinetic energy is at its maximum and is equal to the decrease of potential energy. If one unrealistically assumes that there is no friction, the energy will be conserved and the pendulum will continue swinging forever. (In practice, available energy is never perfectly conserved when a system changes state; otherwise, the creation of perpetual motion machines would be possible.) Another example is a chemical explosion in which potential chemical energy is converted to kinetic energy and heat in a very short time.

Types of energy

All forms of energy: thermal, chemical, electrical, radiant, nuclear etc. can be in fact reduced to kinetic energy or potential energy. For example thermal energy is essentially kinetic energy of atoms and molecules; chemical energy can be visualized to be the potential energy of atoms within molecules; electrical energy can be visualized to be the potential and kinetic energy of electrons; similarly nuclear energy is the potential energy of nucleons in atomic nucleii.

Kinetic energy

Main article: Kinetic energy. Kinetic energy is the portion of energy related to the motion. :

E_k = \int \mathbf \cdot \mathrm\mathbf

The equation above says that the kinetic energy (

E_k

) is equal to the integral of the dot product of the velocity (

\mathbf

) of a body and the infinitesimal of the body's momentum (

\mathbf

). For non-relativistic velocities, that is velocities much smaller than the speed of light, we can use the Newtonian approximation :

E_k = \begin \frac \end mv^2

where E_k is kinetic energy m is mass of the body v is velocity of the body At near-light velocities, we use the correct relativistic formula: :

E_k = m c^2 (\gamma - 1) = \gamma m c^2 - m c^2 \;\!

\gamma = \frac

where v is the velocity of the body m is its rest mass c is the speed of light in a vacuum, which is approximately 300,000 kilometers per second

\gamma m c^2 \,

is the total energy of the body

m c^2 \,

is again the rest mass energy. See also, E=mc². In the form of a Taylor series, the relativistic formula for can be written as: :

E_k = \frac mv^2 - \frac \frac  + \cdots

Hence, the second and higher terms in the series correspond with the "inaccuracy" of the Newtonian approximation for kinetic energy in relation to the relativistic formula. However, the phrase "conservation of energy" is often confusing to a non scientist. This is so, because of the common usage of the terms "save energy" or conserve energy" used in campaigns for conservation of energy resources like electricity or fossil fuels.

Potential energy

Main article: Potential energy. In contrast to kinetic energy, which is the energy of a system due to its motion, or the internal motion of its particles, the potential energy of a system is the energy associated with the spatial configuration of its components and their interaction with each other. Any number of particles which exert forces on each other automatically constitute a system with potential energy. Such forces, for example, may arise from electrostatic interaction (see Coulomb's law), or gravity. In an isolated system consisting of two stationary objects that exert a force

f(x)

on each other and lay on the x-axis, their potential energy is most generally defined as :

E_p = -\int f(x) \, dx

where the force between the objects varies only with distance

x

and is integrated along the line connecting the two objects. To further illustrate the relationship between force and potential energy, consider the same system of two objects situated along the x-axis. If the potential energy due to one of the objects at any point

x

U(x)

, then the force on the that object

x

is :

f(x) = -\frac

This mathematical relationship demonstrates the direct connection between force and potential energy: the force between two objects is in the direction of decreasing potential energy, and the magnitude of the force is proportional to the extent to which potential energy decreases. A large force is associated with a large decrease in potential energy, while a small force is associated with a small decrease in potential energy. Notice how, in this case, the force on an object depends entirely on its potential energy. These two relationships – the definition of potential energy based on force, and the dependence of force on potential energy – show how the concepts of force and potential energy are intimately linked: if two objects do not exert forces on each other, there is no potential energy between them. If two objects do exert forces on each other, then potential energy naturally arises in the system as part of the system's total energy. Since potential energy arises from forces, any change in the system's spatial configuration will either increase or decrease the system's potential energy as the objects are repositioned. When a system moves to a lower potential energy state, energy is either released in some form or converted into another form of energy, such as kinetic energy. The potential energy can be "stored" as gravitational energy, elastic energy, chemical energy, rest mass energy or electrical energy, but arises in all cases from the spatial positioning and interaction of objects within a system. Unlike kinetic energy, which exists in any moving body, potential energy exists in any body which is interacting with another object. For example a mass released above the Earth initially has potential energy resulting from the gravitational attraction of the Earth, which is transferred to kinetic energy as the gravitational force acts on the object and its potential energy is decreased as it falls. Equation: :

E_p = mgh \;

where m is the mass, h is the height and g is the value of acceleration due to gravity at the Earth's surface (see gee).

Internal energy

Main article: Internal energy. Internal energy is the kinetic energy associated with the motion of molecules, and the potential energy associated with the rotational, vibrational and electric energy of atoms within molecules. Internal energy, like energy, is a quantifiable state function of a system.

History

In the past, energy was discussed in terms of easily observable effects it has on the properties of objects or changes in state of various systems. Basically, if something changed, some sort of energy was involved in that change. As it was realized that energy could be stored in objects, the concept of energy came to embrace the idea of the potential for change as well as change itself. Such effects (both potential and realized) come in many different forms; examples are the electrical energy stored in a battery, the chemical energy stored in a piece of food, the thermal energy of a water heater, or the kinetic energy of a moving train. To simply say energy is "change or the potential for change", however, misses many important examples of energy as it exists in the physical world. The concept of energy and work are relatively new additions to the physicist’s toolbox. Neither Galileo nor Newton made any contributions to the theoretical model of energy, and it was not until the middle of the 19th century that these concepts were introduced. The development of steam engines required engineers to develop concepts and formulas that would allow them to describe the mechanical and thermal efficiencies of their systems. Engineers such as Sadi Carnot and James Prescott Joule, mathematicians such as Émile Claperyon and Hermann von Helmholtz , and amateurs such as Julius Robert von Mayer all contibuted to the notions that the ability to perform certain tasks, called work, was somehow related to the amount of energy in the system. The nature of energy was elusive, however, and it was argued for some years whether energy was a substance (the caloric) or merely a physical quantity, such as momentum. William Thomson (Lord Kelvin) amalgamated all of these laws into his laws of thermodynamics, which aided in the rapid development of energetic descriptions of chemical processes by Rudolf Clausius, Josiah Willard Gibbs, Walther Nernst. In addition, this allowed Ludwig Boltzmann to describe entropy in mathematical terms, and to discuss, along with Jožef Stefan, the laws of radiant energy. For further information, see the Timeline of thermodynamics.

Energy and Economy

Main articles: energy development, energy policy The way in which humans use energy is one of the defining characteristics of an economy. The progression from animal power to steam power, then the internal combustion engine and electricity, are key elements in the development of modern civilization. Future energy development, for example of renewable energy, may be key to avoiding the effects of global warming.

Notes

This definition is one of the most common; e.g. [http://observe.arc.nasa.gov/nasa/space/stellardeath/stellardeath_6.html Glossary at the NASA homepage]

External links

- [http://www.unitconversion.org/unit_converter/energy.html Online Energy and Work Converter] - convert between various units of energy and work, such as joule, erg, gigawatt-hour, newton meter, calorie, Btu, and so on
- [http://www.unitconversion.org/unit_converter/energy-v.html Interactive Energy and Work Conversion Table] - convert selected unit to all other units of energy and work
- [http://jumk.de/calc/energy.shtml Conversions of energy units]
- [http://www.physicsweb.org/article/world/15/7/2 What does energy really mean? From Physics World]
- [http://www.energy.ca.gov/glossary/ Glossary of Energy Terms]
- [http://www.iea.org International Energy Agency IEA - OECD] Category:Introductory physics Category:Fundamental physics concepts Category:Physical quantity ko:에너지 ms:Tenaga ja:エネルギー simple:Energy th:พลังงาน

Signal velocity

The signal velocity of a wave is the speed at which a pulse travels through a medium. The signal velocity is usually defined from the position of half-maximum intensity of the pulse. For electromagnetic waves such as light, the signal velocity is identical to the group velocity of the wave except when it is travelling through a medium with an absorption resonance close to the frequency of the wave. In this case, the group velocity v_g may exceed the speed of light c, but the signal velocity v_s will always be less than or equal to c.

References

Brillouin, Léon. Wave propagation and group velocity. Academic Press Inc., New York (1960).

Laser

glass lasers (bottom) used for inertial confinement fusion.]] A LASER (Light Amplification by Stimulated Emission of Radiation) is an optical source that emits photons in a coherent beam. Laser light is typically near-monochromatic, i.e. consisting of a single wavelength or hue, and emitted in a narrow beam. This is in contrast to common light sources, such as the incandescent light bulb, which emit incoherent photons in almost all directions, usually over a wide spectrum of wavelengths. Laser action is understood by application of quantum mechanics and thermodynamics theory (see laser science). The verb "to lase" means "to produce coherent light" or possibly "to cut or otherwise treat with coherent light", and is a back-formation of the term laser.

Physics

back-formation A laser is composed of an active laser medium and a resonant optical cavity. The gain medium is a material of controlled purity, size, and shape, which uses a quantum mechanical effect called stimulated emission (discovered by Einstein while researching the photoelectric effect) to amplify the beam. For a laser to operate, the gain medium must be "pumped" by an external energy source, such as electricity or light (from a classical source such as a flash lamp, or another laser). The pump energy is absorbed by the laser medium to produce excited states in the medium. When the number of particles in one excited state exceeds the number of particles in some lower state, population inversion is achieved. In this condition, an optical beam passing through the medium produces more stimulated emission than stimulated absorption so the beam is amplified. An excited laser medium can also function as an optical amplifier. The light generated by stimulated emission is very similar to the input signal in terms of wavelength, phase, and polarization. This gives laser light its characteristic coherence, and allows it to maintain the uniform polarization and monochromaticity established by the optical cavity design. The resonant cavity (see also cavity resonator) contains a coherent beam of light between reflective surfaces so that each photon passes through the gain medium multiple times before being emitted from the output aperture or lost to diffraction or absorption. As light circulates through the cavity, passing through the gain medium, if the gain (amplification) in the medium is stronger than the resonator losses, the power of the circulating light can rise exponentially. However, each stimulated emission event returns a particle from its excited state to the ground state, reducing the capacity of the gain medium for further amplification. When this effect becomes strong, the gain is said to be saturated. The balance of pump power against gain saturation and cavity losses produces an equilibrium value of the intracavity laser power which determines the operating point of the laser. If the pump power is chosen too small (below the "laser threshold"), the gain is not sufficient to overcome the resonator losses, and the laser will emit only very small light powers. The beam in the cavity and the output beam of the laser, if they occur in free space rather than waveguides (as in an optical fiber laser), are often Gaussian beams. If the beam is not a pure Gaussian shape, the transverse modes of the beam may be analyzed as a superposition of Hermite-Gaussian or Laguerre-Gaussian beams. The beam often has a very small divergence (highly collimated), but a perfectly collimated beam cannot be created, due to the effect of diffraction. Nonetheless, a laser beam will spread much less than a beam of incoherent light. The distance over which the beam remains collimated increases with the square of the beam diameter, and the angle at which the beam eventually diverges varies inversely with the diameter. Thus, a beam generated by a small laboratory laser such as a helium-neon (HeNe) laser spreads to approximately 1 mile (1.6 kilometres) in diameter if shone from the Earth's surface to the Moon. By comparison, the output of a typical semiconductor laser, due to its small diameter, diverges almost immediately on exiting the aperture, at an angle that may be as high as 50°. However, such a divergent beam can be transformed into a collimated beam by means of a lens. In contrast, the light from non-laser light sources cannot be collimated by optics as well or much. lens at Univ. Paris 6. The glowing ray in the middle is an electric discharge producing light in much the same way as a neon light; though it is the gain medium through which the laser passes, it is not the laser beam itself which is visible there. The laser beam crosses the air and marks a red point on the screen to the right.]] The output of a laser may be a continuous, constant-amplitude output (known as CW or continuous wave), or pulsed, by using the techniques of Q-switching, modelocking, or gain-switching. In pulsed operation, much higher peak powers can be achieved. Some types of lasers, such as dye lasers and vibronic solid-state lasers can produce light over a broad range of wavelengths; this property makes them suitable for the generation of extremely short pulses of light, on the order of a femtosecond (10^-15 seconds). It should be understood that the word light in the acronym LASER is meant in the expansive sense, as photons of any energy; and is not limited to photons in the visible spectrum. Hence there are X-ray lasers, infrared lasers, ultraviolet lasers, etc. Because the microwave equivalent of the laser, the maser, was developed first, devices that emit microwave and radio frequencies are usually called masers. In early literature, particularly from researchers at Bell Telephone Laboratories, the laser was often called the optical maser. This usage has since become uncommon, and as of 1998 even Bell Labs uses the term laser [http://www.bell-labs.com/about/history/laser/].

History

In 1916, Albert Einstein laid the foundation for the invention of the laser and its predecessor, the maser, in a ground-breaking rederivation of Max Planck's law of radiation based on the concepts of spontaneous and induced emission. The theory was forgotten until after World War II. In 1953, Charles H. Townes and graduate students James P. Gordon and Herbert J. Zeiger produced the first maser, a device operating on similar principles to the laser, but producing microwave rather than optical radiation. Townes' maser was incapable of continuous output. Nikolay Basov and Aleksandr Prokhorov of the Soviet Union worked independently on the quantum oscillator and solved the problem of continuous output systems by using more than two energy levels. These systems could release stimulated emission without falling to the ground state, thus maintaining a population inversion. Townes, Basov and Prokhorov shared the Nobel Prize in Physics in 1964 "for fundamental work in the field of quantum electronics, which has led to the construction of oscillators and amplifiers based on the maser-laser principle." In 1957 Charles Townes and Arthur Leonard Schawlow, then at Bell Labs, began a serious study of the infrared maser. As ideas were developed, infrared frequencies were abandoned with focus on visible light instead. The concept was originally known as an "optical maser". Bell Labs filed a patent application for their proposed optical maser a year later. Schawlow and Townes sent a manuscript of their theoretical calculations to Physical Review, which published their paper that year (Volume 112, Issue 6). Simultanously, Gordon Gould, a graduate student at Columbia University, was working on a doctoral thesis on the energy levels of excited thallium. Gould and Townes met and had conversations on the general subject of radiation emission. After that meeting, Gould made notes about his ideas for a "laser" in November 1957. In 1958, Prokhorov proposed an open resonator which became an important ingredient of future lasers. The first introduction of the term "laser" to the public was in Gould's 1959 paper "The LASER, Light Amplification by Stimulated Emission of Radiation". Gould intended "aser" to be a suffix, to be used with an appropriate prefix for the spectra of light emitted by the device (e.g. X-ray laser = xaser, UltraViolet laser = uvaser). None of the other terms became popular, although "raser" is sometimes used for radio-frequency emitting devices. Gould's notes included possible applications for a laser, such as spectrometry, interferometry, radar, and nuclear fusion. He continued working on his idea and filed a patent application in April 1959. The U.S. Patent Office denied his application and awarded it to Bell Labs in 1960. This sparked a legal battle that spanned three decades, with scientific prestige and much money at stake. Gould won his first minor patent in 1977, but it was not until 1987 that he could claim his first significant patent victory when a federal judge ordered the government to issue a patent to him for each of the optically pumped and the gas discharge laser. The first working laser was made by Theodore H. Maiman in 1960 at Hughes Research Laboratories in Malibu, California, beating several research teams including those of Townes at Columbia University, and Arthur L. Schawlow at Bell Labs. Maiman used a solid-state flashlamp-pumped synthetic ruby crystal to produce red laser light at 694 nanometres wavelength. Maiman's laser, however, was only capable of pulsed operation due to its three energy level transitions. Later in the same year the Iranian physicist Ali Javan, together with William Bennet and Donald Herriot, made the first gas laser using helium and neon. Javan later received the Albert Einstein Award in 1993. The concept of the semiconductor laser was proposed by Basov and Javan; and the first laser diode was demonstrated by Robert N. Hall in 1962. Hall's device was constructed in the GaAs material system and produced emission at 850 nm, in the near-infrared region of the spectrum. The first semiconductor laser with visible emission was demonstrated later the same year by Nick Holonyak, Jr. As with the first gas lasers, these early semiconductor lasers could be used only in pulsed operation, and indeed only when cooled to liquid nitrogen temperatures (77 K). In 1970, Zhores Alferov in the Soviet Union and Hayashi and Panish of Bell Telephone Laboratories independently developed continuously operating laser diodes at room temperature, using the heterojunction structure. The first application of lasers visible in the daily lives of the general population was the supermarket barcode scanner, introduced in 1974. The laserdisc player, introduced in 1978, was the first successful consumer product to include a laser, but the compact disc player was the first laser-equipped device to become truly common in consumers' homes, beginning in 1982.

Recent innovations

1982 Since the early period of laser history, laser research has produced a variety of improved and specialized laser types, optimized for different performance goals, including
- new wavelength bands
- maximum average output power
- maximum peak output power
- minimum output pulse duration
- maximum power efficiency and this research continues to this day.
Lasing without maintaining the medium excited into a population inversion, was discovered in 1992 in sodium gas and again in 1995 in rubidium gas by various international teams. This was accomplished by using an external maser to induce "optical transparency" in the medium by introducing and destructively interfering the ground electron transitions between two paths, so that the likelihood for the ground electrons to absorb any energy has been cancelled. In 1985 at the University of Rochester's Laboratory for Laser Energetics a breakthrough in creating ultrashort-pulse, very high-intensity (terawatts) laser pulses became available using a technique called chirped pulse amplification, or CPA, discovered by Gérard Mourou. These high intensity pulses can produce filament propagation in the atmosphere.

Uses of lasers

At the time of their invention in 1960, lasers were called "a solution looking for a problem". Since then, they have become ubiquitous, finding utility in thousands of highly varied applications in every section of modern society, including consumer electronics, information technology, science, medicine, industry, law enforcement and the military. They have been widely regarded as one of the most influential technological achievements of the 20th century. The benefits of lasers in various applications stems from their properties such as coherency, high monochromaticity, capability for reaching extremely high powers. For instance, a highly coherent laser beam can be focused down to its diffraction limit, which at visible wavelengths corresponds to only a few hundred nanometers. This property allows a laser to record gigabytes of information in the microscopic pits of a DVD. It also allows a laser of modest power to be focused to very high intensities and used for cutting, burning or even vaporizing materials. For example, a frequency doubled neodymium yttrium aluminum garnet (Nd:YAG) laser emitting 532 nanometer (green) light at 10 watts output power is theoretically capable of achieving an intensity of megawatts per square centimeter. In reality however, perfect focusing of a beam to its diffraction limit is very difficult. square centimeter In consumer electronics, telecommunications, and data communications, lasers are used as the transmitters in optical communications over optical fiber and free space. They are used to store and retrieve data from compact discs and DVDs, as well as magneto-optical discs. Laser lighting displays (pictured) accompany many music concerts. In science, lasers are employed in a wide variety of interferometric techniques, and for Raman spectroscopy and laser induced breakdown spectroscopy . Other uses include atmospheric remote sensing, and investigation of nonlinear optics phenomena. Holographic techniques employing lasers also contribute to a number of measurement techniques. Lasers have also been used aboard scientific spacecraft. In medicine, the laser scalpel is used for laser vision correction and other surgical techniques. Lasers are also used for dermatological procedures including removal of tattoos, birthmarks, and hair; laser types used in dermatology include ruby (694 nm), alexandrite (755 nm), pulsed diode array (810 nm), Nd:YAG (1064 nm), Ho:YAG (2090 nm), and Er:YAG (2940 nm). In industry, laser cutting is used to cut steel and other metals. Laser line levels are used in surveying and construction. Lasers are also used for guidance for aircraft. Lasers are used in certain types of thermonuclear fusion reactors. In law enforcement the most widely known use of lasers is for lidar to detect the speed of vehicles. Military uses of lasers include use as target designators for other weapons; their use as directed-energy weapons is currently under research. Laser weapon systems under development include the airborne laser, the airborne tactical laser, the Tactical High Energy Laser, the High Energy Liquid Laser Area Defense System, and the MIRACL, or Mid-Infrared Advanced Chemical Laser.

Popular misconceptions

The representation of lasers in popular culture, especially in science fiction and action movies, is generally very misleading. For instance, contrary to their portrayal in movies such as Star Wars, a laser beam is never visible in the vacuum of space. In air the ray can hit dust and other particles in its path and scatter producing a glowing "ray", in much the same way that a sunbeam glows in dusty air. This effect can be intensified to make the beam more visible by increasing the amount of suspended particles in the air. Very high intensity beams can be visible in air due to Rayleigh scattering or Raman scattering. With even higher intensity beams, the air can heat up to the point where it becomes a plasma, which would be visible. This would however cause a loud explosion, and will cause a reflection of the ray back into the laser, probably damaging it (depending on the laser design). Furthermore, science fiction film special effects often depict laser beams propagating at only a few metres per second—i.e., slowly enough to see their progress, in a manner reminiscent of conventional tracer ammunition—whereas in reality a laser beam travels at the speed of light, and would be instantly visible along its entire length. Some action movies depict security systems using red lasers (and being foiled by the hero, typically using mirrors); the hero may see the path of the beam by sprinkling some white dust in the air. It is actually easier and cheaper to build infrared laser diodes rather than visible light laser diodes, therefore such systems would almost certainly not use visible light.

Laser safety

Even low-power lasers with only a few milliwatts of output power can be hazardous to a person's eyesight. At wavelengths which the cornea and the lens can focus well, the coherence and low divergence of laser light means that it can be focused by the eye into an extremely small spot on the retina, resulting in localised burning and permanent damage in seconds or even faster. Lasers are classified into safety classes numbered I, inherently safe, to IV, even scattered light can cause eye and/or skin damage. Laser products available for consumers, such as CD players and laser pointers are usually in class I, II, or III.

Common laser types

For a more complete list of laser types see list of laser types. list of laser types

Color	Wavelength interval	Frequency interval
red	~ 625 to 740 nm	~ 480 to 405 THz
orange	~ 590 to 625 nm	~ 510 to 480 THz
yellow	~ 565 to 590 nm	~ 530 to 510 THz
green	~ 520 to 565 nm	~ 580 to 530 THz
cyan	~ 500 to 520 nm	~ 600 to 580 THz
blue	~ 430 to 500 nm	~ 700 to 600 THz
violet	~ 380 to 430 nm	~ 790 to 700 THz

- Gas lasers
- HeNe (543 nm and 633 nm)
- Argon-Ion (458 nm, 488 nm or 514.5 nm)
- Carbon dioxide lasers (9.6 µm and 10.6 µm) used in industry for cutting and welding, up to 100 kW possible
- TEA Laser (UV Light, 337.1 nm)
- Nitrogen laser
- Carbon monoxide lasers, must be cooled, but extremely powerful, up to 500 kW possible
- Chemical lasers
- Chemical oxygen iodine laser (1315 nm)
- Hydrogen fluoride laser (2700-2900 nm)
- Deuterium fluoride laser (3800 nm)
- Excimer gas lasers, producing ultraviolet light, used in semiconductor manufacturing and in LASIK eye surgery; F₂ (157 nm), ArF (193 nm), KrCl (222 nm), KrF (248 nm), XeCl (308 nm), XeF (351 nm)
- Semiconductor lasers
- Laser diodes produce wavelengths from 405 nm to 1550 nm. Low power laser diodes are used in laser pointers, laser printers, and CD/DVD players. More powerful laser diodes are frequently used to optically pump other lasers with high efficiency. The highest power industrial laser diodes, with power up to 10 kW, are used in industry for cutting and welding.
- External-cavity semiconductor lasers have a semiconductor active medium in a larger cavity. These devices can generate high power outputs with good beam quality, wavelength-tunable narrow-linewidth radiation, or ultrashort laser pulses.
- VCSELs are semiconductor lasers whose emission direction is perpendicular to the surface of the wafer. VCSEL devices can achieve better beam quality and higher output power than conventional laser diodes, and potentially could be much cheaper to manufacture. The technology is not (as of 2005) as well developed, however.
- VECSELs are external-cavity VCSELs.
- Quantum cascade lasers are semiconductor lasers that have an active transition between energy sub-bands of an electron in a structure containing several quantum wells.
- Solid-state lasers
- Neodymium-doped YAG lasers (Nd:YAG), a high-power laser operating in the infrared spectrum at 1064nm, used for cutting, welding and marking of metals and other materials also used in spectroscopy and for pumping dye lasers. Can be frequency doubled from 1064nm to 532nm to produce a green laser.
- Ytterbium-doped lasers with crystals such as Yb:YAG, Yb:KGW, Yb:KYW, Yb:SYS, Yb:BOYS, Yb:CaF2, or Yb-doped glasses (e.g. fibers); typically operating around 1020-1050 nm; potentially very high efficiency and high powers due to a small quantum defect; extremely high powers in ultrashort pulses can be achieved with Yb:YAG
- Erbium-doped YAG, 1645 nm, 2940 nm
- Thulium-doped YAG, 2015 nm
- Holmium-doped YAG, 2097 nm; an efficient laser operating in the infrared spectrum, it is strongly absorbed by water-bearing tissues in sections less than a millimeter thick. It is usually operated in a pulsed mode, and passed through optical fiber surgical devices to resurface joints, remove rot from teeth, vaporize cancers, and pulverize kidney and gall stones.
- Titanium-doped sapphire (Ti:sapphire) lasers, a highly tunable infrared laser, used for spectroscopy
- Erbium-doped fiber lasers, a type of laser formed from a specially made optical fiber, which is used as an amplifier for optical communications.
- Dye lasers
- Hollow cathode sputtering metal ion lasers, generating deep ultraviolet wavelengths, of which there are two examples; Helium-Silver (HeAg) 224 nm and Neon-Copper (NeCu) 248 nm. These lasers have particularly narrow oscillation linewidths of less than 0.01 cm^-1 making them good candidates for use in fluorescence suppressed Raman spectroscopy.

External links

- [http://www.repairfaq.org/sam/lasersam.htm Sam's Laser FAQ] by Samuel M. Goldwasser
- [http://www.rp-photonics.com/encyclopedia.html Encyclopedia of laser physics and technology] by [http://www.rp-photonics.com/paschotta.html Dr. Rüdiger Paschotta]
- [http://www.aip.org/pnu/2002/596.html Liquid Light] by Phil Schewe, James Riordon, and Ben Stein
- [http://www.newscientist.com/article.ns?id=dn2497 Light turns into glowing liquid] by Eugenie Samuel
- [http://www.aip.org/pt/vol-54/iss-8/p17.html Experiments Detail How Powerful Ultrashort Laser Pulses Propagate through Air]
- [http://www.nrl.navy.mil/content.php?P=03REVIEW59 Filamentation and Propagation of Ultra-Short, Intense Laser Pulses in Air]
- [http://www.aip.org/pnu/1992/physnews.100.htm Lasing Activity without Population Inversion] by Phillip F. Schewe and Ben Stein
- [http://www.aip.org/pnu/1995/physnews.240.htm Lasing without Inversion] by Phillip F. Schewe and Ben Stein Category:Optical devices Lasers Category:Lighting Category:Acronyms ko:레이저 ms:laser ja:レーザー

Dispersion relation

See also

References

External links

Wave

The medium which carries a wave

Examples of waves

Characteristic properties

Transverse and longitudinal waves

Polarization

Physical description of a wave

Travelling waves

Propagation through strings

The wave equation

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Velocity

Explanation

Formal description

Polar coordinates

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Wave number

Circular wavenumber

Wavenumber

Atmospheric science

Energy

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Transfer of energy

Work

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Signal velocity

See also

References

Laser

Physics

History

Recent innovations

Uses of lasers

Popular misconceptions

Laser safety

Common laser types

See also

External links

Laser

Physics

History

Recent innovations

Uses of lasers

Popular misconceptions