1.
Sine wave
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A sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. It is named after the sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time is, y = A sin = A sin where, A = the amplitude, F = the ordinary frequency, the number of oscillations that occur each second of time. ω = 2πf, the frequency, the rate of change of the function argument in units of radians per second φ = the phase. When φ is non-zero, the entire waveform appears to be shifted in time by the amount φ /ω seconds, a negative value represents a delay, and a positive value represents an advance. The sine wave is important in physics because it retains its shape when added to another sine wave of the same frequency and arbitrary phase. It is the only periodic waveform that has this property and this property leads to its importance in Fourier analysis and makes it acoustically unique. The wavenumber is related to the frequency by. K = ω v =2 π f v =2 π λ where λ is the wavelength, f is the frequency, and v is the linear speed. This equation gives a wave for a single dimension, thus the generalized equation given above gives the displacement of the wave at a position x at time t along a single line. This could, for example, be considered the value of a wave along a wire, in two or three spatial dimensions, the same equation describes a travelling plane wave if position x and wavenumber k are interpreted as vectors, and their product as a dot product. For more complex such as the height of a water wave in a pond after a stone has been dropped in. This wave pattern occurs often in nature, including wind waves, sound waves, a cosine wave is said to be sinusoidal, because cos = sin , which is also a sine wave with a phase-shift of π/2 radians. Because of this start, it is often said that the cosine function leads the sine function or the sine lags the cosine. The human ear can recognize single sine waves as sounding clear because sine waves are representations of a frequency with no harmonics. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, on the other hand, if the sound contains aperiodic waves along with sine waves, then the sound will be perceived noisy as noise is characterized as being aperiodic or having a non-repetitive pattern. In 1822, French mathematician Joseph Fourier discovered that sinusoidal waves can be used as building blocks to describe and approximate any periodic waveform
2.
Phase (waves)
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Phase is the position of a point in time on a waveform cycle. A complete cycle is defined as the interval required for the waveform to return to its initial value. The graphic to the right shows how one cycle constitutes 360° of phase, the graphic also shows how phase is sometimes expressed in radians, where one radian of phase equals approximately 57. 3°. Phase can also be an expression of relative displacement between two corresponding features of two waveforms having the same frequency, in sinusoidal functions or in waves phase has two different, but closely related, meanings. One is the angle of a sinusoidal function at its origin and is sometimes called phase offset or phase difference. Another usage is the fraction of the cycle that has elapsed relative to the origin. Phase shift is any change that occurs in the phase of one quantity and this symbol, φ is sometimes referred to as a phase shift or phase offset because it represents a shift from zero phase. For infinitely long sinusoids, a change in φ is the same as a shift in time, if x is delayed by 14 of its cycle, it becomes, x = A ⋅ cos = A ⋅ cos whose phase is now φ − π2. It has been shifted by π2 radians, Phase difference is the difference, expressed in degrees or time, between two waves having the same frequency and referenced to the same point in time. Two oscillators that have the frequency and no phase difference are said to be in phase. Two oscillators that have the frequency and different phases have a phase difference. The amount by which such oscillators are out of phase with each other can be expressed in degrees from 0° to 360°, if the phase difference is 180 degrees, then the two oscillators are said to be in antiphase. If two interacting waves meet at a point where they are in antiphase, then interference will occur. It is common for waves of electromagnetic, acoustic or other energy to become superposed in their transmission medium, when that happens, the phase difference determines whether they reinforce or weaken each other. Complete cancellation is possible for waves with equal amplitudes, time is sometimes used to express position within the cycle of an oscillation. A phase difference is analogous to two athletes running around a track at the same speed and direction but starting at different positions on the track. They pass a point at different instants in time, but the time difference between them is a constant - same for every pass since they are at the same speed and in the same direction. If they were at different speeds, the difference is undefined
3.
Zero crossing
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A zero-crossing is a point where the sign of a mathematical function changes, represented by a crossing of the axis in the graph of the function. It is a commonly used term in electronics, mathematics, sound, in alternating current, the zero-crossing is the instantaneous point at which there is no voltage present. In a sine wave or other simple waveform, this occurs twice during each cycle. Counting zero-crossings is also a used in speech processing to estimate the fundamental frequency of speech. Artifacts are disconcerting and clearly not desirable, if changes are made only at zero-crossings of the input signal, then no matter how the amplifier gain setting changes, the output also remains at zero, thereby minimizing the change. If electrical power is to be switched, no interference is generated if switched at an instant when there is no current—a zero crossing. Early light dimmers and similar devices generated interference, later versions were designed to switch at the zero crossing, in the field of Digital Image Processing, great emphasis is placed on operators which seek out edges within an image. They are called Edge Detection or Gradient filters, a gradient filter is a filter which seeks out areas of rapid change in pixel value. These points usually mark an edge or a boundary, a Laplace filter is a filter which fits in this family, though it sets about the task in a different way. It seeks out points in the stream where the digital signal of an image passes through a pre-set 0 value. Because the signal has crossed through the point of zero, it is called a zero-crossing, an example can be found here, including the source in Java. Reconstruction from zero crossings Zero-crossing rate Root of a function Sign function
4.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
5.
Multiplicative inverse
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In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity,1. The multiplicative inverse of a fraction a/b is b/a, for the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, the reciprocal function, the function f that maps x to 1/x, is one of the simplest examples of a function which is its own inverse. In the phrase multiplicative inverse, the qualifier multiplicative is often omitted, multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba, then inverse typically implies that an element is both a left and right inverse. The notation f −1 is sometimes used for the inverse function of the function f. For example, the multiplicative inverse 1/ = −1 is the cosecant of x, only for linear maps are they strongly related. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, in the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1. With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, the property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no other than 1 and −1 has an integer reciprocal. In modular arithmetic, the multiplicative inverse of a is also defined. This multiplicative inverse exists if and only if a and n are coprime, for example, the inverse of 3 modulo 11 is 4 because 4 ·3 ≡1. The extended Euclidean algorithm may be used to compute it, the sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i. e. nonzero elements x, y such that xy =0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring, the linear map that has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case. A ring in which every element has a multiplicative inverse is a division ring. As mentioned above, the reciprocal of every complex number z = a + bi is complex. In particular, if ||z||=1, then 1 / z = z ¯, consequently, the imaginary units, ±i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property
6.
Standing wave
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In physics, a standing wave – also known as a stationary wave – is a wave in a medium in which each point on the axis of the wave has an associated constant amplitude. The locations at which the amplitude is minimum are called nodes, standing waves were first noticed by Michael Faraday in 1831. Faraday observed standing waves on the surface of a liquid in a vibrating container, franz Melde coined the term standing wave around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings. For waves of equal amplitude traveling in opposing directions, there is on average no net propagation of energy, as an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the lee of mountain ranges. Such waves are often exploited by glider pilots, standing waves and hydraulic jumps also form on fast flowing river rapids and tidal currents such as the Saltstraumen maelstrom. Many standing river waves are popular river surfing breaks, the effect is a series of nodes and anti-nodes at fixed points along the transmission line. The failure of the line to power at the standing wave frequency will usually result in attenuation distortion. In practice, losses in the line and other components mean that a perfect reflection. The result is a standing wave, which is a superposition of a standing wave. The degree to which the wave resembles either a standing wave or a pure traveling wave is measured by the standing wave ratio. Another example is standing waves in the ocean formed by waves with the same wave period moving in opposite directions. These may form near storm centres, or from reflection of a swell at the shore, in one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. The reflected wave has to have the amplitude and frequency as the incoming wave. At locations which are multiples of a quarter wavelength x =. −3 λ2, − λ, − λ2,0, λ2, λ,3 λ2, called the nodes, the amplitude is always zero, whereas at locations which are odd multiples of a quarter wavelength x =. −5 λ4, −3 λ4, − λ4, λ4,3 λ4,5 λ4, called the anti-nodes, the amplitude is maximum, with a value of twice the amplitude of the original waves. The distance between two nodes or anti-nodes is λ/2. Standing waves can occur in two- or three-dimensional resonators
7.
Lambda
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Lambda is the 11th letter of the Greek alphabet. In the system of Greek numerals lambda has a value of 30, Lambda is related to the Phoenician letter Lamed. Letters in other alphabets that stemmed from lambda include the Latin L, the ancient grammarians and dramatists give evidence to the pronunciation as in Classical Greek times. In Modern Greek the name of the letter, Λάμδα, is pronounced, in early Greek alphabets, the shape and orientation of lambda varied. Most variants consisted of two strokes, one longer than the other, connected at their ends. The angle might be in the upper-left, lower-left, or top, other variants had a vertical line with a horizontal or sloped stroke running to the right. With the general adoption of the Ionic alphabet, Greek settled on an angle at the top, the HTML4 character entity references for the Greek capital and small letter lambda are Λ, and λ, respectively. The Unicode code points for lambda are U+039B and U+03BB, the lambda particle is a type of subatomic particle in subatomic particle physics. Lambda is the set of axioms in the axiomatic method of logical deduction in first-order logic. Lambda was used as a pattern by the Spartan army. This stood for Lacedaemon, the name of the polis of the Spartans, Lambda is the von Mangoldt function in mathematical number theory. Lambda is a associated with the Identitarian movement, intended to emulate a Spartan shield pattern. In statistics, Wilkss lambda is used in analysis of variance to compare group means on a combination of dependent variables. In the spectral decomposition of matrices, lambda indicates the diagonal matrix of the eigenvalues of the matrix, in computer science, lambda is the time window over which a process is observed for determining the working memory set for a digital computers virtual memory management. In astrophysics, lambda represents the likelihood that a body will encounter a planet or a dwarf planet leading to a deflection of a significant magnitude. An object with a value of lambda is expected to have cleared its neighborhood. In crystal optics, lambda is used to represent the period of a lattice, in NATO military operations, a chevron is painted on the vehicles of this military alliance for identification. In chemistry there are Δ and Λ isomers, see, coordination complex In electrochemistry, in cosmology, lambda is the symbol for the cosmological constant, a term added to some dynamical equations to account for the acceleration of the universe
8.
Modulation
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Most radio systems in the 20th century used frequency modulation or amplitude modulation to make the carrier carry the radio broadcast. Modulation of a sine waveform transforms a narrow frequency range baseband message signal into a passband signal, a modulator is a device that performs modulation. A demodulator is a device that performs demodulation, the inverse of modulation, a modem can perform both operations. The aim of digital modulation is to transfer a digital bit stream over an analog bandpass channel, in music synthesizers, modulation may be used to synthesise waveforms with an extensive overtone spectrum using a small number of oscillators. In this case the carrier frequency is typically in the order or much lower than the modulating waveform. In analog modulation, the modulation is applied continuously in response to the information signal. Where an AM carrier also carries a variable phase f. TM is f In digital modulation, a carrier signal is modulated by a discrete signal. Digital modulation methods can be considered as digital-to-analog conversion, and the corresponding demodulation or detection as analog-to-digital conversion, the changes in the carrier signal are chosen from a finite number of M alternative symbols. A simple example, A telephone line is designed for transferring audible sounds, for example tones, computers may however communicate over a telephone line by means of modems, which are representing the digital bits by tones, called symbols. If there are four symbols, the first symbol may represent the bit sequence 00, the second 01, the third 10. If the modem plays a melody consisting of 1000 tones per second, since each tone represents a message consisting of two digital bits in this example, the bit rate is twice the symbol rate, i. e.2000 bits per second. This is similar to the used by dialup modems as opposed to DSL modems. According to one definition of digital signal, the signal is a digital signal. According to another definition, the modulation is a form of digital-to-analog conversion, most textbooks would consider digital modulation schemes as a form of digital transmission, synonymous to data transmission, very few would consider it as analog transmission. The most fundamental digital modulation techniques are based on keying, PSK, FSK, a finite number of frequencies are used. ASK, a number of amplitudes are used. QAM, a number of at least two phases and at least two amplitudes are used
9.
Envelope (mathematics)
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In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two adjacent curves, meaning the limit of intersections of nearby curves and this idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions. Let each curve Ct in the family be given as the solution of an equation ft=0, write F=ft and assume F is differentiable. Letting u→t gives the definition above, an important special case is when F is a polynomial in t. This includes, by clearing denominators, the case where F is a function in t. In this case, the amounts to t being a double root of F. For example, let Ct be the line whose x and y intercepts are t and 1−t, this is shown in the animation above. The equation of Ct is x t + y 1 − t =1 or, clearing fractions, the equation of the envelope is then 2 −4 x =2 −2 +1 =0. Often when F is not a function of the parameter it may be reduced to this case by an appropriate substitution. The equation of the envelope is given by setting the discriminant to 0, − w 2 =0 or u 2 + v 2 = w 2. The envelope E1 is the limit of intersections of nearby curves Ct, the envelope E2 is a curve tangent to all of the Ct. The envelope E3 is the boundary of the region filled by the curves Ct, then E1 ⊆ D, E2 ⊆ D and E3 ⊆ D, where D is the set of curves given by the first definition at the beginning of this document. These four definitions of the discriminant, E1, E2, and E3, consider for instance the curve y = x3 parametrised by γ, R → R2 where γ =. The one-parameter family of curves will be given by the tangent lines to γ, first we calculate the discriminant D. The generating function is F =3 t 2 x − y −2 t 3, calculating the partial derivative Ft = 6t. It follows that either x = t or t =0, first assume that x = t and t ≠0. Substituting into F, F = t 3 − y and so, assuming that t ≠0, it follows that F = Ft =0 if, next, assuming that t =0 and substituting into F gives F = −y. So, assuming t =0, it follows that F = Ft =0 if, thus the discriminant is the original curve and its tangent line at γ, D = ∪
10.
Interference (wave propagation)
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In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves or matter waves. If a crest of a wave meets a crest of wave of the same frequency at the same point. If a crest of one wave meets a trough of another wave, constructive interference occurs when the phase difference between the waves is an even multiple of π, whereas destructive interference occurs when the difference is an odd multiple of π. If the difference between the phases is intermediate between two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values. Consider, for example, what happens when two identical stones are dropped into a pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped, when the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, and will produce a maximum displacement, in other places, the waves will be in anti-phase, and there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above, prime examples of light interference are the famous Double-slit experiment, laser speckle, optical thin layers and films and interferometers. Dark areas in the slit are not available to the photons. Thin films also behave in a quantum manner, the above can be demonstrated in one dimension by deriving the formula for the sum of two waves. Suppose a second wave of the frequency and amplitude but with a different phase is also traveling to the right W2 = A cos where ϕ is the phase difference between the waves in radians. Constructive interference, If the phase difference is a multiple of pi. Interference is essentially a redistribution process. The energy which is lost at the interference is regained at the constructive interference. One wave is travelling horizontally, and the other is travelling downwards at an angle θ to the first wave, assuming that the two waves are in phase at the point B, then the relative phase changes along the x-axis. Constructive interference occurs when the waves are in phase, and destructive interference when they are half a cycle out of phase. Thus, a fringe pattern is produced, where the separation of the maxima is d f = λ sin θ
11.
Frequency
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Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as frequency, which emphasizes the contrast to spatial frequency. The period is the duration of time of one cycle in a repeating event, for example, if a newborn babys heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as vibrations, audio signals, radio waves. For cyclical processes, such as rotation, oscillations, or waves, in physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter f or by the Greek letter ν or ν. For a simple motion, the relation between the frequency and the period T is given by f =1 T. The SI unit of frequency is the hertz, named after the German physicist Heinrich Hertz, a previous name for this unit was cycles per second. The SI unit for period is the second, a traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. As a matter of convenience, longer and slower waves, such as ocean surface waves, short and fast waves, like audio and radio, are usually described by their frequency instead of period. Spatial frequency is analogous to temporal frequency, but the axis is replaced by one or more spatial displacement axes. Y = sin = sin d θ d x = k Wavenumber, in the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has a relationship to the wavelength. Even in dispersive media, the frequency f of a wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave. In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes, f = c λ. When waves from a monochrome source travel from one medium to another, their remains the same—only their wavelength. For example, if 71 events occur within 15 seconds the frequency is, the latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an error in the calculated frequency of Δf = 1/, or a fractional error of Δf / f = 1/ where Tm is the timing interval. This error decreases with frequency, so it is a problem at low frequencies where the number of counts N is small, an older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope
12.
Sound
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In physics, sound is a vibration that propagates as a typically audible mechanical wave of pressure and displacement, through a transmission medium such as air or water. In physiology and psychology, sound is the reception of such waves, humans can hear sound waves with frequencies between about 20 Hz and 20 kHz. Sound above 20 kHz is ultrasound and below 20 Hz is infrasound, other animals have different hearing ranges. Acoustics is the science that deals with the study of mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound. A scientist who works in the field of acoustics is an acoustician, an audio engineer, on the other hand, is concerned with the recording, manipulation, mixing, and reproduction of sound. Auditory sensation evoked by the oscillation described in, sound can propagate through a medium such as air, water and solids as longitudinal waves and also as a transverse wave in solids. The sound waves are generated by a source, such as the vibrating diaphragm of a stereo speaker. The sound source creates vibrations in the surrounding medium, as the source continues to vibrate the medium, the vibrations propagate away from the source at the speed of sound, thus forming the sound wave. At a fixed distance from the source, the pressure, velocity, at an instant in time, the pressure, velocity, and displacement vary in space. Note that the particles of the medium do not travel with the sound wave and this is intuitively obvious for a solid, and the same is true for liquids and gases. During propagation, waves can be reflected, refracted, or attenuated by the medium, the behavior of sound propagation is generally affected by three things, A complex relationship between the density and pressure of the medium. This relationship, affected by temperature, determines the speed of sound within the medium, if the medium is moving, this movement may increase or decrease the absolute speed of the sound wave depending on the direction of the movement. For example, sound moving through wind will have its speed of propagation increased by the speed of the if the sound and wind are moving in the same direction. If the sound and wind are moving in opposite directions, the speed of the wave will be decreased by the speed of the wind. Medium viscosity determines the rate at which sound is attenuated, for many media, such as air or water, attenuation due to viscosity is negligible. When sound is moving through a medium that does not have constant physical properties, the mechanical vibrations that can be interpreted as sound can travel through all forms of matter, gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium, sound cannot travel through a vacuum. Sound is transmitted through gases, plasma, and liquids as longitudinal waves and it requires a medium to propagate
13.
Light
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Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to light, which is visible to the human eye and is responsible for the sense of sight. Visible light is defined as having wavelengths in the range of 400–700 nanometres, or 4.00 × 10−7 to 7.00 × 10−7 m. This wavelength means a range of roughly 430–750 terahertz. The main source of light on Earth is the Sun, sunlight provides the energy that green plants use to create sugars mostly in the form of starches, which release energy into the living things that digest them. This process of photosynthesis provides virtually all the used by living things. Historically, another important source of light for humans has been fire, with the development of electric lights and power systems, electric lighting has effectively replaced firelight. Some species of animals generate their own light, a process called bioluminescence, for example, fireflies use light to locate mates, and vampire squids use it to hide themselves from prey. Visible light, as all types of electromagnetic radiation, is experimentally found to always move at this speed in a vacuum. In physics, the term sometimes refers to electromagnetic radiation of any wavelength. In this sense, gamma rays, X-rays, microwaves and radio waves are also light, like all types of light, visible light is emitted and absorbed in tiny packets called photons and exhibits properties of both waves and particles. This property is referred to as the wave–particle duality, the study of light, known as optics, is an important research area in modern physics. Generally, EM radiation, or EMR, is classified by wavelength into radio, microwave, infrared, the behavior of EMR depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths, when EMR interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. There exist animals that are sensitive to various types of infrared, infrared sensing in snakes depends on a kind of natural thermal imaging, in which tiny packets of cellular water are raised in temperature by the infrared radiation. EMR in this range causes molecular vibration and heating effects, which is how these animals detect it, above the range of visible light, ultraviolet light becomes invisible to humans, mostly because it is absorbed by the cornea below 360 nanometers and the internal lens below 400. Furthermore, the rods and cones located in the retina of the eye cannot detect the very short ultraviolet wavelengths and are in fact damaged by ultraviolet. Many animals with eyes that do not require lenses are able to detect ultraviolet, by quantum photon-absorption mechanisms, various sources define visible light as narrowly as 420 to 680 to as broadly as 380 to 800 nm
14.
Water wave
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In fluid dynamics, wind waves, or wind-generated waves, are surface waves that occur on the free surface of bodies of water. They result from the wind blowing over an area of fluid surface, Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft high, when directly generated and affected by local winds, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells, more generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago, wind waves in the ocean are called ocean surface waves. Wind waves have an amount of randomness, subsequent waves differ in height, duration. The key statistics of wind waves in evolving sea states can be predicted with wind wave models, although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves. The great majority of large breakers seen at a result from distant winds. Water depth All of these work together to determine the size of wind waves. Further exposure to that wind could only cause a dissipation of energy due to the breaking of wave tops. Waves in an area typically have a range of heights. For weather reporting and for analysis of wind wave statistics. This figure represents an average height of the highest one-third of the waves in a time period. The significant wave height is also the value a trained observer would estimate from visual observation of a sea state, given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm. Wave formation on a flat water surface by wind is started by a random distribution of normal pressure of turbulent wind flow over the water. This pressure fluctuation produces normal and tangential stresses in the surface water and it is assumed that, The water is originally at rest. There is a distribution of normal pressure to the water surface from the turbulent wind. Correlations between air and water motions are neglected, the second mechanism involves wind shear forces on the water surface
15.
Electrical conductor
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In physics and electrical engineering, a conductor is an object or type of material that allows the flow of an electrical current in one or more directions. Materials made of metal are common electrical conductors, Electrical current is generated by the flow of negatively charged electrons, positively charged holes, and positive or negative ions in some cases. In order for current to flow, it is not necessary for one charged particle to travel from the producing the current to that consuming it. Instead, the particle simply needs to nudge its neighbor a finite amount who will nudge its neighbor and on and on until a particle is nudged into the consumer. Essentially what is occurring here is a chain of momentum transfer between mobile charge carriers, the Drude model of conduction describes this process more rigorously. Insulators are non-conducting materials with few mobile charges that support only insignificant electric currents, the resistance of a given conductor depends on the material it is made of, and on its dimensions. For a given material, the resistance is proportional to the cross-sectional area. For example, a copper wire has lower resistance than an otherwise-identical thin copper wire. Also, for a material, the resistance is proportional to the length, for example. The resistance R and conductance G of a conductor of uniform cross section, therefore, the resistivity and conductivity are proportionality constants, and therefore depend only on the material the wire is made of, not the geometry of the wire. Resistivity and conductivity are reciprocals, ρ =1 / σ, resistivity is a measure of the materials ability to oppose electric current. This formula is not exact, It assumes the current density is uniform in the conductor. However, this still provides a good approximation for long thin conductors such as wires. Another situation this formula is not exact for is with alternating current, then, the geometrical cross-section is different from the effective cross-section in which current actually flows, so the resistance is higher than expected. Similarly, if two conductors are each other carrying AC current, their resistances increase due to the proximity effect. Aside from the geometry of the wire, temperature also has a significant effect on the efficacy of conductors, temperature affects conductors in two main ways, the first is that materials may expand under the application of heat. The amount that the material will expand is governed by the expansion coefficient specific to the material. Such an expansion will change the geometry of the conductor and therefore its characteristic resistance, however, this effect is generally small, on the order of 10−6
16.
Sound pressure
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Sound pressure or acoustic pressure is the local pressure deviation from the ambient atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, the SI unit of sound pressure is the pascal. A sound wave in a medium causes a deviation in the local ambient pressure. Sound pressure, denoted p, is defined by p t o t a l = p s t a t + p, in a sound wave, the complementary variable to sound pressure is the particle velocity. Together they determine the intensity of the wave. Sound intensity, denoted I and measured in W·m−2 in SI units, is defined by I = p v. Consequently, the amplitude of the displacement is related to that of the acoustic velocity. This relationship is an inverse-proportional law, if the sound pressure p1 is measured at a distance r1 from the centre of the sphere, the sound pressure p2 at another position r2 can be calculated, p 2 = r 1 r 2 p 1. The inverse-proportional law for sound pressure comes from the law for sound intensity. Indeed, I = p v = p ∝ p 2, the sound pressure may vary in direction from the centre of the sphere as well, so measurements at different angles may be necessary, depending on the situation. An obvious example of a sound source whose spherical sound wave varies in level in different directions is a bullhorn, Sound pressure level or acoustic pressure level is a logarithmic measure of the effective pressure of a sound relative to a reference value. The commonly used reference sound pressure in air is p 0 =20 μ P a, which is often considered as the threshold of human hearing. The proper notations for sound pressure level using this reference are Lp/ or Lp, most sound level measurements will be made relative to this reference, meaning 1 Pa will equal an SPL of 94 dB. In other media, such as underwater, a level of 1 μPa is used. These references are defined in ANSI S1. 1-1994, the lower limit of audibility is defined as SPL of 0 dB, but the upper limit is not as clearly defined. Ears detect changes in sound pressure, human hearing does not have a flat spectral sensitivity relative to frequency versus amplitude. Humans do not perceive low- and high-frequency sounds as well as they perceive sounds between 3,000 and 4,000 Hz, as shown in the equal-loudness contour. Because the frequency response of human hearing changes with amplitude, three weightings have been established for measuring pressure, A, B and C. A-weighting applies to sound pressures levels up to 55 dB, B-weighting applies to sound levels between 55 dB and 85 dB, and C-weighting is for measuring sound pressure levels above 85 dB
17.
Electromagnetic radiation
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In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating through space carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, light, ultraviolet, X-, classically, electromagnetic radiation consists of electromagnetic waves, which are synchronized oscillations of electric and magnetic fields that propagate at the speed of light through a vacuum. The oscillations of the two fields are perpendicular to other and perpendicular to the direction of energy and wave propagation. The wavefront of electromagnetic waves emitted from a point source is a sphere, the position of an electromagnetic wave within the electromagnetic spectrum can be characterized by either its frequency of oscillation or its wavelength. Electromagnetic waves are produced whenever charged particles are accelerated, and these waves can interact with other charged particles. EM waves carry energy, momentum and angular momentum away from their source particle, quanta of EM waves are called photons, whose rest mass is zero, but whose energy, or equivalent total mass, is not zero so they are still affected by gravity. Thus, EMR is sometimes referred to as the far field, in this language, the near field refers to EM fields near the charges and current that directly produced them, specifically, electromagnetic induction and electrostatic induction phenomena. In the quantum theory of electromagnetism, EMR consists of photons, quantum effects provide additional sources of EMR, such as the transition of electrons to lower energy levels in an atom and black-body radiation. The energy of a photon is quantized and is greater for photons of higher frequency. This relationship is given by Plancks equation E = hν, where E is the energy per photon, ν is the frequency of the photon, a single gamma ray photon, for example, might carry ~100,000 times the energy of a single photon of visible light. The effects of EMR upon chemical compounds and biological organisms depend both upon the power and its frequency. EMR of visible or lower frequencies is called non-ionizing radiation, because its photons do not individually have enough energy to ionize atoms or molecules, the effects of these radiations on chemical systems and living tissue are caused primarily by heating effects from the combined energy transfer of many photons. In contrast, high ultraviolet, X-rays and gamma rays are called ionizing radiation since individual photons of high frequency have enough energy to ionize molecules or break chemical bonds. These radiations have the ability to cause chemical reactions and damage living cells beyond that resulting from simple heating, Maxwell derived a wave form of the electric and magnetic equations, thus uncovering the wave-like nature of electric and magnetic fields and their symmetry. Because the speed of EM waves predicted by the wave equation coincided with the speed of light. Maxwell’s equations were confirmed by Heinrich Hertz through experiments with radio waves, according to Maxwells equations, a spatially varying electric field is always associated with a magnetic field that changes over time. Likewise, a varying magnetic field is associated with specific changes over time in the electric field. In an electromagnetic wave, the changes in the field are always accompanied by a wave in the magnetic field in one direction
18.
Electric field
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An electric field is a vector field that associates to each point in space the Coulomb force that would be experienced per unit of electric charge, by an infinitesimal test charge at that point. Electric fields are created by electric charges and can be induced by time-varying magnetic fields, the electric field combines with the magnetic field to form the electromagnetic field. The electric field, E, at a point is defined as the force, F. A particle of charge q would be subject to a force F = q E and its SI units are newtons per coulomb or, equivalently, volts per metre, which in terms of SI base units are kg⋅m⋅s−3⋅A−1. Electric fields are caused by electric charges or varying magnetic fields, in the special case of a steady state, the Maxwell-Faraday inductive effect disappears. The resulting two equations, taken together, are equivalent to Coulombs law, written as E =14 π ε0 ∫ d r ′ ρ r − r ′ | r − r ′ |3 for a charge density ρ. Notice that ε0, the permittivity of vacuum, must be substituted if charges are considered in non-empty media, the equations of electromagnetism are best described in a continuous description. A charge q located at r 0 can be described mathematically as a charge density ρ = q δ, conversely, a charge distribution can be approximated by many small point charges. Electric fields satisfy the principle, because Maxwells equations are linear. This principle is useful to calculate the field created by point charges. Q n are stationary in space at r 1, r 2, in that case, Coulombs law fully describes the field. If a system is static, such that magnetic fields are not time-varying, then by Faradays law, in this case, one can define an electric potential, that is, a function Φ such that E = − ∇ Φ. This is analogous to the gravitational potential, Coulombs law, which describes the interaction of electric charges, F = q = q E is similar to Newtons law of universal gravitation, F = m = m g. This suggests similarities between the electric field E and the gravitational field g, or their associated potentials, mass is sometimes called gravitational charge because of that similarity. Electrostatic and gravitational forces both are central, conservative and obey an inverse-square law, a uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conducting plates parallel to other and maintaining a voltage between them, it is only an approximation because of boundary effects. Assuming infinite planes, the magnitude of the electric field E is, electrodynamic fields are E-fields which do change with time, for instance when charges are in motion. The electric field cannot be described independently of the field in that case
19.
Magnetic field
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A magnetic field is the magnetic effect of electric currents and magnetic materials. The magnetic field at any point is specified by both a direction and a magnitude, as such it is represented by a vector field. The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter in the SI, B is measured in teslas and newtons per meter per ampere in the SI. B is most commonly defined in terms of the Lorentz force it exerts on moving electric charges, Magnetic fields can be produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. In quantum physics, the field is quantized and electromagnetic interactions result from the exchange of photons. Magnetic fields are used throughout modern technology, particularly in electrical engineering. The Earth produces its own field, which is important in navigation. Rotating magnetic fields are used in electric motors and generators. Magnetic forces give information about the carriers in a material through the Hall effect. The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits, noting that the resulting field lines crossed at two points he named those points poles in analogy to Earths poles. He also clearly articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them, almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus work and was the first to state explicitly that Earth is a magnet. Published in 1600, Gilberts work, De Magnete, helped to establish magnetism as a science, in 1750, John Michell stated that magnetic poles attract and repel in accordance with an inverse square law. Charles-Augustin de Coulomb experimentally verified this in 1785 and stated explicitly that the north and south poles cannot be separated, building on this force between poles, Siméon Denis Poisson created the first successful model of the magnetic field, which he presented in 1824. In this model, a magnetic H-field is produced by magnetic poles, three discoveries challenged this foundation of magnetism, though. First, in 1819, Hans Christian Ørsted discovered that an electric current generates a magnetic field encircling it, then in 1820, André-Marie Ampère showed that parallel wires having currents in the same direction attract one another. Finally, Jean-Baptiste Biot and Félix Savart discovered the Biot–Savart law in 1820, extending these experiments, Ampère published his own successful model of magnetism in 1825. This has the benefit of explaining why magnetic charge can not be isolated. Also in this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism, in 1831, Michael Faraday discovered electromagnetic induction when he found that a changing magnetic field generates an encircling electric field
20.
Phonon
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In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, like solids and some liquids. Often designated a quasiparticle, it represents a state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles. Phonons play a role in many of the physical properties of condensed matter, like thermal conductivity. The study of phonons is an important part of condensed matter physics, the concept of phonons was introduced in 1932 by Soviet physicist Igor Tamm. The name phonon comes from the Greek word φωνή, which translates to sound or voice because long-wavelength phonons give rise to sound, shorter-wavelength higher-frequency phonons are responsible for the majority of the thermal capacity of solids. A phonon is a quantum description of an elementary vibrational motion in which a lattice of atoms or molecules uniformly oscillates at a single frequency. In classical mechanics this designates a normal mode of vibration, normal modes are important because any arbitrary lattice vibration can be considered to be a superposition of these elementary vibration modes. While normal modes are wave-like phenomena in classical mechanics, phonons have particle-like properties too, the equations in this section do not use axioms of quantum mechanics but instead use relations for which there exists a direct correspondence in classical mechanics. For example, a regular, crystalline, lattice is composed of N particles. These particles may be atoms or molecules, N is a large number, say of the order of 1023, or on the order of Avogadros number for a typical sample of a solid. Since the lattice is rigid, the atoms must be exerting forces on one another to keep each atom near its equilibrium position. These forces may be Van der Waals forces, covalent bonds, electrostatic attractions, magnetic and gravitational forces are generally negligible. The forces between each pair of atoms may be characterized by an energy function V that depends on the distance of separation of the atoms. The potential energy of the lattice is the sum of all pairwise potential energies, ∑ i ≠ j V where ri is the position of the ith atom. It is difficult to solve this many-body problem explicitly in either classical or quantum mechanics, in order to simplify the task, two important approximations are usually imposed. First, the sum is only performed over neighboring atoms, although the electric forces in real solids extend to infinity, this approximation is still valid because the fields produced by distant atoms are effectively screened. Secondly, the potentials V are treated as harmonic potentials and this is permissible as long as the atoms remain close to their equilibrium positions. Formally, this is accomplished by Taylor expanding V about its value to quadratic order, giving V proportional to the displacement x2
21.
Spectrum
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A spectrum is a condition that is not limited to a specific set of values but can vary, without steps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors in visible light after passing through a prism, as scientific understanding of light advanced, it came to apply to the entire electromagnetic spectrum. Spectrum has since been applied by analogy to topics outside of optics, thus, one might talk about the spectrum of political opinion, or the spectrum of activity of a drug, or the autism spectrum. In these uses, values within a spectrum may not be associated with precisely quantifiable numbers or definitions, such uses imply a broad range of conditions or behaviors grouped together and studied under a single title for ease of discussion. Nonscientific uses of the spectrum are sometimes misleading. For instance, a single left–right spectrum of opinion does not capture the full range of peoples political beliefs. Political scientists use a variety of biaxial and multiaxial systems to accurately characterize political opinion. In most modern usages of spectrum there is a theme between the extremes at either end. This was not always true in older usage, in Latin spectrum means image or apparition, including the meaning spectre. Spectral evidence is testimony about what was done by spectres of persons not present physically and it was used to convict a number of persons of witchcraft at Salem, Massachusetts in the late 17th century. The word spectrum was used to designate a ghostly optical afterimage by Goethe in his Theory of Colors and Schopenhauer in On Vision. The prefix spectro- is used to form words relating to spectra, for example, a spectrometer is a device used to record spectra and spectroscopy is the use of a spectrometer for chemical analysis. In the 17th century the word spectrum was introduced into optics by Isaac Newton, soon the term referred to a plot of light intensity or power as a function of frequency or wavelength, also known as a spectral density plot. The term spectrum was expanded to apply to other waves, such as sound waves that could also be measured as a function of frequency, frequency spectrum and power spectrum of a signal. The term now applies to any signal that can be measured or decomposed along a variable such as energy in electron spectroscopy or mass to charge ratio in mass spectrometry. Spectrum is also used to refer to a representation of the signal as a function of the dependent variable. Devices used to measure an electromagnetic spectrum are called spectrograph or spectrometer, the visible spectrum is the part of the electromagnetic spectrum that can be seen by the human eye. The wavelength of light ranges from 390 to 700 nm
22.
Visible spectrum
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The visible spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. Electromagnetic radiation in this range of wavelengths is called light or simply light. A typical human eye will respond to wavelengths from about 390 to 700 nm, in terms of frequency, this corresponds to a band in the vicinity of 430–770 THz. The spectrum does not, however, contain all the colors that the human eyes, unsaturated colors such as pink, or purple variations such as magenta, are absent, for example, because they can be made only by a mix of multiple wavelengths. Colors containing only one wavelength are called pure colors or spectral colors. Visible wavelengths pass through the window, the region of the electromagnetic spectrum that allows wavelengths to pass largely unattenuated through the Earths atmosphere. An example of this phenomenon is that clean air scatters blue light more than red wavelengths, the optical window is also referred to as the visible window because it overlaps the human visible response spectrum. The near infrared window lies just out of the vision, as well as the Medium Wavelength IR window. In the 13th century, Roger Bacon theorized that rainbows were produced by a process to the passage of light through glass or crystal. In the 17th century, Isaac Newton discovered that prisms could disassemble and reassemble white light and he was the first to use the word spectrum in this sense in print in 1671 in describing his experiments in optics. The result is red light is bent less sharply than violet as it passes through the prism. Newton divided the spectrum into seven named colors, red, orange, yellow, green, blue, indigo, the human eye is relatively insensitive to indigos frequencies, and some people who have otherwise-good vision cannot distinguish indigo from blue and violet. For this reason, some commentators, including Isaac Asimov, have suggested that indigo should not be regarded as a color in its own right. However, the evidence indicates that what Newton meant by indigo, comparing Newtons observation of prismatic colors to a color image of the visible light spectrum shows that indigo corresponds to what is today called blue, whereas blue corresponds to cyan. In the 18th century, Goethe wrote about optical spectra in his Theory of Colours, Goethe used the word spectrum to designate a ghostly optical afterimage, as did Schopenhauer in On Vision and Colors. Goethe argued that the spectrum was a compound phenomenon. Where Newton narrowed the beam of light to isolate the phenomenon, Goethe observed that a wider aperture produces not a spectrum but rather reddish-yellow, the spectrum appears only when these edges are close enough to overlap. Young was the first to measure the wavelengths of different colors of light, the connection between the visible spectrum and color vision was explored by Thomas Young and Hermann von Helmholtz in the early 19th century
23.
Electromagnetic spectrum
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The electromagnetic spectrum is the collective term for all known frequencies and their linked wavelengths of the known photons. The electromagnetic spectrum of an object has a different meaning, and is instead the characteristic distribution of radiation emitted or absorbed by that particular object. Visible light lies toward the end, with wavelengths from 400 to 700 nanometres. The limit for long wavelengths is the size of the universe itself, until the middle of the 20th century it was believed by most physicists that this spectrum was infinite and continuous. Nearly all types of radiation can be used for spectroscopy, to study. Other technological uses are described under electromagnetic radiation, for most of history, visible light was the only known part of the electromagnetic spectrum. The ancient Greeks recognized that light traveled in straight lines and studied some of its properties, the study of light continued, and during the 16th and 17th centuries conflicting theories regarded light as either a wave or a particle. The first discovery of electromagnetic radiation other than visible light came in 1800 and he was studying the temperature of different colors by moving a thermometer through light split by a prism. He noticed that the highest temperature was beyond red and he theorized that this temperature change was due to calorific rays that were a type of light ray that could not be seen. The next year, Johann Ritter, working at the end of the spectrum. These behaved similarly to visible light rays, but were beyond them in the spectrum. They were later renamed ultraviolet radiation, during the 1860s James Maxwell developed four partial differential equations for the electromagnetic field. Two of these equations predicted the possibility of, and behavior of, analyzing the speed of these theoretical waves, Maxwell realized that they must travel at a speed that was about the known speed of light. This startling coincidence in value led Maxwell to make the inference that light itself is a type of electromagnetic wave, maxwells equations predicted an infinite number of frequencies of electromagnetic waves, all traveling at the speed of light. This was the first indication of the existence of the electromagnetic spectrum. Maxwells predicted waves included waves at very low compared to infrared. Hertz found the waves and was able to infer that they traveled at the speed of light, Hertz also demonstrated that the new radiation could be both reflected and refracted by various dielectric media, in the same manner as light. For example, Hertz was able to focus the waves using a lens made of tree resin, in a later experiment, Hertz similarly produced and measured the properties of microwaves
24.
Harmonic series (music)
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A harmonic series is the sequence of sounds where the base frequency of each sound is an integer multiple of the lowest base frequency. Pitched musical instruments are based on an approximate harmonic oscillator such as a string or a column of air. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves, interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the spacing of the resonances, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest frequency. The musical timbre of a tone from such an instrument is determined by the relative strengths of each harmonic. A complex tone can be described as a combination of many simple periodic waves or partials, each with its own frequency of vibration, amplitude, a partial is any of the sine waves of which a complex tone is composed. A harmonic is any member of the series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The fundamental is also considered a harmonic because it is 1 times itself, a harmonic partial is any real partial component of a complex tone that matches an ideal harmonic. An inharmonic partial is any partial that does not match an ideal harmonic, Inharmonicity is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial. Unpitched, or indefinite-pitched instruments, such as cymbals, gongs, or tam-tams make sounds that are rich in inharmonic partials, an overtone is any partial except the lowest partial. The term overtone does not imply harmonicity or inharmonicity and has no special meaning other than to exclude the fundamental. It is the relative strengths of the different overtones that gives an instrument its particular timbre, some electronic instruments, such as synthesizers, can play a pure frequency with no overtones. Synthesizers can also combine pure frequencies into more complex tones, such as to other instruments. Certain flutes and ocarinas are very nearly without overtones, in most pitched musical instruments, the fundamental is accompanied by other, higher-frequency harmonics. Thus shorter-wavelength, higher-frequency waves occur with varying prominence and give each instrument its characteristic tone quality, the fact that a string is fixed at each end means that the longest allowed wavelength on the string is twice the length of the string. Other allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, theoretically, these shorter wavelengths correspond to vibrations at frequencies that are 2,3,4,5,6, etc. times the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator it vibrates against often alter these frequencies, the harmonic series is an arithmetic series. In terms of frequency, the difference between consecutive harmonics is therefore constant and equal to the fundamental, but because human ears respond to sound nonlinearly, higher harmonics are perceived as closer together than lower ones
25.
Molecular vibration
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A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. In general, a molecule with N atoms has 3N –6 normal modes of vibration, a diatomic molecule has one normal mode of vibration. A molecular vibration is excited when the molecule absorbs a quantum of energy, E, corresponding to the frequency, ν. A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state, when two quanta are absorbed the first overtone is excited, and so on to higher overtones. To a first approximation, the motion in a vibration can be described as a kind of simple harmonic motion. In this approximation, the energy is a quadratic function with respect to the atomic displacements. In reality, vibrations are anharmonic and the first overtone has a frequency that is lower than twice that of the fundamental. The vibrational states of a molecule can be probed in a variety of ways, the most direct way is through infrared spectroscopy, as vibrational transitions typically require an amount of energy that corresponds to the infrared region of the spectrum. Raman spectroscopy, which uses visible light, can also be used to measure vibration frequencies directly. The two techniques are complementary and comparison between the two can provide structural information such as in the case of the rule of mutual exclusion for centrosymmetric molecules. Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region, the combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state. When the vibration is excited the coordinate changes sinusoidally with a frequency ν, out-of-plane, a change in the angle between any one of the C-H bonds and the plane defined by the remaining atoms of the ethylene molecule. Another example is in BF3 when the boron atom moves in, in a rocking, wagging or twisting coordinate the bond lengths within the groups involved do not change. Rocking is distinguished from wagging by the fact that the atoms in the stay in the same plane. In ethene there are 12 internal coordinates,4 C-H stretching,1 C-C stretching,2 H-C-H bending,2 CH2 rocking,2 CH2 wagging,1 twisting. Note that the H-C-C angles cannot be used as internal coordinates as the angles at each atom cannot all increase at the same time. The atoms in a CH2 group, commonly found in compounds, can vibrate in six different ways. Symmetry-adapted coordinates may be created by applying a projection operator to a set of internal coordinates, the projection operator is constructed with the aid of the character table of the molecular point group
26.
Dispersion relation
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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. Dispersion may be caused either by geometric boundary conditions or by interaction of the waves with the transmitting medium, elementary particles, considered as matter waves, have a nontrivial dispersion relation even in the absence of geometric constraints and other media. In the presence of dispersion, wave velocity is no uniquely defined, giving rise to the distinction of phase velocity. Dispersion occurs when pure plane waves of different wavelengths have different propagation velocities, the speed of a plane wave, v, is a function of the waves wavelength λ, v = v. The waves speed, wavelength, and frequency, f, are related by the identity v = λ f, the function f expresses the dispersion relation of the given medium. Dispersion relations are more commonly expressed in terms of the angular frequency ω =2 π f, rewriting the relation above in these variables gives ω = v k. where we now view f as a function of k. The use of ω to describe the relation has become standard because both the phase velocity ω/k and the group velocity dω/dk have convenient representations via this function. Plane waves in vacuum are the simplest case of propagation, no geometric constraint. For electromagnetic waves in vacuum, the frequency is proportional to the wavenumber. This is a dispersion relation. In this case, the velocity and the group velocity are the same, v = ω k = d ω d k = c, they are given by c, the speed of light in vacuum. In the nonrelativistic limit, m c 2 is a constant and p 2 /2 m is the familiar kinetic energy expressed in terms of the momentum p = m v. According to the de Broglie relations, their kinetic energy E can be expressed as a frequency ω, accordingly, angular frequency and wavenumber are connected through a dispersion relation, which in the nonrelativistic limit reads ω = ℏ k 22 m. For particles, this translates to a knowledge of energy as a function of momentum, the name dispersion relation originally comes from optics. In this case, the waveform will spread over time, such that a pulse will become an extended pulse. In these materials, ∂ ω ∂ k is known as the velocity and corresponds to the speed at which the peak of the pulse propagates. The dispersion relation for water waves is often written as ω = g k
27.
Vacuum
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Vacuum is space void 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. In engineering and applied physics on the 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, the quality of a partial vacuum refers to how closely it approaches a perfect vacuum. Other things equal, lower gas pressure means higher-quality vacuum, for example, a typical vacuum cleaner produces enough suction to reduce air pressure by around 20%. Ultra-high vacuum chambers, common in chemistry, physics, and engineering, operate below one trillionth of atmospheric pressure, outer space is an even higher-quality vacuum, with the equivalent of just a few hydrogen atoms per cubic meter on average. In the electromagnetism in the 19th century, vacuum was thought to be filled with a medium called aether, in modern particle physics, the vacuum state is considered the ground state of matter. Vacuum has been a frequent topic of debate since ancient Greek times. Evangelista Torricelli produced the first laboratory vacuum in 1643, and other techniques were developed as a result of his theories of atmospheric pressure. A torricellian vacuum is created by filling a glass container closed at one end with mercury. Vacuum became an industrial tool in the 20th century with the introduction of incandescent light bulbs and vacuum tubes. The recent development of human spaceflight has raised interest in the impact of vacuum on human health, the word vacuum comes from Latin an empty space, void, noun use of neuter of vacuus, meaning empty, related to vacare, meaning be empty. Vacuum is one of the few words in the English language that contains two consecutive letters u. Historically, there has been dispute over whether such a thing as a vacuum can exist. Ancient Greek philosophers debated the existence of a vacuum, or void, in the context of atomism, Aristotle believed that no void could occur naturally, because the denser surrounding material continuum would immediately fill any incipient rarity that might give rise to a void. Almost two thousand years after Plato, René Descartes also proposed a geometrically based alternative theory of atomism, without the problematic nothing–everything dichotomy of void, by the ancient definition however, directional information and magnitude were conceptually distinct. The explanation of a clepsydra or water clock was a topic in the Middle Ages. Although a simple wine skin sufficed to demonstrate a partial vacuum, in principle and he concluded that airs volume can expand to fill available space, and he suggested that the concept of perfect vacuum was incoherent. However, according to Nader El-Bizri, the physicist Ibn al-Haytham and the Mutazili theologians disagreed with Aristotle and Al-Farabi, using geometry, Ibn al-Haytham mathematically demonstrated that place is the imagined three-dimensional void between the inner surfaces of a containing body
28.
Speed of light
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The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299792458 metres per second, it is exact because the unit of length, the metre, is defined from this constant, according to special relativity, c is the maximum speed at which all matter and hence information in the universe can travel. It is the speed at which all particles and changes of the associated fields travel in vacuum. Such particles and waves travel at c regardless of the motion of the source or the reference frame of the observer. In the theory of relativity, c interrelates space and time, the speed at which light propagates through transparent materials, such as glass or air, is less than c, similarly, the speed of radio waves in wire cables is slower than c. The ratio between c and the speed v at which light travels in a material is called the index n of the material. In communicating with distant space probes, it can take minutes to hours for a message to get from Earth to the spacecraft, the light seen from stars left them many years ago, allowing the study of the history of the universe by looking at distant objects. The finite speed of light limits the theoretical maximum speed of computers. The speed of light can be used time of flight measurements to measure large distances to high precision. Ole Rømer first demonstrated in 1676 that light travels at a speed by studying the apparent motion of Jupiters moon Io. In 1865, James Clerk Maxwell proposed that light was an electromagnetic wave, in 1905, Albert Einstein postulated that the speed of light c with respect to any inertial frame is a constant and is independent of the motion of the light source. He explored the consequences of that postulate by deriving the theory of relativity and in doing so showed that the parameter c had relevance outside of the context of light and electromagnetism. After centuries of increasingly precise measurements, in 1975 the speed of light was known to be 299792458 m/s with a measurement uncertainty of 4 parts per billion. In 1983, the metre was redefined in the International System of Units as the distance travelled by light in vacuum in 1/299792458 of a second, as a result, the numerical value of c in metres per second is now fixed exactly by the definition of the metre. The speed of light in vacuum is usually denoted by a lowercase c, historically, the symbol V was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used c for a different constant later shown to equal √2 times the speed of light in vacuum, in 1894, Paul Drude redefined c with its modern meaning. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to c, sometimes c is used for the speed of waves in any material medium, and c0 for the speed of light in vacuum. This article uses c exclusively for the speed of light in vacuum, since 1983, the metre has been defined in the International System of Units as the distance light travels in vacuum in 1⁄299792458 of a second
29.
Red
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Red is the color at the longer-wavelengths end of the spectrum of visible light next to orange, at the opposite end from violet. Red color has a predominant light wavelength of roughly 620–740 nanometers, light with a longer wavelength than red but shorter than terahertz radiation and microwave is called infrared. Red is one of the secondary colors, resulting from the combination of yellow. Traditionally, it was viewed as a primary colour, along with yellow and blue, in the RYB color space and traditional color wheel formerly used by painters. Reds can vary in shade from light pink to very dark maroon or burgundy. Red is the color of cyan. In nature, the red color of blood comes from hemoglobin, the red color of the Grand Canyon and other geological features is caused by hematite or red ochre, both forms of iron oxide. It also causes the red color of the planet Mars, the color of autumn leaves is caused by pigments called anthocyanins, which are produced towards the end of summer, when the green chlorophyll is no longer produced. One to two percent of the population has red hair, the color is produced by high levels of the reddish pigment pheomelanin. Since red is the color of blood, it has historically been associated with sacrifice, danger, modern surveys in the United States and Europe show red is also the color most commonly associated with heat, activity, passion, sexuality, anger, love and joy. In China, India and many other Asian countries it is the color of symbolizing happiness, since the 19th century, red has also been associated with socialism and communism. The word red is derived from the Old English rēad, the word can be further traced to the Proto-Germanic rauthaz and the Proto-Indo European root rewdʰ-. In Sanskrit, the word means red or blood. In the Akkadian language of Ancient Mesopotamia and in the modern Inuit language of Inuit, the words for colored in Latin and Spanish both also mean red. In Portuguese the word for red is vermelho, which comes from Latin vermiculus, in the Russian language, the word for red, Кра́сный, comes from the same old Slavic root as the words for beautiful—красивый and excellent—прекрасный. Thus Red Square in Moscow, named long before the Russian Revolution, in heraldry, the word gules is used for red. Red can vary in hue from orange-red to violet-red, and for each hue there is a variety of shades and tints. Red hematite powder was found scattered around the remains at a grave site in a Zhoukoudian cave complex near Beijing
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Violet (color)
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Violet is the color of amethyst, lavender and beautyberries. It takes its name from the violet flower, Violet is the color at the end of the visible spectrum of light between blue and the invisible ultraviolet. Violet color has a predominant light wavelength of roughly 380-450 nanometers, light with a shorter wavelength than violet but longer than X-rays and gamma rays is called ultraviolet. In the color wheel used by painters, it is located between blue and purple. This is not true violet, since it is composed of multiple longer wavelengths rather than a wavelength shorter than that of blue light. Violet and purple look very similar, but violet is a color, with its own set of wavelengths on the spectrum of visible light, while purple is a composite color, made by combining blue. In history, violet and purple have long associated with royalty and majesty. The emperors of Rome wore purple togas, as did the Byzantine emperors, during the Middle Ages violet was worn by bishops and university professors and was often used in art as the color of the robes of the Virgin Mary. According to surveys in Europe and the United States, violet is the people most often associate with extravagance and individualism, the unconventional, the artificial. In Chinese painting, the color represents the harmony of the universe because it is a combination of red. In Hinduism and Buddhism violet is associated with the Crown Chakra, from the Middle English and old French violette, and from the Latin viola, the names of the violet flower. The first recorded use of violet as a name in English was in 1370. Violet can also refer to the first violas which were painted a similar color. In the traditional color used by painters, violet and purple are both placed between red and blue. Purple occupies the space closer to red, between crimson and violet, Violet is closer to blue, and usually less intense and bright than purple. The range of purples is created by combining blue and red light of any intensities, Violet is one of the oldest colors used by man. It has also found in the cave of Altamira and Lascaux. It was sometimes used an alternative to black charcoal, sticks of manganese, used for drawing, have been found at sites occupied by Neanderthal man in France and Israel
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Speed of sound
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The speed of sound is the distance travelled per unit time by a sound wave as it propagates through an elastic medium. In dry air at 20 °C, the speed of sound is 343 metres per second, the speed of sound in an ideal gas depends only on its temperature and composition. The speed has a dependence on frequency and pressure in ordinary air. In common everyday speech, speed of sound refers to the speed of waves in air. However, the speed of sound varies from substance to substance, sound travels most slowly in gases, it travels faster in liquids, and faster still in solids. For example, sound travels at 343 m/s in air, it travels at 1,484 m/s in water, in an exceptionally stiff material such as diamond, sound travels at 12,000 m/s, which is around the maximum speed that sound will travel under normal conditions. Sound waves in solids are composed of waves, and a different type of sound wave called a shear wave. Shear waves in solids usually travel at different speeds, as exhibited in seismology, the speed of compression waves in solids is determined by the mediums compressibility, shear modulus and density. The speed of waves is determined only by the solid materials shear modulus. In fluid dynamics, the speed of sound in a medium is used as a relative measure for the speed of an object moving through the medium. The ratio of the speed of an object to the speed of sound in the fluid is called the objects Mach number, objects moving at speeds greater than Mach1 are said to be traveling at supersonic speeds. During the 17th century, there were attempts to measure the speed of sound accurately, including attempts by Marin Mersenne in 1630, Pierre Gassendi in 1635. In 1709, the Reverend William Derham, Rector of Upminster, published an accurate measure of the speed of sound. Measurements were made of gunshots from a number of local landmarks, the distance was known by triangulation, and thus the speed that the sound had travelled was calculated. The transmission of sound can be illustrated by using a model consisting of an array of balls interconnected by springs, for real material the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighbouring balls, which transmit energy to their springs, the speed of sound through the model depends on the stiffness of the springs, and the mass of the balls. As long as the spacing of the balls remains constant, stiffer springs transmit energy more quickly, effects like dispersion and reflection can also be understood using this model. In a real material, the stiffness of the springs is called the modulus
32.
Hertz
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The hertz is the unit of frequency in the International System of Units and is defined as one cycle per second. It is named for Heinrich Rudolf Hertz, the first person to provide proof of the existence of electromagnetic waves. Hertz are commonly expressed in SI multiples kilohertz, megahertz, gigahertz, kilo means thousand, mega meaning million, giga meaning billion and tera for trillion. Some of the units most common uses are in the description of waves and musical tones, particularly those used in radio-. It is also used to describe the speeds at which computers, the hertz is equivalent to cycles per second, i. e. 1/second or s −1. In English, hertz is also used as the plural form, as an SI unit, Hz can be prefixed, commonly used multiples are kHz, MHz, GHz and THz. One hertz simply means one cycle per second,100 Hz means one hundred cycles per second, and so on. The unit may be applied to any periodic event—for example, a clock might be said to tick at 1 Hz, the rate of aperiodic or stochastic events occur is expressed in reciprocal second or inverse second in general or, the specific case of radioactive decay, becquerels. Whereas 1 Hz is 1 cycle per second,1 Bq is 1 aperiodic radionuclide event per second, the conversion between a frequency f measured in hertz and an angular velocity ω measured in radians per second is ω =2 π f and f = ω2 π. This SI unit is named after Heinrich Hertz, as with every International System of Units unit named for a person, the first letter of its symbol is upper case. Note that degree Celsius conforms to this rule because the d is lowercase. — Based on The International System of Units, the hertz is named after the German physicist Heinrich Hertz, who made important scientific contributions to the study of electromagnetism. The name was established by the International Electrotechnical Commission in 1930, the term cycles per second was largely replaced by hertz by the 1970s. One hobby magazine, Electronics Illustrated, declared their intention to stick with the traditional kc. Mc. etc. units, sound is a traveling longitudinal wave which is an oscillation of pressure. Humans perceive frequency of waves as pitch. Each musical note corresponds to a frequency which can be measured in hertz. An infants ear is able to perceive frequencies ranging from 20 Hz to 20,000 Hz, the range of ultrasound, infrasound and other physical vibrations such as molecular and atomic vibrations extends from a few femtoHz into the terahertz range and beyond. Electromagnetic radiation is described by its frequency—the number of oscillations of the perpendicular electric and magnetic fields per second—expressed in hertz. Radio frequency radiation is measured in kilohertz, megahertz, or gigahertz
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Metre
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The metre or meter, is the base unit of length in the International System of Units. The metre is defined as the length of the path travelled by light in a vacuum in 1/299792458 seconds, the metre was originally defined in 1793 as one ten-millionth of the distance from the equator to the North Pole. In 1799, it was redefined in terms of a metre bar. In 1960, the metre was redefined in terms of a number of wavelengths of a certain emission line of krypton-86. In 1983, the current definition was adopted, the imperial inch is defined as 0.0254 metres. One metre is about 3 3⁄8 inches longer than a yard, Metre is the standard spelling of the metric unit for length in nearly all English-speaking nations except the United States and the Philippines, which use meter. Measuring devices are spelled -meter in all variants of English, the suffix -meter has the same Greek origin as the unit of length. This range of uses is found in Latin, French, English. Thus calls for measurement and moderation. In 1668 the English cleric and philosopher John Wilkins proposed in an essay a decimal-based unit of length, as a result of the French Revolution, the French Academy of Sciences charged a commission with determining a single scale for all measures. In 1668, Wilkins proposed using Christopher Wrens suggestion of defining the metre using a pendulum with a length which produced a half-period of one second, christiaan Huygens had observed that length to be 38 Rijnland inches or 39.26 English inches. This is the equivalent of what is now known to be 997 mm, no official action was taken regarding this suggestion. In the 18th century, there were two approaches to the definition of the unit of length. One favoured Wilkins approach, to define the metre in terms of the length of a pendulum which produced a half-period of one second. The other approach was to define the metre as one ten-millionth of the length of a quadrant along the Earths meridian, that is, the distance from the Equator to the North Pole. This means that the quadrant would have defined as exactly 10000000 metres at that time. To establish a universally accepted foundation for the definition of the metre, more measurements of this meridian were needed. This portion of the meridian, assumed to be the length as the Paris meridian, was to serve as the basis for the length of the half meridian connecting the North Pole with the Equator
34.
Node (physics)
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A node is a point along a standing wave where the wave has minimum amplitude. For instance, in a guitar string, the ends of the string are nodes. By changing the position of the end node through frets, the guitarist changes the length of the vibrating string. The opposite of a node is an anti-node, a point where the amplitude of the wave is a maximum. These occur midway between the nodes, standing waves result when two sinusoidal wave trains of the same frequency are moving in opposite directions in the same space and interfere with each other. In a standing wave the nodes are a series of locations at equally spaced intervals where the amplitude is zero. At these points the two waves add with opposite phase and cancel each other out and they occur at intervals of half a wavelength. Midway between each pair of nodes are locations where the amplitude is maximum, at these points the two waves add with the same phase and reinforce each other. In cases where the two opposite wave trains are not the amplitude, they do not cancel perfectly, so the amplitude of the standing wave at the nodes is not zero. This occurs when the reflection at the boundary is imperfect and this is indicated by a finite standing wave ratio, the ratio of the amplitude of the wave at the antinode to the amplitude at the node. These can be visible by sprinkling sand on the surface. In transmission lines a voltage node is a current antinode, in this type the derivative of the waves amplitude is forced to zero at the boundary. So there is a maximum at the boundary, the first node occurs a quarter wavelength from the end. A sound wave consists of alternating cycles of compression and expansion of the wave medium, during compression, the molecules of the medium are forced together, resulting in the increased pressure and density. During expansion the molecules are forced apart, resulting in the decreased pressure, the number of nodes in a specified length is directly proportional to the frequency of the wave. Occasionally on a guitar, violin, or other stringed instrument, when the finger is placed on top of the string at a certain point, but does not push the string all the way down to the fretboard, a third node is created and a harmonic is sounded. During normal play when the frets are used, the harmonics are always present, with the artificial node method, the overtone is louder and the fundamental tone is quieter. If the finger is placed at the midpoint of the string, the first overtone is heard, when two additional nodes divide the string into thirds, this creates an octave and a perfect fifth
35.
Boundary value problem
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In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them, problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems, the analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed and this means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of differential equations is devoted to proving that boundary value problems arising from scientific. Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions, boundary value problems are similar to initial value problems. Finding the temperature at all points of a bar with one end kept at absolute zero. If the problem is dependent on both space and time, one could specify the value of the problem at a point for all time or at a given time for all space. Concretely, an example of a value is the problem y ″ + y =0 to be solved for the unknown function y with the boundary conditions y =0, y =2. Without the boundary conditions, the solution to this equation is y = A sin + B cos . From the boundary condition y =0 one obtains 0 = A ⋅0 + B ⋅1 which implies that B =0, from the boundary condition y =2 one finds 2 = A ⋅1 and so A =2. One sees that imposing boundary conditions allowed one to determine a unique solution, a boundary condition which specifies the value of the function itself is a Dirichlet boundary condition, or first-type boundary condition. For example, if one end of a rod is held at absolute zero. A boundary condition which specifies the value of the derivative of the function is a Neumann boundary condition. For example, if there is a heater at one end of a rod, then energy would be added at a constant rate. If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a Cauchy boundary condition. Summary of boundary conditions for the function, y, constants c 0 and c 1 specified by the boundary conditions
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Amplitude
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The amplitude of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the extreme values. In older texts the phase is called the amplitude. Peak-to-peak amplitude is the change between peak and trough, with appropriate circuitry, peak-to-peak amplitudes of electric oscillations can be measured by meters or by viewing the waveform on an oscilloscope. Peak-to-peak is a measurement on an oscilloscope, the peaks of the waveform being easily identified and measured against the graticule. This remains a common way of specifying amplitude, but sometimes other measures of amplitude are more appropriate. In audio system measurements, telecommunications and other areas where the measurand is a signal that swings above and below a value but is not sinusoidal. If the reference is zero, this is the absolute value of the signal, if the reference is a mean value. Semi-amplitude means half the peak-to-peak amplitude, some scientists use amplitude or peak amplitude to mean semi-amplitude, that is, half the peak-to-peak amplitude. It is the most widely used measure of orbital wobble in astronomy, the RMS of the AC waveform. For complicated waveforms, especially non-repeating signals like noise, the RMS amplitude is used because it is both unambiguous and has physical significance. For example, the power transmitted by an acoustic or electromagnetic wave or by an electrical signal is proportional to the square of the RMS amplitude. For alternating current electric power, the practice is to specify RMS values of a sinusoidal waveform. One property of root mean square voltages and currents is that they produce the same heating effect as direct current in a given resistance, the peak-to-peak value is used, for example, when choosing rectifiers for power supplies, or when estimating the maximum voltage that insulation must withstand. Some common voltmeters are calibrated for RMS amplitude, but respond to the value of a rectified waveform. Many digital voltmeters and all moving coil meters are in this category, the RMS calibration is only correct for a sine wave input since the ratio between peak, average and RMS values is dependent on waveform. If the wave shape being measured is greatly different from a sine wave, true RMS-responding meters were used in radio frequency measurements, where instruments measured the heating effect in a resistor to measure current. The advent of microprocessor controlled meters capable of calculating RMS by sampling the waveform has made true RMS measurement commonplace
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Wave vector
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In physics, a wave vector is a vector which helps describe a wave. In the context of relativity the wave vector can also be defined as a four-vector. There are two definitions of wave vector, which differ by a factor of 2π in their magnitudes. One definition is preferred in physics and related fields, while the definition is preferred in crystallography. For this article, they will be called the physics definition, a perfect one-dimensional traveling wave follows the equation, ψ = A cos where, x is position, t is time, ψ is the disturbance describing the wave. This wave travels in the +x direction with speed ω / k, in crystallography, the same waves are described using slightly different equations. In one and three respectively, ψ = A cos ψ = A cos The differences are. They are related by 2 π ν = ω and this substitution is not important for this article, but reflects common practice in crystallography. The wavenumber k and wave vector k are defined in a different way, here, k = | k | =1 / λ, while in the physics definition above, k = | k | =2 π / λ. The direction of k is discussed below, the direction in which the wave vector points must be distinguished from the direction of wave propagation. The direction of propagation is the direction of a waves energy flow. For light waves, this is also the direction of the Poynting vector, on the other hand, the wave vector points in the direction of phase velocity. In other words, the vector points in the normal direction to the surfaces of constant phase. In a lossless isotropic medium such as air, any gas, any liquid, or some solids, if the medium is lossy, the wave vector in general points in directions other than that of wave propagation. The condition for wave vector to point in the direction in which the wave propagates is that the wave has to be homogeneous. In a homogeneous wave, the surfaces of constant phase are also surfaces of constant amplitude, in case of inhomogeneous waves, these two species of surfaces differ in orientation. Wave vector is perpendicular to surfaces of constant phase. In solid-state physics, the wavevector of an electron or hole in a crystal is the wavevector of its quantum-mechanical wavefunction, see Bloch wave for further details
38.
Plane wave
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In the physics of wave propagation, a plane wave is a wave whose wavefronts are infinite parallel planes. The solutions in x → of n → c ⋅ x → − t = c o n s t. comprise the plane with normal vector n →, thus, the points of equal field value of A always form a plane in space. This plane then shifts with time t, along the direction of propagation n → with velocity c, the term is often used to mean the special case of a monochromatic, homogeneous plane wave. A monochromatic plane wave is one in which the amplitude is a function of x and t. A homogeneous plane wave is one in which the planes of constant phase are perpendicular to the direction of propagation n →, however, many waves are approximately plane waves in a localized region of space. For example, a source such as an antenna produces a field that is approximately a plane wave far from the antenna in its far-field region. Two functions that meet the criteria of having a constant frequency. One of the simplest ways to use such a sinusoid involves defining it along the direction of the x-axis, the equation below, which is illustrated toward the right, uses the cosine function to represent a harmonic and homogeneous plane wave travelling in the positive x direction. A = A o cos In the above equation, A is the magnitude or disturbance of the wave at a point in space. An example would be to let A represent the variation of air pressure relative to the norm in the case of a sound wave, a o is the amplitude of the wave which is the peak magnitude of the oscillation. K is the wave number or more specifically the angular wave number and equals 2π/λ. K has the units of radians per unit distance and is a measure of how rapidly the disturbance changes over a distance at a particular point in time. X is a point along the x-axis, Y and z are not part of the equation because the waves magnitude and phase are the same at every point on any given y-z plane. This equation defines what that magnitude and phase are, ω is the wave’s angular frequency which equals 2π/T, where T is the period of the wave. ω has the units of radians per unit time and is a measure of how rapidly the disturbance changes over a length of time at a particular point in space. T is a point in time φ is the phase shift of the wave and has the units of radians. Note that a phase shift, at a given moment of time. A phase shift of 2π radians shifts it exactly one wavelength, other formalisms which directly use the wave’s wavelength λ, period T, frequency f and velocity c are below
39.
Three-dimensional space
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Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
40.
Trigonometric functions
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In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
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