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Sine Wave
A SINE WAVE or SINUSOID is a mathematical curve that describes a smooth repetitive oscillation . A sine wave is a continuous wave . It is named after the function 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 (_t_) is: y ( t ) = A sin ( 2 f t + ) = A sin ( t + ) {displaystyle y(t)=Asin(2pi ft+varphi )=Asin(omega t+varphi )} where: * _A_ = the _amplitude _, the peak deviation of the function from zero. * _f_ = the _ordinary frequency _, the _number _ of oscillations (cycles) that occur each second of time. * _ω_ = 2π_f_, the _angular frequency _, the rate of change of the function argument in units of radians per second* _ {displaystyle varphi } _ = the _phase _, specifies (in radians) where in its cycle the oscillation is at _t_ = 0. * When _ {displaystyle varphi } _ is non-zero, the entire waveform appears to be shifted in time by the amount _ {displaystyle varphi } _/_ω_ seconds
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Sinusoid (blood Vessel)
Blood vessels are the tubes which are two types arteries and veins. A SINUSOID is a small blood vessel that is a type of capillary similar to a fenestrated endothelium . Sinusoids are actually classified as a type of open pore capillary (or discontinuous) as opposed to continuous and fenestrated types. Fenestrated capillaries have diaphragms that cover the pores whereas open pore capillaries lack a diaphragm and just have an open pore. The open pores of endothelial cells greatly increase their permeability . In addition, permeability is increased by large inter-cellular clefts and fewer tight junctions. The level of permeability is such as to allow small and medium-sized proteins such as albumin to readily enter and leave the blood stream. Sinusoids are found in the liver , lymphoid tissue , endocrine organs , and hematopoietic organs such as the bone marrow and the spleen . Sinusoids found within terminal villi of the placenta are not comparable to these because they possess a continuous endothelium and complete basal lamina. This word was first used in 1893
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Curve
In mathematics , a CURVE (also called a CURVED LINE in older texts) is, generally speaking, an object similar to a line but that need not be straight . Thus, a curve is a generalization of a line, in that curvature is not necessarily zero. Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola , shown to the right. A large number of other curves have been studied in multiple mathematical fields. A CLOSED CURVE is a curve that forms a path whose starting point is also its ending point—that is, a path from any of its points to the same point. Closely related meanings include the graph of a function (as in Phillips curve ) and a two-dimensional graph
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Oscillation
OSCILLATION is the repetitive variation, typically in time , of some measure about a central value (often a point of equilibrium ) or between two or more different states. The term vibration is precisely used to describe mechanical oscillation. Familiar examples of oscillation include a swinging pendulum and alternating current power. Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example the beating human heart , business cycles in economics , predator–prey population cycles in ecology , geothermal geysers in geology , vibrating strings in musical instruments , periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy . CONTENTS * 1 Simple harmonic oscillator * 2 Damped and driven oscillations * 3 Coupled oscillations * 4 Continuous systems – waves * 5 Mathematics * 6 Examples * 6.1 Mechanical * 6.2 Electrical * 6.3 Electro-mechanical * 6.4 Optical * 6.5 Biological * 6.6 Human * 6.7 Economic and social * 6.8 Climate and geophysics * 6.9 Astrophysics * 6.10 Quantum mechanical * 6.11 Chemical * 6.12 Computing * 7 See also * 8 References * 9 External links SIMPLE HARMONIC OSCILLATOR Main article: Simple harmonic motion The simplest mechanical oscillating system is a weight attached to a linear spring subject to only weight and tension
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Continuous Wave
A CONTINUOUS WAVE or CONTINUOUS WAVEFORM (CW) is an electromagnetic wave of constant amplitude and frequency ; almost always a sine wave , that for mathematical analysis is considered to be of infinite duration. Continuous wave
Continuous wave
is also the name given to an early method of radio transmission , in which a sinusoidal carrier wave is switched on and off. Information is carried in the varying duration of the on and off periods of the signal, for example by Morse code in early radio. In early wireless telegraphy radio transmission, CW waves were also known as "undamped waves", to distinguish this method from damped wave signals produced by earlier spark gap type transmitters. CONTENTS* 1 Radio
Radio
* 1.1 Transmissions before CW * 1.2 Transition to CW * 1.3 Key clicks * 1.4 Persistence of radio telegraphy * 2 Radar * 3 Laser physics * 4 See also * 5 References RADIOTRANSMISSIONS BEFORE CWVery early radio transmitters used a spark gap to produce radio-frequency oscillations in the transmitting antenna. The signals produced by these spark-gap transmitters consisted of strings of brief pulses of sinusoidal radio frequency oscillations which died out rapidly to zero, called damped waves . The disadvantage of damped waves was that their energy was spread over an extremely wide band of frequencies ; they had wide bandwidth
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Sine
In mathematics , the SINE is a trigonometric function of an angle . The sine of an acute angle is defined in the context of a right triangle : for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse ). More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle . More modern definitions express the sine as an infinite series or as the solution of certain differential equations , allowing their extension to arbitrary positive and negative values and even to complex numbers . The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. The function sine can be traced to the _jyā_ and _koṭi-jyā_ functions used in Gupta period Indian astronomy (_ Aryabhatiya _, _ Surya Siddhanta _), via translation from Sanskrit to Arabic and then from Arabic to Latin. The word "sine" comes from a Latin mistranslation of the Arabic _jiba_, which is a transliteration of the Sanskrit word for half the chord, _jya-ardha_
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Graph Of A Function
In mathematics, the GRAPH of a function _f_ is the collection of all ordered pairs (_x_, _f_(_x_)). If the function input _x_ is a scalar , the graph is a two-dimensional graph , and for a continuous function is a curve . If the function input _x_ is an ordered pair (_x_1, _x_2) of real numbers, the graph is the collection of all ordered triples (_x_1, _x_2, _f_(_x_1, _x_2)), and for a continuous function is a surface . Informally, if _x_ is a real number and _f_ is a real function , _graph_ may mean the graphical representation of this collection, in the form of a line chart : a curve on a Cartesian plane , together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as _curve sketching_. The graph of a function on real numbers may be mapped directly to the graphic representation of the function. For general functions, a graphic representation cannot necessarily be found and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis . The concept of the graph of a function is generalized to the graph of a relation . Note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers but it is not if its codomain is the complex field
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Mathematics
MATHEMATICS (from Greek μάθημα _máthēma_, “knowledge, study, learning”) is the study of topics such as quantity (numbers ), structure , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics . Mathematicians seek out patterns and use them to formulate new conjectures . Mathematicians resolve the truth or falsity of conjectures by mathematical proof . When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic , mathematics developed from counting , calculation , measurement , and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics , most notably in Euclid 's _Elements _. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century , it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions
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Physics
PHYSICS (from Ancient Greek : φυσική (ἐπιστήμη) _phusikḗ (epistḗmē)_ "knowledge of nature", from φύσις _phúsis_ "nature" ) 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 scientific 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 . Over the last two millennia, physics was a part of natural philosophy along with chemistry , biology , and certain branches of mathematics , but during the scientific revolution in the 17th century, the natural sciences emerged as unique research programs in their own right. 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 fundamental mechanisms of other sciences while opening new avenues of research in areas such as mathematics and philosophy . Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs
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Engineering
ENGINEERING is the application of mathematics , as well as scientific , economic , social, and practical knowledge to invent , innovate , design , build, maintain , research , and improve structures , machines , tools , systems , components , materials , processes , solutions, and organizations . The discipline of engineering is extremely broad and encompasses a range of more specialized fields of engineering , each with a more specific emphasis on particular areas of applied science, technology and types of application. The term _Engineering_ is derived from the Latin _ingenium_, meaning "cleverness" and _ingeniare_, meaning "to contrive, devise"
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Signal Processing
SIGNAL PROCESSING concerns the analysis, synthesis, and modification of signals , which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound , images , and biological measurements. For example, signal processing techniques are used to improve signal transmission fidelity, storage efficiency, and subjective quality, and to emphasize or detect components of interest in a measured signal. CONTENTS * 1 History * 2 Application fields * 3 Typical devices * 4 Mathematical methods applied * 5 Categories * 5.1 Analog signal processing * 5.2 Continuous-time signal processing * 5.3 Discrete-time signal processing * 5.4 Digital signal processing * 5.5 Nonlinear signal processing * 6 See also * 7 Notes and references * 8 External links HISTORYAccording to Alan V. Oppenheim and Ronald W. Schafer , the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. Oppenheim and Schafer further state that the "digitalization" or digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s. APPLICATION FIELDS _ This article IS IN A LIST FORMAT THAT MAY BE BETTER PRESENTED USING PROSE . You can help by converting this article to prose, if appropriate . Editing help is available
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Amplitude
The AMPLITUDE of a periodic variable is a measure of its change over a single period (such as time or spatial period ). There are various definitions of amplitude (see below), which are all functions of the magnitude of the difference between the variable's extreme values . In older texts the phase is sometimes called the amplitude. CONTENTS* 1 Definitions * 1.1 Peak-to-peak amplitude * 1.2 Peak amplitude * 1.3 Semi-amplitude * 1.4 Root mean square amplitude * 1.5 Ambiguity * 1.6 Pulse amplitude * 2 Formal representation * 3 Units * 4 Waveform and envelope * 5 See also * 6 Notes DEFINITIONS A sinusoidal curve * Peak amplitude ( U {displaystyle scriptstyle {hat {U}}} ), * Peak-to-peak amplitude ( 2 U {displaystyle scriptstyle 2{hat {U}}} ), * Root mean square amplitude ( U / 2 {displaystyle scriptstyle {hat {U}}/{sqrt {2}}} ), * Wave period (not an amplitude)PEAK-TO-PEAK AMPLITUDE "Peak to peak" redirects here. For the school, see Peak to Peak Charter School . Peak-to-peak amplitude is the change between peak (highest amplitude value) and trough (lowest amplitude value, which can be negative). With appropriate circuitry, peak-to-peak amplitudes of electric oscillations can be measured by meters or by viewing the waveform on an oscilloscope
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Frequency
FREQUENCY is the number of occurrences of a repeating event per unit time . It is also referred to as TEMPORAL FREQUENCY, which emphasizes the contrast to spatial frequency and angular frequency . The PERIOD is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example, if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second (that is, 60 seconds divided by 120 beats ). Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio (sound ) signals, radio waves , and light . CONTENTS * 1 Definitions * 2 Units * 3 Period versus frequency * 4 Related types of frequency * 5 In wave propagation * 6 Measurement * 6.1 Counting * 6.2 Stroboscope * 6.3 Frequency counter * 6.4 Heterodyne methods * 7 Examples * 7.1 Light * 7.2 Sound * 7.3 Line current * 8 See also * 9 Notes and references * 10 Further reading * 11 External links DEFINITIONS As time elapses—here moving left to right on the horizontal axis—the five sinusoidal waves vary, or cycle, regularly at different rates
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Real Number
In mathematics , a REAL NUMBER is a value that represents a quantity along a line . The adjective real in this context was introduced in the 17th century by René Descartes , who distinguished between real and imaginary roots of polynomials . The real numbers include all the rational numbers , such as the integer −5 and the fraction 4/3, and all the irrational numbers , such as √2 (1.41421356..., the square root of 2 , an irrational algebraic number ). Included within the irrationals are the transcendental numbers , such as π (3.14159265...). Real numbers can be thought of as points on an infinitely long line called the number line or real line , where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation , such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. The real line can be thought of as a part of the complex plane , and complex numbers include real numbers. Real numbers can be thought of as points on an infinitely long number line These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers – indeed, the realization that a better definition was needed – was one of the most important developments of 19th century mathematics
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Angular Frequency
In physics , ANGULAR FREQUENCY ω (also referred to by the terms ANGULAR SPEED, RADIAL FREQUENCY, CIRCULAR FREQUENCY, ORBITAL FREQUENCY, RADIAN FREQUENCY, and PULSATANCE) is a scalar measure of rotation rate. It refers to the angular displacement per unit time (e.g., in rotation) or the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves), or as the rate of change of the argument of the sine function. Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity . The term ANGULAR FREQUENCY VECTOR {displaystyle {vec {omega }}} is sometimes used as a synonym for the vector quantity angular velocity. One revolution is equal to 2π radians , hence = 2 T = 2 f , {displaystyle omega ={{2pi } over T}={2pi f},} where: ω is the angular frequency or angular speed (measured in radians per second ), T is the period (measured in seconds ), f is the ordinary frequency (measured in hertz ) (sometimes symbolised with ν ). CONTENTS * 1 Units * 2 Circular motion * 2.1 Oscillations of a spring * 2.2 LC circuits * 3 See also * 4 References and notes * 5 External links UNITSIn SI units , angular frequency is normally presented in radians per second , even when it does not express a rotational value
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Radian
The RADIAN is the standard unit of angular measure, used in many areas of mathematics . The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at  A072097 ). The unit was formerly an SI supplementary unit , but this category was abolished in 1995 and the radian is now considered an SI derived unit . Separately, the SI unit of solid angle measurement is the steradian . The radian is represented by the symbol RAD. An alternative symbol is c, the superscript letter c, for "circular measure", or the letter r, but both of those symbols are infrequently used as it can be easily mistaken for a degree symbol (°) or a radius (r). So for example, a value of 1.2 radians could be written as 1.2 rad, 1.2 r, 1.2rad, or 1.2c. CONTENTS * 1 Definition * 2 History * 3 Conversions * 3.1 Conversion between radians and degrees * 3.1.1 Radian to degree conversion derivation * 3.2 Conversion between radians and gradians * 4 Advantages of measuring in radians * 5 Dimensional analysis * 6 Use in physics * 7 SI multiples * 8 See also * 9 Notes and references * 10 External links DEFINITION Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc
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