1.
United States Army
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The United States Armed Forces are the federal armed forces of the United States. They consist of the Army, Marine Corps, Navy, Air Force, from the time of its inception, the military played a decisive role in the history of the United States. A sense of unity and identity was forged as a result of victory in the First Barbary War. Even so, the Founders were suspicious of a permanent military force and it played an important role in the American Civil War, where leading generals on both sides were picked from members of the United States military. Not until the outbreak of World War II did a standing army become officially established. The National Security Act of 1947, adopted following World War II and during the Cold Wars onset, the U. S. military is one of the largest militaries in terms of number of personnel. It draws its personnel from a pool of paid volunteers. As of 2016, the United States spends about $580.3 billion annually to fund its military forces, put together, the United States constitutes roughly 40 percent of the worlds military expenditures. For the period 2010–14, the Stockholm International Peace Research Institute found that the United States was the worlds largest exporter of major arms, the United States was also the worlds eighth largest importer of major weapons for the same period. The history of the U. S. military dates to 1775 and these forces demobilized in 1784 after the Treaty of Paris ended the War for Independence. All three services trace their origins to the founding of the Continental Army, the Continental Navy, the United States President is the U. S. militarys commander-in-chief. Rising tensions at various times with Britain and France and the ensuing Quasi-War and War of 1812 quickened the development of the U. S. Navy, the reserve branches formed a military strategic reserve during the Cold War, to be called into service in case of war. Time magazines Mark Thompson has suggested that with the War on Terror, Command over the armed forces is established in the United States Constitution. The sole power of command is vested in the President by Article II as Commander-in-Chief, the Constitution also allows for the creation of executive Departments headed principal officers whose opinion the President can require. This allowance in the Constitution formed the basis for creation of the Department of Defense in 1947 by the National Security Act, the Defense Department is headed by the Secretary of Defense, who is a civilian and member of the Cabinet. The Defense Secretary is second in the chain of command, just below the President. Together, the President and the Secretary of Defense comprise the National Command Authority, to coordinate military strategy with political affairs, the President has a National Security Council headed by the National Security Advisor. The collective body has only power to the President
2.
Swell (ocean)
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A swell, in the context of an ocean, sea or lake, is a series of mechanical waves that propagate along the interface between water and air and so they are often referred to as surface gravity waves. These series of gravity waves are not generated by the immediate local wind, instead by distant weather systems. This is the definition of a swell as opposed to a locally generated wind wave. More generally, a swell consists of wind-generated waves that are not—or are hardly—affected by the wind at that time. Swell wavelength, also, varies from event to event, occasionally, swells which are longer than 700 m occur as a result of the most severe storms. Swell direction is the direction from which the swell is coming and it is measured in degrees, and often referred to in general directions, such as a NNW or SW swell. Large breakers one observes on a beach may result from distant weather systems over a fetch of ocean, further exposure to that specific wind could only cause a loss of energy due to the breaking of wave tops and formation of whitecaps. 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. Sea water wave is generated by many kinds of such as Seismic events, gravity. The generation of wave is initiated by the disturbances of cross wind field on the surface of the sea water. Two major mechanisms of surface wave formation by winds and other sources of wave formation can explain the generation of wind waves. 1) Starts from Fluctuations of wind, the wind wave formation on water surface by wind is started by a distribution of normal pressure acting on the water from the wind. By the mechanism developed by O. M and this pressure fluctuation arise normal and tangential stresses to the surface water, and generates wave behavior on the water surface. Since the wind profile Ua is logarithmic to the water surface and this relations show the wind flow transferring its kinetic energy to the water surface at their interface, and arises wave speed, c. For example, If we suppose a very flat sea surface, turbulent wind flows form random pressure fluctuations at the sea surface
3.
Panama
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Panama, officially called the Republic of Panama, is a country usually considered to be entirely in North America or Central America. It is bordered by Costa Rica to the west, Colombia to the southeast, the Caribbean Sea to the north, the capital and largest city is Panama City, whose metropolitan area is home to nearly half of the countrys 4.1 million people. Panama was inhabited by indigenous tribes prior to settlement by the Spanish in the 16th century. Panama broke away from Spain in 1821 and joined a union of Nueva Granada, Ecuador, when Gran Colombia dissolved in 1831, Panama and Nueva Granada remained joined, eventually becoming the Republic of Colombia. With the backing of the United States, Panama seceded from Colombia in 1903, in 1977 an agreement was signed for the total transfer of the Canal from the United States to Panama by the end of the 20th century, which culminated on 31 December 1999. Revenue from canal tolls continues to represent a significant portion of Panamas GDP, although commerce, banking, in 2015 Panama ranked 60th in the world in terms of the Human Development Index. Since 2010, Panama remains the second most competitive economy in Latin America, covering around 40 percent of its land area, Panamas jungles are home to an abundance of tropical plants and animals – some of them to be found nowhere else on the planet. There are several theories about the origin of the name Panama, some believe that the country was named after a commonly found species of tree. Others believe that the first settlers arrived in Panama in August, when butterflies abound, the best-known version is that a fishing village and its nearby beach bore the name Panamá, which meant an abundance of fish. Captain Antonio Tello de Guzmán, while exploring the Pacific side in 1515, in 1517 Don Gaspar De Espinosa, a Spanish lieutenant, decided to settle a post there. In 1519 Pedrarias Dávila decided to establish the Empires Pacific city in this site, the new settlement replaced Santa María La Antigua del Darién, which had lost its function within the Crowns global plan after the beginning of the Spanish exploitation of the riches in the Pacific. Blending all of the above together, Panamanians believe in general that the word Panama means abundance of fish and this is the official definition given in social studies textbooks approved by the Ministry of Education in Panama. However, others believe the word Panama comes from the Kuna word bannaba which means distant or far away, at the time of the arrival of the Spanish in the 16th century, the known inhabitants of Panama included the Cuevas and the Coclé tribes. These people have disappeared, as they had no immunity from European infectious diseases. The earliest discovered artifacts of indigenous peoples in Panama include Paleo-Indian projectile points, later central Panama was home to some of the first pottery-making in the Americas, for example the cultures at Monagrillo, which date back to 2500–1700 BC. These evolved into significant populations best known through their spectacular burials at the Monagrillo archaeological site, the monumental monolithic sculptures at the Barriles site are also important traces of these ancient isthmian cultures. Before Europeans arrived Panama was widely settled by Chibchan, Chocoan, the largest group were the Cueva. The size of the population of the isthmus at the time of European colonization is uncertain
4.
Fluid dynamics
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In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids. It has several subdisciplines, including aerodynamics and hydrodynamics, before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, the foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy. These are based on mechanics and are modified in quantum mechanics. They are expressed using the Reynolds transport theorem, in addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects, however, the continuum assumption assumes that fluids are continuous, rather than discrete. The fact that the fluid is made up of molecules is ignored. The unsimplified equations do not have a general solution, so they are primarily of use in Computational Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier to solve, some of the simplifications allow some simple fluid dynamics problems to be solved in closed form. Three conservation laws are used to solve fluid dynamics problems, the conservation laws may be applied to a region of the flow called a control volume. A control volume is a volume in space through which fluid is assumed to flow. The integral formulations of the laws are used to describe the change of mass, momentum. Mass continuity, The rate of change of fluid mass inside a control volume must be equal to the net rate of flow into the volume. Mass flow into the system is accounted as positive, and since the vector to the surface is opposite the sense of flow into the system the term is negated. The first term on the right is the net rate at which momentum is convected into the volume, the second term on the right is the force due to pressure on the volumes surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, the third term on the right is the net acceleration of the mass within the volume due to any body forces. Surface forces, such as forces, are represented by F surf. The following is the form of the momentum conservation equation
5.
Nonlinear system
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In mathematics and physical sciences, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, physicists and mathematicians, nonlinear systems may appear chaotic, unpredictable or counterintuitive, contrasting with the much simpler linear systems. In other words, in a system of equations, the equation to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as non-linear, regardless of whether or not known linear functions appear in the equations. In particular, an equation is linear if it is linear in terms of the unknown function and its derivatives. As nonlinear equations are difficult to solve, nonlinear systems are approximated by linear equations. This works well up to some accuracy and some range for the input values and it follows that some aspects of the behavior of a nonlinear system appear commonly to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is not random. For example, some aspects of the weather are seen to be chaotic and this nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. Some authors use the term nonlinear science for the study of nonlinear systems and this is disputed by others, Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals. In mathematics, a function f is one which satisfies both of the following properties, Additivity or superposition, f = f + f, Homogeneity. Additivity implies homogeneity for any rational α, and, for continuous functions, for a complex α, homogeneity does not follow from additivity. For example, a map is additive but not homogeneous. The equation is called homogeneous if C =0, if f contains differentiation with respect to x, the result will be a differential equation. Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials to zero, for example, x 2 + x −1 =0. For a single equation, root-finding algorithms can be used to find solutions to the equation. However, systems of equations are more complicated, their study is one motivation for the field of algebraic geometry. It is even difficult to decide whether a given system has complex solutions
6.
Periodic function
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In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, any function which is not periodic is called aperiodic. A function f is said to be periodic with period P if we have f = f for all values of x in the domain, geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P and this definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane. A function that is not periodic is called aperiodic, for example, the sine function is periodic with period 2 π, since sin = sin x for all values of x. This function repeats on intervals of length 2 π, everyday examples are seen when the variable is time, for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position of the system are expressible as periodic functions, for a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals. A simple example of a function is the function f that gives the fractional part of its argument. In particular, f = f = f =, =0.5 The graph of the function f is the sawtooth wave. The trigonometric functions sine and cosine are periodic functions, with period 2π. The subject of Fourier series investigates the idea that a periodic function is a sum of trigonometric functions with matching periods. According to the definition above, some functions, for example the Dirichlet function, are also periodic, in the case of Dirichlet function. For example, f = sin has period 2 π therefore sin will have period 2 π5, a function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions, if L is the period of the function then, L =2 π / k One common generalization of periodic functions is that of antiperiodic functions. This is a function f such that f = −f for all x, for example, the sine or cosine function is π-antiperiodic and 2π-periodic. A further generalization appears in the context of Bloch waves and Floquet theory, in this context, the solution is typically a function of the form, f = e i k P f where k is a real or complex number. Functions of this form are sometimes called Bloch-periodic in this context, a periodic function is the special case k =0, and an antiperiodic function is the special case k = π/P
7.
Wave
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In physics, a wave is an oscillation accompanied by a transfer of energy that travels through a medium. Frequency refers to the addition of time, wave motion transfers energy from one point to another, which displace particles of the transmission medium–that is, with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations, around almost fixed locations, there are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed, restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbors, when the molecules collide, they also bounce away from each other. This keeps the molecules from continuing to travel in the direction of the wave, the second main type, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields generated by charged particles. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this varies depending on the type of wave. Further, the behavior of particles in quantum mechanics are described by waves, in addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields. While mechanical waves can be transverse and longitudinal, all electromagnetic waves are transverse in free space. A single, all-encompassing definition for the wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value, however, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon as a results in a blurred line. The term wave is often understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium and it may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a rather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, or elasticity and this difference in origin introduces certain wave characteristics particular to the properties of the medium involved
8.
Jacobi elliptic functions
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In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that are of historical importance. Many of their features show up in important structures and have direct relevance to some applications and they also have useful analogies to the functions of trigonometry, as indicated by the matching notation sn for sin. The Jacobi elliptic functions are used often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi, Jacobian elliptic functions are doubly periodic meromorphic functions on the complex plane. Since they are periodic, they factor through a torus – in effect, their domain can be taken to be a torus. Instead of having only one circle, we now have the product of two circles, one real and the other imaginary, the complex plane can be replaced by a complex torus. The circumference of the first circle is 4K and the second 4K′, each function has two zeroes and two poles at opposite positions on the torus. Among the points 0, K, K + iK′, iK′ there is one zero, so an arrow can be drawn from a zero to a pole. So there are twelve Jacobian elliptic functions, each of the twelve corresponds to an arrow drawn from one corner of a rectangle to another. The corners of the rectangle are labeled, by convention, s, c, d and n. S is at the origin, c is at the point K on the real axis/loop, d is at the point K + iK′ and n is at point iK′ on the imaginary axis/loop. The twelve Jacobian elliptic functions are then pq, where each of p and q is a different one of the s, c, d, n. The step from p to q is equal to half the period of the function pq u, that is, the function pq u is periodic in the direction pq, with the period being twice the distance from p to q. The function pq u is periodic in the other two directions, with a period such that the distance from p to one of the other corners is a quarter period. If the function pq u is expanded in terms of u at one of the corners, more generally, there is no need to impose a rectangle, a parallelogram will do. However, if K and iK are kept on the real and imaginary axis respectively, the elliptic functions can be given in a variety of notations, which can make the subject unnecessarily confusing. Elliptic functions are functions of two variables, the first variable might be given in terms of the amplitude φ, or more commonly, in terms of u given below. The second variable might be given in terms of the m, or as the elliptic modulus k, where k2 = m, or in terms of the modular angle α
9.
Gravity wave
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In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere and the ocean, which rise to wind waves. A gravity wave results when fluid is displaced from a position of equilibrium, the restoration of the fluid to equilibrium will produce a movement of the fluid back and forth, called a wave orbit. Gravity waves on an interface of the ocean are called surface gravity waves or surface waves. Wind-generated waves on the surface are examples of gravity waves, as are tsunamis. Wind-generated gravity waves on the surface of the Earths ponds, lakes, seas. Shorter waves are affected by surface tension and are called gravity–capillary waves. Alternatively, so-called infragravity waves, which are due to nonlinear wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves. In the Earths atmosphere, gravity waves are a mechanism that produce the transfer of momentum from the troposphere to the stratosphere and mesosphere, Gravity waves are generated in the troposphere by frontal systems or by airflow over mountains. At first, waves propagate through the atmosphere without appreciable change in mean velocity, but as the waves reach more rarefied air at higher altitudes, their amplitude increases, and nonlinear effects cause the waves to break, transferring their momentum to the mean flow. This transfer of momentum is responsible for the forcing of the many large-scale dynamical features of the atmosphere, thus, this process plays a key role in the dynamics of the middle atmosphere. The effect of gravity waves in clouds can look like altostratus undulatus clouds, and are confused with them. The phase velocity c of a gravity wave with wavenumber k is given by the formula c = g k. When surface tension is important, this is modified to c = g k + σ k ρ, where σ is the surface tension coefficient and ρ is the density. Since c = ω / k is the speed in terms of the angular frequency ω and the wavenumber. The group velocity of a wave is given by c g = d ω d k, the group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive, Gravity waves traveling in shallow water, are nondispersive, the phase and group velocities are identical and independent of wavelength and frequency. When the water depth is h, c p = c g = g h, wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the oceans surface, and capillary-gravity waves play an essential role in this effect
10.
Wavelength
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In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the waves shape repeats, and thus the inverse of the spatial frequency. Wavelength is commonly designated by the Greek letter lambda, the concept can also be applied to periodic waves of non-sinusoidal shape. The term wavelength is also applied to modulated waves. Wavelength depends on the medium that a wave travels through, examples of wave-like phenomena are sound waves, light, water waves and periodic electrical signals in a conductor. A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric, water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary, wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in waves over deep water a particle near the waters surface moves in a circle of the same diameter as the wave height. The range of wavelengths or frequencies for wave phenomena is called a spectrum, the name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, in a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the speed is the speed of light. Thus the wavelength of a 100 MHz electromagnetic wave is about, the wavelength of visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm. For sound waves in air, the speed of sound is 343 m/s, the wavelengths of sound frequencies audible to the human ear are thus between approximately 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light, a standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed, the stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Consequently, wavelength, period, and wave velocity are related just as for a traveling wave, for example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. In that case, the k, the magnitude of k, is still in the same relationship with wavelength as shown above
11.
Diederik Korteweg
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Diederik Johannes Korteweg was a Dutch mathematician. He is now best remembered for his work on the Korteweg–de Vries equation, Diederik Kortewegs father was a judge in s-Hertogenbosch, Netherlands. Korteweg received his schooling there, studying at an academy which prepared students for a military career. However, he decided against a career and, making the first of his changes of direction. He decided to terminate his course and pull out of his studies so that he could concentrate on mathematics and he then enrolled in mathematics and mechanics courses qualifying him to become a high school teacher. In 1878, Korteweg received a Ph. D. from the University of Amsterdam and his dissertation was titled On the Propagation of Waves in Elastic Tubes. He was the first Ph. D. recipient from that University after it received authority to grant the doctorate, in 1881, Korteweg joined the University of Amsterdam as Professor of Mathematics, Mechanics and Astronomy. While there he published a paper in Philosophical Magazine titled On the Change of Form of Long Waves. Some of his famous students were Gustav de Vries, Gerrit Mannoury, Korteweg was a member of the Royal Netherlands Academy of Arts and Sciences for 60 years. He was a member of the Dutch Mathematical Society for 75 years and he was editor of Nieuw Archief voor Wiskunde from 1897 to his death in 1941. An experiment conducted aboard the International Space Station in 2003 was mounted to one of Kortewegs theories. The asteroid 9685 Korteweg is named after him, Korteweg, D. J. & de Vries, G. On the Change of Form of Long Waves advancing in a Rectangular Canal and on a New Type of Long Stationary Waves, Philosophical Magazine, 5th series,39, 422–443, doi,10. 1080/14786449508620739. OConnor, John J. Robertson, Edmund F. Diederik Korteweg, MacTutor History of Mathematics archive, Diederik Korteweg at the Mathematics Genealogy Project
12.
Gustav de Vries
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Gustav de Vries was a Dutch mathematician, who is best remembered for his work on the Korteweg–de Vries equation with Diederik Korteweg. He was born on 22 January 1866 in Amsterdam, and studied at the University of Amsterdam with the physical chemist Johannes van der Waals. While doing his doctoral research De Vries supported himself by teaching at the Royal Military Academy in Breda and at the cadettenschool in Alkmaar. Under Kortewegs supervision de Vries completed his dissertation, Bijdrage tot de kennis der lange golven, Acad. proefschrift, Universiteit van Amsterdam,1894,95 pp, Loosjes. In 1894 de Vries worked as a school teacher at the HBS en Handelsschool in Haarlem. He died in Haarlem on 16 December 1934, cnoidal wave Korteweg–de Vries equation Bastiaan Willink, The collaboration between Korteweg and de Vries — An enquiry into personalities, History of Physics,16 p. October 2007. OConnor, John J. Robertson, Edmund F. Gustav de Vries, MacTutor History of Mathematics archive, Gustav de Vries at the Mathematics Genealogy Project
13.
Dispersion (water waves)
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In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in context, are waves propagating on the water surface, with gravity. As a result, water with a surface is generally considered to be a dispersive medium. For a certain depth, surface gravity waves – i. e. waves occurring at the air–water interface. On the other hand, for a wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves propagate faster for shorter wavelengths, besides frequency dispersion, water waves also exhibit amplitude dispersion. This is an effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves. This section is about frequency dispersion for waves on a fluid layer forced by gravity, for surface tension effects on frequency dispersion, see surface tension effects in Airy wave theory and capillary wave. The simplest propagating wave of unchanging form is a sine wave. Characteristic phases of a wave are, the upward zero-crossing at θ =0, the wave crest at θ = ½ π, the downward zero-crossing at θ = π. A certain phase repeats itself after an integer m multiple of 2π, the dispersion relation has two solutions, ω = +Ω and ω = −Ω, corresponding to waves travelling in the positive or negative x–direction. The dispersion relation will in general depend on other parameters in addition to the wavenumber k. For gravity waves, according to theory, these are the acceleration by gravity g. The dispersion relation for these waves is, an equation with tanh denoting the hyperbolic tangent function. An initial wave phase θ = θ0 propagates as a function of space and its subsequent position is given by, x = λ T t + λ2 π θ0 = ω k t + θ0 k. This shows that the moves with the velocity, c p = λ T = ω k = Ω k. A sinusoidal wave, of small amplitude and with a constant wavelength, propagates with the phase velocity. While the phase velocity is a vector and has an associated direction, according to linear theory for waves forced by gravity, the phase speed depends on the wavelength and the water depth
14.
Infinity
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Infinity is an abstract concept describing something without any bound or larger than any number. In mathematics, infinity is treated as a number but it is not the same sort of number as natural or real numbers. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th, in the theory he developed, there are infinite sets of different sizes. For example, the set of integers is countably infinite, while the set of real numbers is uncountable. Ancient cultures had various ideas about the nature of infinity, the ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus and he used the word apeiron which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea, aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as immeasurably subtle, however, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. The Jain mathematical text Surya Prajnapti classifies all numbers into three sets, enumerable, innumerable, and infinite, on both physical and ontological grounds, a distinction was made between asaṃkhyāta and ananta, between rigidly bounded and loosely bounded infinities. European mathematicians started using numbers in a systematic fashion in the 17th century. John Wallis first used the notation ∞ for such a number, euler used the notation i for an infinite number, and exploited it by applying the binomial formula to the i th power, and infinite products of i factors. In 1699 Isaac Newton wrote about equations with an number of terms in his work De analysi per aequationes numero terminorum infinitas. The infinity symbol ∞ is a symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ infinity and in LaTeX as \infty and it was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology. Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers, in real analysis, the symbol ∞, called infinity, is used to denote an unbounded limit. X → ∞ means that x grows without bound, and x → − ∞ means the value of x is decreasing without bound. ∑ i =0 ∞ f = ∞ means that the sum of the series diverges in the specific sense that the partial sums grow without bound. Infinity can be used not only to define a limit but as a value in the real number system
15.
John Scott Russell
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John Scott Russell FRSE FRS was a Scottish civil engineer, naval architect and shipbuilder who built the Great Eastern in collaboration with Isambard Kingdom Brunel. He made the discovery of the wave of translation that gave birth to the study of solitons. Russell was a promoter of the Great Exhibition of 1851, John Russell was born on 9 May 1808 in Parkhead, Glasgow, the son of Reverend David Russell and Agnes Clark Scott. He spent one year at St. Andrews University before transferring to Glasgow University and it was while at Glasgow University that he added his mothers maiden name, Scott, to his own, to become John Scott Russell. Arthur Sullivan and his friend Frederic Clay were frequent visitors at the Scott Russell home in the mid-1860s, Clay became engaged to Alice, at some point in 1868, Sullivan started a simultaneous affair with Louise. Both relationships had ceased by early 1869, the American engineer Alexander Lyman Holley befriended Scott Russell and his family on his various visits to London at the time of the construction of Great Eastern. Holley also visited Scott Russells house in Sydenham, as a result of this, Holley, and his colleague Zerah Colburn, travelled on the maiden voyage of Great Eastern from Southampton to New York in June 1860. Scott Russells son, Norman, stayed with Holley at his house in Brooklyn — Norman also travelled on the maiden voyage and his son, Norman, followed his father in becoming a naval architect, contributing to the Institution of Naval Architects which his father had founded. While in Edinburgh he experimented with engines, using a square boiler for which he developed a method of staying the surface of the boiler which became universal. The Scottish Steam Carriage Company was formed producing a steam carriage with two cylinders developing 12 horsepower each, six were constructed in 1834, well-sprung and fitted out to high standard, which ran between Glasgow and Paisley at hourly intervals at 15 mph. The road trustees objected that it wore out the road and placed various obstructions of logs and stones in the road, but eventually one of the carriages was overturned and the boiler smashed, causing the death of several passengers. Two of the coaches were sent to London where they ran for a time between London and Greenwich. In 1834, while conducting experiments to determine the most efficient design for canal boats, in fluid dynamics the wave is now called Russells solitary wave. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular, Scott Russell spent some time making practical and theoretical investigations of these waves. Unlike normal waves they will never merge—so a small wave is overtaken by a large one, if a wave is too big for the depth of water, it splits into two, one big and one small. Scott Russells experimental work seemed at contrast with Isaac Newtons and Daniel Bernoullis theories of hydrodynamics and his contemporaries spent some time attempting to extend the theory but it would take until the 1870s before an explanation was provided. Lord Rayleigh published a paper in Philosophical Magazine in 1876 to support John Scott Russells experimental observation with his mathematical theory, in his 1876 paper, Lord Rayleigh mentioned Scott Russells name and also admitted that the first theoretical treatment was by Joseph Valentin Boussinesq in 1871
16.
Boussinesq approximation (water waves)
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In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation, the 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations. The Boussinesq approximation for water waves takes into account the structure of the horizontal and vertical flow velocity. This results in partial differential equations, called Boussinesq-type equations. In coastal engineering, Boussinesq-type equations are used in computer models for the simulation of water waves in shallow seas. While the Boussinesq approximation is applicable to fairly long waves – that is and this is useful because the waves propagate in the horizontal plane and have a different behaviour in the vertical direction. Often, as in Boussinesqs case, the interest is primarily in the wave propagation and this elimination of the vertical coordinate was first done by Joseph Boussinesq in 1871, to construct an approximate solution for the solitary wave. Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations, thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate. As a result, the partial differential equations are in terms of functions of the horizontal coordinates. As an example, consider potential flow over a bed in the plane, with x the horizontal. The bed is located at z = −h, where h is the water depth. Invoking Laplaces equation for φ, as valid for incompressible flow, gives, φ = + = and this series may subsequently be truncated to a finite number of terms. Now the Boussinesq approximation for the velocity potential φ, as given above, is applied in these boundary conditions, further, in the resulting equations only the linear and quadratic terms with respect to η and ub are retained. The cubic and higher terms are assumed to be negligible. This set of equations has been derived for a horizontal bed. When the right-hand sides of the equations are set to zero. From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the Ursell number, for the case of infinitesimal wave amplitude, the terminology is linear frequency dispersion. The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation
17.
Crest (physics)
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A crest is the point on a wave with the maximum value or upward displacement within a cycle. A crest is a point on the wave where the displacement of the medium is at a maximum, a trough is the opposite of a crest, so the minimum or lowest point in a cycle. When in antiphase – 180° out of phase – the result is destructive interference, superposition principle Wave Kinsman, Blair, Wind Waves, Their Generation and Propagation on the Ocean Surface, Dover Publications, ISBN 0-486-49511-6,704 pages
18.
Sine
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In mathematics, the sine is a trigonometric function of an angle. More generally, the definition of sine can be extended to any value in terms of the length of a certain line segment in a unit circle. The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy, via translation from Sanskrit to Arabic and then from Arabic to Latin. The word sine comes from a Latin mistranslation of the Arabic jiba, to define the trigonometric functions for an acute angle α, start with any right triangle that contains an angle of measure α, in the accompanying figure, angle A in triangle ABC has measure α. The three sides of the triangle are named as follows, The opposite side is the side opposite to the angle of interest, the hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle, the adjacent side is the remaining side, in this case side b. It forms a side of both the angle of interest and the right angle, once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse. As stated, the value sin appears to depend on the choice of right triangle containing an angle of measure α, however, this is not the case, all such triangles are similar, and so the ratio is the same for each of them. The trigonometric functions can be defined in terms of the rise, run, when the length of the line segment is 1, sine takes an angle and tells the rise. Sine takes an angle and tells the rise per unit length of the line segment, rise is equal to sin θ multiplied by the length of the line segment. In contrast, cosine is used for telling the run from the angle, arctan is used for telling the angle from the slope. The line segment is the equivalent of the hypotenuse in the right-triangle, in trigonometry, a unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. Let a line through the origin, making an angle of θ with the half of the x-axis. The x- and y-coordinates of this point of intersection are equal to cos θ and sin, the points distance from the origin is always 1. Unlike the definitions with the triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function. Exact identities, These apply for all values of θ. sin = cos =1 csc The reciprocal of sine is cosecant, i. e. the reciprocal of sin is csc, or cosec. Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side, the inverse function of sine is arcsine or inverse sine
19.
Atlantic Ocean
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The Atlantic Ocean is the second largest of the worlds oceans with a total area of about 106,460,000 square kilometres. It covers approximately 20 percent of the Earths surface and about 29 percent of its surface area. It separates the Old World from the New World, the Atlantic Ocean occupies an elongated, S-shaped basin extending longitudinally between Eurasia and Africa to the east, and the Americas to the west. The Equatorial Counter Current subdivides it into the North Atlantic Ocean, in contrast, the term Atlantic originally referred specifically to the Atlas Mountains in Morocco and the sea off the Strait of Gibraltar and the North African coast. The Greek word thalassa has been reused by scientists for the huge Panthalassa ocean that surrounded the supercontinent Pangaea hundreds of years ago. The term Aethiopian Ocean, derived from Ancient Ethiopia, was applied to the Southern Atlantic as late as the mid-19th century, many Irish or British people refer to the United States and Canada as across the pond, and vice versa. The Black Atlantic refers to the role of ocean in shaping black peoples history. Irish migration to the US is meant when the term The Green Atlantic is used, the term Red Atlantic has been used in reference to the Marxian concept of an Atlantic working class, as well as to the Atlantic experience of indigenous Americans. Correspondingly, the extent and number of oceans and seas varies, the Atlantic Ocean is bounded on the west by North and South America. It connects to the Arctic Ocean through the Denmark Strait, Greenland Sea, Norwegian Sea, to the east, the boundaries of the ocean proper are Europe, the Strait of Gibraltar and Africa. In the southeast, the Atlantic merges into the Indian Ocean, the 20° East meridian, running south from Cape Agulhas to Antarctica defines its border. In the 1953 definition it extends south to Antarctica, while in later maps it is bounded at the 60° parallel by the Southern Ocean, the Atlantic has irregular coasts indented by numerous bays, gulfs, and seas. Including these marginal seas the coast line of the Atlantic measures 111,866 km compared to 135,663 km for the Pacific. Including its marginal seas, the Atlantic covers an area of 106,460,000 km2 or 23. 5% of the ocean and has a volume of 310,410,900 km3 or 23. 3%. Excluding its marginal seas, the Atlantic covers 81,760,000 km2 and has a volume of 305,811,900 km3, the North Atlantic covers 41,490,000 km2 and the South Atlantic 40,270,000 km2. The average depth is 3,646 m and the maximum depth, the bathymetry of the Atlantic is dominated by a submarine mountain range called the Mid-Atlantic Ridge. It runs from 87°N or 300 km south of the North Pole to the subantarctic Bouvet Island at 42°S, the MAR divides the Atlantic longitudinally into two halves, in each of which a series of basins are delimited by secondary, transverse ridges. The MAR reaches above 2000 m along most of its length, the MAR is a barrier for bottom water, but at these two transform faults deep water currents can pass from one side to the other
20.
Airy wave theory
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In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the layer has a uniform mean depth. This theory was first published, in form, by George Biddell Airy in the 19th century. Further, several second-order nonlinear properties of gravity waves, and their propagation. Airy wave theory is also a good approximation for tsunami waves in the ocean and this linear theory is often used to get a quick and rough estimate of wave characteristics and their effects. This approximation is accurate for small ratios of the height to water depth. Airy wave theory uses a potential approach to describe the motion of gravity waves on a fluid surface. This is due to the fact that for the part of the fluid motion. Airy wave theory is used in ocean engineering and coastal engineering. Diffraction is one of the effects which can be described with Airy wave theory. Further, by using the WKBJ approximation, wave shoaling and refraction can be predicted, earlier attempts to describe surface gravity waves using potential flow were made by, among others, Laplace, Poisson, Cauchy and Kelland. But Airy was the first to publish the correct derivation and formulation in 1841, soon after, in 1847, the linear theory of Airy was extended by Stokes for non-linear wave motion – known as Stokes wave theory – correct up to third order in the wave steepness. Even before Airys linear theory, Gerstner derived a nonlinear wave theory in 1802. Airy wave theory is a theory for the propagation of waves on the surface of a potential flow. The waves propagate along the surface with the phase speed cp. The angular wavenumber k and frequency ω are not independent parameters, surface gravity waves on a fluid are dispersive waves – exhibiting frequency dispersion – meaning that each wavenumber has its own frequency and phase speed. Note that in engineering the wave height H – the difference in elevation between crest and trough – is often used, H =2 a and a =12 H, underneath the surface, there is a fluid motion associated with the free surface motion. While the surface shows a propagating wave, the fluid particles are in an orbital motion
21.
Stokes wave
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In fluid dynamics, a Stokes wave is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth. Stokes wave theory is of practical use for waves on intermediate. It is used in the design of coastal and offshore structures, the wave kinematics are subsequently needed in the design process to determine the wave loads on a structure. For long waves – and using only a few terms in the Stokes expansion – its applicability is limited to waves of small amplitude, in such shallow water, a cnoidal wave theory often provides better periodic-wave approximations. While, in the sense, Stokes wave refers to progressive periodic waves of permanent form. The examples below describe Stokes waves under the action of gravity in case of wave motion. The phase speed increases with increasing non-linearity ka of the waves, the wave height H, being the difference between the surface elevation η at a crest and a trough, is, H =2 a. Note that the second- and third-order terms in the velocity potential Φ are zero, only at fourth order contributions deviating from first-order theory – i. e. Airy wave theory – appear. Up to third order the orbital velocity field u = ∇Φ consists of a motion of the velocity vector at each position. As a result, the elevation of deep-water waves is to a good approximation trochoidal. Stokes further observed, that although the orbital velocity field consists of a circular motion at each point. This is due to the reduction of the velocity amplitude at increasing depth below the surface and this Lagrangian drift of the fluid parcels is known as the Stokes drift. Observe that for finite depth the velocity potential Φ contains a drift in time. Both this temporal drift and the term in Φ vanish for deep-water waves. The ratio S of the free-surface amplitudes at second or and first order – according to Stokes second-order theory – is, in deep water, for large kh the ratio S has the asymptote lim k h → ∞ S =12 k a. Here U is the Ursell parameter, for long waves of small height H, i. e. U ≪ 32π2/3 ≈100, second-order Stokes theory is applicable. Otherwise, for long waves of appreciable height H a cnoidal wave description is more appropriate. According to Hedges, fifth-order Stokes theory is applicable for U <40, for Stokes waves under the action of gravity, the third-order dispersion relation is – according to Stokes first definition of celerity, ω2 = + O, with σ = tanh k h
22.
Stream function
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The stream function is defined for incompressible flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can then be expressed as the derivatives of the stream function. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow, the two-dimensional Lagrange stream function was introduced by Joseph Louis Lagrange in 1781. The Stokes stream function is for axisymmetrical three-dimensional flow, and is named after George Gabriel Stokes, considering the particular case of fluid dynamics, the difference between the stream function values at any two points gives the volumetric flow rate through a line connecting the two points. Since streamlines are tangent to the velocity vector of the flow. A stream function may be defined for any flow of dimensions greater than or equal to two, however the case is generally the easiest to visualize and derive. For two-dimensional potential flow, streamlines are perpendicular to equipotential lines, taken together with the velocity potential, the stream function may be used to derive a complex potential. In other words, the stream function accounts for the part of a two-dimensional Helmholtz decomposition. Lamb and Batchelor define the stream function ψ – in the point P with two-dimensional coordinates and as a function of time t – for a flow by. So the stream function ψ is the flux through the curve A P, that is, the integral of the dot product of the flow velocity vector. The point A is a reference point defining where the function is zero. This is the condition of zero divergence resulting from flow incompressibility, the sign of the stream function depends on the definition used. One way is to define the stream function ψ for a two-dimensional flow such that the velocity can be expressed through the vector potential ψ, u = ∇ × ψ Where ψ = if the flow velocity vector u =. In Cartesian coordinate system this is equivalent to u = ∂ ψ ∂ y, v = − ∂ ψ ∂ x Where u and v are the velocity components in the cartesian x and y coordinate directions. Another definition is u = z × ∇ ψ ′ ≡, where z = is a vector in the + z direction. Note that this definition has the sign to that given above. Note that ψ = ψ z in this two-dimensional flow, Consider two points A and B in two-dimensional plane flow. Suppose an observer looks along an axis in the direction of increase
23.
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
24.
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
25.
Average
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In colloquial language, an average is the sum of a list of numbers divided by the number of numbers in the list. In mathematics and statistics, this would be called the arithmetic mean, in statistics, mean, median, and mode are all known as measures of central tendency. The most common type of average is the arithmetic mean, one may find that A = /2 =5. Switching the order of 2 and 8 to read 8 and 2 does not change the value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list to 2,8, and 11, one finds that A = /3 =7. Along with the arithmetic mean above, the mean and the harmonic mean are known collectively as the Pythagorean means. The geometric mean of n numbers is obtained by multiplying them all together. See Inequality of arithmetic and geometric means, thus for the above harmonic mean example, AM =50, GM ≈49, and HM =48 km/h. The mode, the median, and the mid-range are often used in addition to the mean as estimates of central tendency in descriptive statistics, the most frequently occurring number in a list is called the mode. For example, the mode of the list is 3 and it may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there is no agreed definition of mode, some authors say they are all modes and some say there is no mode. The median is the number of the group when they are ranked in order. Thus to find the median, order the list according to its elements magnitude, if exactly one value is left, it is the median, if two values, the median is the arithmetic mean of these two. This method takes the list 1,7,3,13, then the 1 and 13 are removed to obtain the list 3,7. Since there are two elements in this remaining list, the median is their arithmetic mean, /2 =5, the table of mathematical symbols explains the symbols used below. Other more sophisticated averages are, trimean, trimedian, and normalized mean, one can create ones own average metric using the generalized f-mean, y = f −1 where f is any invertible function. The harmonic mean is an example of this using f = 1/x, however, this method for generating means is not general enough to capture all averages
26.
Gravity of Earth
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The gravity of Earth, which is denoted by g, refers to the acceleration that is imparted to objects due to the distribution of mass within the Earth. In SI units this acceleration is measured in metres per second squared or equivalently in newtons per kilogram and this quantity is sometimes referred to informally as little g. The precise strength of Earths gravity varies depending on location, the nominal average value at the Earths surface, known as standard gravity is, by definition,9.80665 m/s2. This quantity is denoted variously as gn, ge, g0, gee, the weight of an object on the Earths surface is the downwards force on that object, given by Newtons second law of motion, or F = ma. Gravitational acceleration contributes to the acceleration, but other factors, such as the rotation of the Earth, also contribute. The Earth is not spherically symmetric, but is slightly flatter at the poles while bulging at the Equator, there are consequently slight deviations in both the magnitude and direction of gravity across its surface. The net force as measured by a scale and plumb bob is called effective gravity or apparent gravity, effective gravity includes other factors that affect the net force. These factors vary and include such as centrifugal force at the surface from the Earths rotation. Effective gravity on the Earths surface varies by around 0. 7%, in large cities, it ranges from 9.766 in Kuala Lumpur, Mexico City, and Singapore to 9.825 in Oslo and Helsinki. The surface of the Earth is rotating, so it is not a frame of reference. At latitudes nearer the Equator, the centrifugal force produced by Earths rotation is larger than at polar latitudes. This counteracts the Earths gravity to a small degree – up to a maximum of 0. 3% at the Equator –, the same two factors influence the direction of the effective gravity. Gravity decreases with altitude as one rises above the Earths surface because greater altitude means greater distance from the Earths centre, all other things being equal, an increase in altitude from sea level to 9,000 metres causes a weight decrease of about 0. 29%. It is a misconception that astronauts in orbit are weightless because they have flown high enough to escape the Earths gravity. In fact, at an altitude of 400 kilometres, equivalent to an orbit of the Space Shuttle. Weightlessness actually occurs because orbiting objects are in free-fall, the effect of ground elevation depends on the density of the ground. A person flying at 30000 ft above sea level over mountains will feel more gravity than someone at the same elevation, however, a person standing on the earths surface feels less gravity when the elevation is higher. The following formula approximates the Earths gravity variation with altitude, g h = g 02 Where gh is the acceleration at height h above sea level
27.
Shallow water equations
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The shallow water equations are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid. The shallow water equations can also be simplified to the commonly used 1-D Saint-Venant equation, the equations are derived from depth-integrating the Navier–Stokes equations, in the case where the horizontal length scale is much greater than the vertical length scale. Under this condition, conservation of mass implies that the velocity of the fluid is small. Vertically integrating allows the velocity to be removed from the equations. The shallow water equations are thus derived, while a vertical velocity term is not present in the shallow water equations, note that this velocity is not necessarily zero. Once a solution has been found, the velocity can be recovered via the continuity equation. Situations in fluid dynamics where the length scale is much greater than the vertical length scale are common. They are used with Coriolis forces in atmospheric and oceanic modeling, shallow water equation models have only one vertical level, so they cannot directly encompass any factor that varies with height. However, in cases where the state is sufficiently simple. Here η is the fluid column height, and the 2D vector is the fluids horizontal flow velocity. Further g is acceleration due to gravity and ρ is the fluid density, the first equation is derived from mass conservation, the second two from momentum conservation. Expanding the derivatives in the using the product rule, the non-conservative form of the shallow-water equations is obtained. Since velocities are not subject to a conservation equation, the non-conservative forms do not hold across a shock or hydraulic jump. This is called geostrophic balance, and is equivalent to saying that the Rossby number is small, the one-dimensional Saint-Venant equations were derived by Adhémar Jean Claude Barré de Saint-Venant, and are commonly used to model transient open-channel flow and surface runoff. They can be viewed as a contraction of the shallow water equations. The 1-D Saint-Venant equations contain to an extent the main characteristics of the channel cross-sectional shape. Common applications of the 1-D Saint-Venant equations include flood routing along rivers, dam break analysis, storm pulses in an open channel, further ρ is the fluid density and g is the gravitational acceleration. For example, for a cross section, with constant channel width B and channel bed elevation zb
28.
Joseph Valentin Boussinesq
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Joseph Valentin Boussinesq was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat. From 1872 to 1886, he was appointed professor at Faculty of Sciences of Lille, from 1896 to his retirement in 1918, he was professor of mechanics at Faculty of Sciences of Paris. John Scott Russell experimentally observed his great solitary wave of translation in 1834, subsequently this was developed into the modern physics of solitons. In 1871, Boussinesq published the first mathematical theory to support Russells experimental observation, in 1876, Lord Rayleigh published his mathematical theory to support Russells experimental observation. At the end of his paper, Lord Rayleigh admitted that Boussinesqs theory came before his, in 1897 he published Théorie de lécoulement tourbillonnant et tumultueux des liquides, a work that greatly contributed to the study of turbulence and hydrodynamics. The word turbulence was never used by Boussinesq and he used sentences such as écoulement tourbillonnant et tumultueux. The first mention of the turbulence in French or English scientific fluid mechanics literature can be found in a paper by Lord Kelvin in 1887. G
29.
John William Strutt, 3rd Baron Rayleigh
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John William Strutt, 3rd Baron Rayleigh OM PC PRS was a physicist who, with William Ramsay, discovered argon, an achievement for which he earned the Nobel Prize for Physics in 1904. He also discovered the now called Rayleigh scattering, which can be used to explain why the sky is blue. Rayleighs textbook, The Theory of Sound, is referred to by acoustic engineers today. John William Strutt, of Terling Place Essex, suffered from frailty and he attended Harrow School, before going on to the University of Cambridge in 1861 where he studied mathematics at Trinity College, Cambridge. He obtained a Bachelor of Arts degree in 1865, and a Master of Arts in 1868 and he was subsequently elected to a Fellowship of Trinity. He held the post until his marriage to Evelyn Balfour, daughter of James Maitland Balfour and he had three sons with her. In 1873, on the death of his father, John Strutt, 2nd Baron Rayleigh and he was the second Cavendish Professor of Physics at the University of Cambridge, from 1879 to 1884. He first described dynamic soaring by seabirds in 1883, in the British journal Nature, from 1887 to 1905 he was Professor of Natural Philosophy at the Royal Institution. Around the year 1900 Lord Rayleigh developed the theory of human sound localisation using two binaural cues, interaural phase difference and interaural level difference. The theory posits that we use two primary cues for sound lateralisation, using the difference in the phases of sinusoidal components of the sound, in 1919, Rayleigh served as President of the Society for Psychical Research. The rayl unit of acoustic impedance is named after him, as an advocate that simplicity and theory be part of the scientific method, Lord Rayleigh argued for the principle of similitude. Lord Rayleigh was elected Fellow of the Royal Society on 12 June 1873, from time to time Lord Rayleigh participated in the House of Lords, however, he spoke up only if politics attempted to become involved in science. He died on 30 June 1919, in Witham, Essex and he was succeeded, as the 4th Lord Rayleigh, by his son Robert John Strutt, another well-known physicist. Lord Rayleigh was buried in the graveyard of All Saints Church in Terling in Essex, though he did not write about the relationship of science and religion, he retained a personal interest in spiritual matters. The Secretary to the Press suggested with many apologies that the reader might suppose that I was the Lord, still, he kept his wish and the quotation was printed in the five-volume collection of scientific papers. What is more, I think that Christ and indeed other spiritually gifted men see further and truer than I do, Lord Rayleigh was the president of the SPR in 1919. He gave an address the year of his death but did not come to any definite conclusions. Craters on Mars and the Moon are named in his honour as well as a type of surface wave known as a Rayleigh wave, the asteroid 22740 Rayleigh was named in his honour on 1 June 2007
30.
Wave height
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In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighbouring trough. Wave height is a used by mariners, as well as in coastal, ocean. At sea, the significant wave height is used as a means to introduce a well-defined and standardized statistic to denote the characteristic height of the random waves in a sea state. It is defined in such a way that it corresponds to what a mariner observes when estimating visually the average wave height. For a sine wave, the wave height H is twice the amplitude, for a periodic wave it is simply the difference between the maximum and minimum of the surface elevation z = η, H = max − min, with cp the phase speed of the wave. The sine wave is a case of a periodic wave. For this to be possible, it is necessary to first split the time series of the surface elevation into individual waves. Commonly, a wave is denoted as the time interval between two successive downward-crossings through the average surface elevation. Then the individual wave height of each wave is again the difference between maximum and minimum elevation in the interval of the wave under consideration. Only the highest one-third is used, since this corresponds best with visual observations of experienced mariners, eyes, Significant wave height Hm0, defined in the frequency domain, is used both for measured and forecasted wave variance spectra. In case of a measurement, the standard deviation ση is the easiest and most accurate statistic to be used, sea state Significant wave height Wind wave
31.
Elliptic integral
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In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler, in general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this rule are when P has repeated roots. However, with the reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions. Besides the Legendre form given below, the integrals may also be expressed in Carlson symmetric form. Additional insight into the theory of the integral may be gained through the study of the Schwarz–Christoffel mapping. Historically, elliptic functions were discovered as inverse functions of elliptic integrals, incomplete elliptic integrals are functions of two arguments, complete elliptic integrals are functions of a single argument. These arguments are expressed in a variety of different but equivalent ways, most texts adhere to a canonical naming scheme, using the following naming conventions. Thus, they can be used interchangeably, the other argument can likewise be expressed as φ, the amplitude, or as x or u, where x = sin φ = sn u and sn is one of the Jacobian elliptic functions. Specifying the value of any one of these determines the others. Note that u also depends on m, some additional relationships involving u include cos φ = cn u, and 1 − m sin 2 φ = dn u. The latter is called the delta amplitude and written as Δ = dn u. Sometimes the literature refers to the complementary parameter, the complementary modulus. These are further defined in the article on quarter periods, the incomplete elliptic integral of the first kind F is defined as F = F = F = ∫0 φ d θ1 − k 2 sin 2 θ. This is the form of the integral, substituting t = sin θ and x = sin φ, one obtains Jacobis form. Equivalently, in terms of the amplitude and modular angle one has, in this notation, the use of a vertical bar as delimiter indicates that the argument following it is the parameter, while the backslash indicates that it is the modular angle. The use of a semicolon implies that the argument preceding it is the sine of the amplitude, with x = sn one has, F = u, thus, the Jacobian elliptic functions are inverses to the elliptic integrals. There are still other conventions for the notation of elliptic integrals employed in the literature, the notation with interchanged arguments, F, is often encountered, and similarly E for the integral of the second kind. e
32.
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
33.
Ursell number
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In fluid dynamics, the Ursell number indicates the nonlinearity of long surface gravity waves on a fluid layer. This dimensionless parameter is named after Fritz Ursell, who discussed its significance in 1953, so the Ursell parameter U is the relative wave height H / h times the relative wavelength λ / h squared. For long waves with small Ursell number, U ≪32 π2 /3 ≈100, otherwise a non-linear theory for fairly long waves – like the Korteweg–de Vries equation or Boussinesq equations – has to be used. The parameter, with different normalisation, was introduced by George Gabriel Stokes in his historical paper on surface gravity waves of 1847. Dingemans, M. W. Water wave propagation over uneven bottoms
34.
Potential flow
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In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function, the velocity potential. As a result, a flow is characterized by an irrotational velocity field. The irrotationality of a flow is due to the curl of the gradient of a scalar always being equal to zero. In the case of an incompressible flow the velocity potential satisfies Laplaces equation, however, potential flows also have been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows, applications of potential flow are for instance, the outer flow field for aerofoils, water waves, electroosmotic flow, and groundwater flow. For flows with strong vorticity effects, the potential flow approximation is not applicable, in fluid dynamics, a potential flow is described by means of a velocity potential φ, being a function of space and time. The flow velocity v is a field equal to the gradient, ∇. Sometimes, also the definition v = −∇φ, with a sign, is used. But here we use the definition above, without the minus sign. From vector calculus it is known, that the curl of a gradient is equal to zero, ∇ × ∇ φ =0, and consequently the vorticity and this implies that a potential flow is an irrotational flow. This has direct consequences for the applicability of potential flow, in flow regions where vorticity is known to be important, such as wakes and boundary layers, potential flow theory is not able to provide reasonable predictions of the flow. Fortunately, there are often large regions of a flow where the assumption of irrotationality is valid, for instance in, flow around aircraft, groundwater flow, acoustics, water waves, and electroosmotic flow. As a result, the velocity potential φ has to satisfy Laplaces equation ∇2 φ =0, in this case the flow can be determined completely from its kinematics, the assumptions of irrotationality and zero divergence of flow. Dynamics only have to be applied afterwards, if one is interested in computing pressures, in two dimensions, potential flow reduces to a very simple system that is analyzed using complex analysis. Potential flow theory can also be used to model irrotational compressible flow, the flow velocity v is again equal to ∇Φ, with Φ the velocity potential. The full potential equation is valid for sub-, trans- and supersonic flow at arbitrary angle of attack and this linear equation is much easier to solve than the full potential equation, it may be recast into Laplaces equation by a simple coordinate stretching in the x-direction. Small-amplitude sound waves can be approximated with the following model, ∂2 φ ∂ t 2 = a ¯2 Δ φ. Note that also the parts of the pressure p and density ρ each individually satisfy the wave equation
35.
Elevation
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GIS or geographic information system is a computer system that allows for visualizing, manipulating, capturing, and storage of data with associated attributes. GIS offers better understanding of patterns and relationships of the landscape at different scales, tools inside the GIS allow for manipulation of data for spatial analysis or cartography. A topographical map is the type of map used to depict elevation. In a Geographic Information System, digital models are commonly used to represent the surface of a place. Digital terrain models are another way to represent terrain in GIS, USGS is developing a 3D Elevation Program to keep up with growing needs for high quality topographic data. 3DEP is a collection of enhanced elevation data in the form of high quality LiDAR data over the conterminous United States, Hawaii, there are three bare earth DEM layers in 3DEP which are nationally seamless at the resolution of 1/3,1, and 2 arcseconds. This map is derived from GTOPO30 data that describes the elevation of Earths terrain at intervals of 30 arcseconds and it uses color and shading instead of contour lines to indicate elevation. Hypsography is the study of the distribution of elevations on the surface of the Earth, the term originates from the Greek word ὕψος hypsos meaning height. Most often it is used only in reference to elevation of land, related to the term hypsometry, the measurement of these elevations of a planets solid surface are taken relative to mean datum, except for Earth which is taken relative to the sea level. In the troposphere, temperatures decrease with altitude and this lapse rate is approximately 6.5 °C/km. S
36.
Partial derivative
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In mathematics, the symmetry of second derivatives refers to the possibility under certain conditions of interchanging the order of taking partial derivatives of a function f of n variables. This is sometimes known as Schwarzs theorem or Youngs theorem, in the context of partial differential equations it is called the Schwarz integrability condition. This matrix of partial derivatives of f is called the Hessian matrix of f. The entries in it off the diagonal are the mixed derivatives. In most real-life circumstances the Hessian matrix is symmetric, although there are a number of functions that do not have this property. Mathematical analysis reveals that symmetry requires a hypothesis on f that goes further than simply stating the existence of the derivatives at a particular point. Schwarz theorem gives a sufficient condition on f for this to occur, in symbols, the symmetry says that, for example, ∂ ∂ x = ∂ ∂ y. This equality can also be written as ∂ x y f = ∂ y x f, alternatively, the symmetry can be written as an algebraic statement involving the differential operator Di which takes the partial derivative with respect to xi, Di. From this relation it follows that the ring of operators with constant coefficients. But one should naturally specify some domain for these operators and it is easy to check the symmetry as applied to monomials, so that one can take polynomials in the xi as a domain. In fact smooth functions are possible, the partial differentiations of this function are commutative at that point. One easy way to establish this theorem is by applying Greens theorem to the gradient of f, a weaker condition than the continuity of second partial derivatives which nevertheless suffices to ensure symmetry is that all partial derivatives are themselves differentiable. The theory of distributions eliminates analytic problems with the symmetry, the derivative of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of integration by parts to define differentiation of distributions puts the symmetry question back onto the test functions. In more detail, = − = f = f = − =, another approach, which defines the Fourier transform of a function, is to note that on such transforms partial derivatives become multiplication operators that commute much more obviously. The symmetry may be if the function fails to have differentiable partial derivatives. An example of non-symmetry is the function, This function is everywhere continuous, however, the second partial derivatives are not continuous at, and the symmetry fails. In fact, along the x-axis the y-derivative is ∂ y f | = x, vice versa, along the y-axis the x-derivative ∂ x f | = − y, and so ∂ y ∂ x f | = −1
37.
Tilde
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The tilde is a grapheme with several uses. The name of the character came into English from Spanish, which in came from the Latin titulus. The reason for the name was that it was written over a letter as a scribal abbreviation, as a mark of suspension. Thus the commonly used words Anno Domini were frequently abbreviated to Ao Dñi, such a mark could denote the omission of one letter or several letters. This saved on the expense of the labour and the cost of vellum. Medieval European charters written in Latin are largely made up of such abbreviated words with suspension marks and other abbreviations, the tilde has since been applied to a number of other uses as a diacritic mark or a character in its own right. These are encoded in Unicode at U+0303 ◌̃ Combining Tilde and U+007E ~ Tilde, in lexicography, the latter kind of tilde and the swung dash are used in dictionaries to indicate the omission of the entry word. This symbol informally means approximately, about, or around, such as ~30 minutes before and it can mean similar to, including of the same order of magnitude as, such as, x ~ y meaning that x and y are of the same order of magnitude. The tilde is used to indicate equal to or approximately equal to by placing it over the = symbol, like so. The text of the Domesday Book of 1086, relating for example, the text with abbreviations expanded is as follows, Mollande tempore regis Edwardi geldabat pro iiii hidis et uno ferling. In dominio sunt iii carucae et x servi et xxx villani et xx bordarii cum xvi carucis, ibi xii acrae prati et xv acrae silvae. Pastura iii leugae in longitudine et latitudine, elwardus tenebat tempore regis Edwardi pro manerio et geldabat pro dimidia hida. Ibi sunt v villani cum i servo, valet xx solidos ad pensam et arsuram. Eidem manerio est injuste adjuncta Nimete et valet xv solidos, ipsi manerio pertinet tercius denarius de Hundredis Nortmoltone et Badentone et Brantone et tercium animal pasturae morarum. The incorporation of the tilde into ASCII is a result of its appearance as a distinct character on mechanical typewriters in the late nineteenth century. Any good typewriter store had a catalog of alternative keyboards that could be specified for machines ordered from the factory, at that time, the tilde was used only in Spanish and Portuguese typewriters. In Modern Spanish, the tilde is used only with n and N, both were conveniently assigned to a single mechanical typebar, which sacrificed a key that was felt to be less important, usually the 1⁄2— 1⁄4 key. Portuguese, however, uses not ñ but nh and it uses the tilde on the vowels a and o
38.
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
39.
Derivative
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The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small
40.
Parameter
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A parameter, generally, is any characteristic that can help in defining or classifying a particular system. That is, a parameter is an element of a system that is useful, or critical, parameter has more specific meanings within various disciplines, including mathematics, computing and computer programming, engineering, statistics, logic and linguistics. Mathematical functions have one or more arguments that are designated in the definition by variables, a function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that the function takes. When parameters are present, the definition actually defines a family of functions. A parameter could be incorporated into the name to indicate its dependence on the parameter. For instance, one may define the base b of a logarithm by log b = log log where b is a parameter that indicates which logarithmic function is being used. It is not an argument of the function, and will, for instance, in some informal situations it is a matter of convention whether some or all of the symbols in a function definition are called parameters. However, changing the status of symbols between parameter and variable changes the function as a mathematical object, for instance, the notation for the falling factorial power n k _ = n ⋯, defines a polynomial function of n, but is not a polynomial function of k. Indeed, in the case, it is only defined for non-negative integer arguments. Sometimes it is useful to all functions with certain parameters as parametric family. Examples from probability theory are given further below, a variable is one of the many things a parameter is not. The dependent variable, the speed of the car, depends on the independent variable, change the lever arms of the linkage. Will still depend on the pedal position and you have changed a parameter A parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied and these parameters each describe some aspect of the response curve seen as a whole, over all frequencies. A graphic equaliser provides individual level controls for various frequency bands, if asked to imagine the graph of the relationship y = ax2, one typically visualizes a range of values of x, but only one value of a. Of course a different value of a can be used, generating a different relation between x and y, thus a is a parameter, it is less variable than the variable x or y, but it is not an explicit constant like the exponent 2. More precisely, changing the parameter a gives a different problem, in calculating income based on wage and hours worked, it is typically assumed that the number of hours worked is easily changed, but the wage is more static
41.
Ordinary differential equation
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In mathematics, an ordinary differential equation is a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the partial differential equation which may be with respect to more than one independent variable. ODEs that are linear equations have exact closed-form solutions that can be added and multiplied by coefficients. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, Ordinary differential equations arise in many contexts of mathematics and science. Mathematical descriptions of change use differentials and derivatives, often, quantities are defined as the rate of change of other quantities, or gradients of quantities, which is how they enter differential equations. Specific mathematical fields include geometry and analytical mechanics, scientific fields include much of physics and astronomy, meteorology, chemistry, biology, ecology and population modelling, economics. Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, dAlembert, in general, F is a function of the position x of the particle at time t. The unknown function x appears on both sides of the equation, and is indicated in the notation F. In what follows, let y be a dependent variable and x an independent variable, the notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. Given F, a function of x, y, and derivatives of y, then an equation of the form F = y is called an explicit ordinary differential equation of order n. The function r is called the term, leading to two further important classifications, Homogeneous If r =0, and consequently one automatic solution is the trivial solution. The solution of a homogeneous equation is a complementary function. The additional solution to the function is the particular integral. The general solution to an equation can be written as y = yc + yp. Non-linear A differential equation that cannot be written in the form of a linear combination, a number of coupled differential equations form a system of equations. In column vector form, = These are not necessarily linear, the implicit analogue is, F =0 where 0 = is the zero vector. In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations and this distinction is not merely one of terminology, DAEs have fundamentally different characteristics and are generally more involved to solve than ODE systems. Given a differential equation F =0 a function u, I ⊂ R → R is called the solution or integral curve for F, if u is n-times differentiable on I, and F =0 x ∈ I
42.
Integral
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed