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Spacetime Triangle Diagram Technique
In physics and mathematics, the spacetime triangle diagram (STTD) technique, also known as the Smirnov method of incomplete separation of variables, is the direct space-time domain method for electromagnetic and scalar wave motion. Basic stages # (Electromagnetics) The system of Maxwell's equations is reduced to a second-order PDE for the field components, or potentials, or their derivatives. # The spatial variables are separated using convenient expansions into series and/or integral transforms—except one that remains bounded with the time variable, resulting in a PDE of hyperbolic type. # The resulting hyperbolic PDE and the simultaneously transformed initial conditions compose a problem, which is solved using thRiemann–Volterra integral formula This yields the generic solution expressed via a double integral over a triangle domain in the bounded-coordinate—time space. Then this domain is replaced by a more complicated but smaller one, in which the integrant is essentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Functions
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where a and q are parameters. They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads.Morse and Feshbach (1953).Brimacombe, Corless and Zamir (2021) They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation boundary value problems possessing elliptic symmetry.Gutiérrez-Vega (2015). Definition Mathieu functions In some usages, ''Mathieu function'' refers to solutions of the Mathieu differential equation for arbitrary values of a and q. When no confusion can arise, other authors use the term to refer specifically to \pi- or 2\pi-periodic solutions, which exist only for special v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-wave
In physics, X-waves are localized solutions of the wave equation that travel at a constant velocity in a given direction. X-waves can be sound, electromagnetic, or gravitational waves. They are built as a non-monochromatic superposition of Bessel beams. Ideal X-waves carry infinite energy, but finite-energy realizations have been observed in various frameworks. Electromagnetic X-waves travel faster than the speed of light, and X-wave pulses can have superluminal phase and group velocity. In optics, X-waves solution have been reported within a quantum mechanical formulation.A. Ciattoni and C. ContiQuantum electromagnetic X-waves''arxiv.org'' 0704.0442v1. See also * Nonlinear X-wave * Droplet-shaped wave References * J. Lu and J. F. Greenleaf, "Nondiffracting X waves: exact solutions to free-space scalar wave equation and their infinite realizations", IEEE Trans. Ultrasonic Ferroelectric Frequency. ''Control Control may refer to: Basic meanings Economics and business * Con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F140910a STTDmethod F RotatedDomain
The Grumman F-14 Tomcat is an American carrier-capable supersonic aircraft, supersonic, twinjet, twin-engine, two-seat, twin-tail, variable-sweep wing fighter aircraft. The Tomcat was developed for the United States Navy's Naval Fighter Experimental (VFX) program after the collapse of the General Dynamics–Grumman F-111B, General Dynamics-Grumman F-111B project. The F-14 was the first of the American Teen Series fighters, which were designed incorporating aerial warfare, air combat experience against Mikoyan, MiG fighters during the Vietnam War. The F-14 first flew on 21 December 1970 and made its first deployment in 1974 with the U.S. Navy aboard , replacing the McDonnell Douglas F-4 Phantom II. The F-14 served as the U.S. Navy's primary maritime air superiority fighter, fleet defense interceptor aircraft, interceptor, and tactical aerial reconnaissance platform into the 2000s. The Low Altitude Navigation and Targeting Infrared for Night (LANTIRN) pod system was added in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press, Inc
An academy ( Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosophy, founded approximately 385 BC at Akademia, a sanctuary of Athena, the goddess of wisdom and skill, north of Athens, Greece. Etymology The word comes from the ''Academy'' in ancient Greece, which derives from the Athenian hero, ''Akademos''. Outside the city walls of Athens, the gymnasium was made famous by Plato as a center of learning. The sacred space, dedicated to the goddess of wisdom, Athena, had formerly been an olive grove, hence the expression "the groves of Academe". In these gardens, the philosopher Plato conversed with followers. Plato developed his sessions into a method of teaching philosophy and in 387 BC, established what is known today as the Old Academy. By extension, ''academia'' has come to mean the accumulation, de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradshteyn And Ryzhik
''Gradshteyn and Ryzhik'' (''GR'') is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is ''Table of Integrals, Series, and Products''. Since its first publication in 1943, it was considerably expanded and it soon became a "classic" and highly regarded reference for mathematicians, scientists and engineers. After the deaths of the original authors, the work was maintained and further expanded by other editors. At some stage a German and English dual-language translation became available, followed by Polish, English-only and Japanese versions. After several further editions, the Russian and German-English versions went out of print and have not been updated after the fall of the Iron Curtain, but the English version is still being actively maintained and refined by new editors, and it has recently been retranslated back into Russian as well. Overview One of the valuab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Polynomial
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions. Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval 1,1/math>. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional standardization condi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Poisson–Darboux Equation
In mathematics, the Euler–Poisson–Darboux equation is the partial differential equation : u_+\frac=0. This equation is named for Siméon Poisson, Leonhard Euler, and Gaston Darboux. It plays an important role in solving the classical wave equation. This equation is related to : u_+\fracu_r-u_=0, by x=r+t , y=r-t , where N=\frac and some sources quote this equation when referring to the Euler–Poisson–Darboux equation. References External links * Differential calculus Partial differential equations Leonhard Euler {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Legendre Polynomial
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently \frac \left \left(1 - x^2\right) \frac P_\ell^m(x) \right+ \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, where the indices ''ℓ'' and ''m'' (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if ''ℓ'' and ''m'' are integers with 0 ≤ ''m'' ≤ ''ℓ'', or with trivially equivalent negative values. When in addition ''m'' is even, the function is a polynomial. When ''m'' is zero and ''ℓ'' integer, these functions are identical to the Legendre polynomials. In general, when ''ℓ'' and ''m'' are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Considerations
A general officer is an officer of high rank in the armies, and in some nations' air forces, space forces, and marines or naval infantry. In some usages the term "general officer" refers to a rank above colonel."general, adj. and n.". OED Online. March 2021. Oxford University Press. https://www.oed.com/view/Entry/77489?rskey=dCKrg4&result=1 (accessed May 11, 2021) The term ''general'' is used in two ways: as the generic title for all grades of general officer and as a specific rank. It originates in the 16th century, as a shortening of '' captain general'', which rank was taken from Middle French ''capitaine général''. The adjective ''general'' had been affixed to officer designations since the late medieval period to indicate relative superiority or an extended jurisdiction. Today, the title of ''general'' is known in some countries as a four-star rank. However, different countries use different systems of stars or other insignia for senior ranks. It has a NATO rank ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Coordinate System
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measured from a fixed zenith direction, and the '' azimuthal angle'' of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is also called the ''radius'' or ''radial coordinate''. The polar angle may be called '' colatitude'', '' zenith angle'', '' normal angle'', or '' inclination angle''. When radius is fixed, the two angular coordinates make a coordinate system on the sphere sometimes called spherical polar coordinates. The use of symbols and the order of the coordinates differs among sources and disciplines. This articl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heaviside Step Function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive arguments. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Oliver Heaviside, who developed the operational calculus as a tool in the analysis of telegraphic communications, represented the function as . The Heaviside function may be defined as: * a piecewise function: H(x) := \begin 1, & x > 0 \\ 0, & x \le 0 \end * using the Iverson bracket notation: H(x) := 0.html" ;"title=">0">>0/math> * an indicator function: H(x) := \mathbf_=\mathbf 1_(x) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
