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Precursor (physics)
Precursors are characteristic wave patterns caused by dispersion of an impulse's frequency components as it propagates through a medium. Classically, precursors precede the main signal, although in certain situations they may also follow it. Precursor phenomena exist for all types of waves, as their appearance is only predicated on the prominence of dispersion effects in a given mode of wave propagation. This non-specificity has been confirmed by the observation of precursor patterns in different types of electromagnetic radiation (microwaves, visible light, and terahertz radiation) as well as in fluid surface waves and seismic waves. History Precursors were first theoretically predicted in 1914 by Arnold Sommerfeld for the case of electromagnetic radiation propagating through a neutral dielectric in a region of normal dispersion.See L. Brillouin, ''Wave Propagation and Group Velocity'' (Academic Press, New York, NY, 1960), Ch. 1. Sommerfeld's work was expanded in the following ye ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dispersion Relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the phase velocity and group velocity of waves in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency dependence of wave propagation and attenuation. Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation even in the absence of geometric constraints and other media. In the presence of dispersion, wave velocity is no longer uniquely defined, giving rise to the distinction of phase ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Phase Approximation
In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to the limit as k \to \infty . This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin. It is closely related to Laplace's method and the method of steepest descent, but Laplace's contribution precedes the others. Basics The main idea of stationary phase methods relies on the cancellation of sinusoids with rapidly varying phase. If many sinusoids have the same phase and they are added together, they will add constructively. If, however, these same sinusoids have phases which change rapidly as the frequency changes, they will add incoherently, varying between constructive and destructive addition at different times. Formula Letting \Sigma denote the set of critical points of the function f (i.e. points where \nabla f =0), under the assumption that g is either compactly supported or has exponential decay, and that all critical poi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Retarded Time
In electromagnetism, electromagnetic waves in vacuum travel at the speed of light ''c'', according to Maxwell's Equations. The retarded time is the time when the field began to propagate from the point where it was emitted to an observer. The term "retarded" is used in this context (and the literature) in the sense of propagation delays. Retarded and advanced times The calculation of the retarded time ''tr'' or ''t''′ is nothing more than a simple " speed-distance-time" calculation for EM fields. If the EM field is radiated at position vector r′ (within the source charge distribution), and an observer at position r measures the EM field at time ''t'', the time delay for the field to travel from the charge distribution to the observer is , r − r′, /''c'', so subtracting this delay from the observer's time ''t'' gives the time when the field ''actually began to propagate'' - the retarded time, ''t''′. The retarded time is: t' = t - \frac which can be rearrang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric constant, or the distributed capacitance of the vacuum. It is an ideal (baseline) physical constant. Its CODATA value is: : ( farads per meter), with a relative uncertainty of It is a measure of how dense of an electric field is "permitted" to form in response to electric charges, and relates the units for electric charge to mechanical quantities such as length and force. For example, the force between two separated electric charges with spherical symmetry (in the vacuum of classical electromagnetism) is given by Coulomb's law: :F_\text = \frac \frac Here, ''q''1 and ''q''2 are the charges, ''r'' is the distance between their centres, and the value of the constant fraction 1/4 \pi \varepsilon_0 (known as the Coulomb constant, ''k''e) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to ''quadrature'') is more or less a synonym for ''numerical integration'', especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take ''quadrature'' to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Form (calculus)
In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another differential form ''β''. Thus, an ''exact'' form is in the '' image'' of ''d'', and a ''closed'' form is in the ''kernel'' of ''d''. For an exact form ''α'', for some differential form ''β'' of degree one less than that of ''α''. The form ''β'' is called a "potential form" or "primitive" for ''α''. Since the exterior derivative of a closed form is zero, ''β'' is not unique, but can be modified by the addition of any closed form of degree one less than that of ''α''. Because , every exact form is necessarily closed. The question of whether ''every'' closed form is exact depends on the topology of the domain of interest. On a contractible domain, every closed form is exact by the Poincaré lemma. More general questions of this ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second. If this wavelet were to be convolved with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the Middle C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar. Correlation is at the core of many practical wavelet applications. As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including but not limited to au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Point Approximation
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals. The integral to be estimated is often of the form :\int_Cf(z)e^\,dz, where ''C'' is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration ''C'' into a new path integration ''C′'' so that the following conditions hold: # ''C′'' passes through one or more zeros of the derivative ''g''′(''z''), # the imaginary part of ''g''(''z'') is constant on ''C′''. The method of steepest descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetic Radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, Light, (visible) light, ultraviolet, X-rays, and gamma rays. All of these waves form part of the electromagnetic spectrum. Classical electromagnetism, Classically, electromagnetic radiation consists of electromagnetic waves, which are synchronized oscillations of electric field, electric and magnetic fields. Depending on the frequency of oscillation, different wavelengths of electromagnetic spectrum are produced. In a vacuum, electromagnetic waves travel at the speed of light, commonly denoted ''c''. In homogeneous, isotropic media, the oscillations of the two fields are perpendicular to each other and perpendicular to the direction of energy and wave propagation, forming a transverse wave. The position of an electromagnetic wave w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Léon Brillouin
Léon Nicolas Brillouin (; August 7, 1889 – October 4, 1969) was a French physicist. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid state physics, and information theory. Early life Brillouin was born in Sèvres, near Paris, France. His father, Marcel Brillouin, grandfather, Éleuthère Mascart, and great-grandfather, Charles Briot, were physicists as well. Education From 1908 to 1912, Brillouin studied physics at the École Normale Supérieure, in Paris. From 1911 he studied under Jean Perrin until he left for the Ludwig Maximilian University of Munich (LMU), in 1912. At LMU, he studied theoretical physics with Arnold Sommerfeld. Just a few months before Brillouin's arrival at LMU, Max von Laue had conducted his experiment showing X-ray diffraction in a crystal lattice. In 1913, he went back to France to study at the University of Paris and it was in this year that Niels Bohr submitted his first paper on the Bohr model of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
