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Fornberg–Whitham Equation
In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves. The equation is notated as follows:This integro-differential equation for the oscillatory variable ''η''(''x'',''t'') is named after Gerald Whitham who introduced it as a model to study breaking of non-linear dispersive water waves in 1967. Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven. For a certain choice of the kernel ''K''(''x'' − ''ξ'') it becomes the Fornberg–Whitham equation. Water waves Using the Fourier transform (and its inverse), with respect to the space coordinate ''x'' and in terms of the wavenumber ''k'': * For surface gravity waves, the phase speed ''c''(''k'') as a function of wavenumber ''k'' is taken as: :: c_\text(k) = \sqrt, while \alpha_\text = \frac \sqrt, :with ''g'' the gravitational acceleration and ''h'' the mean water depth. The associated kernel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, an average might be another statistic such as the median, or mode. For example, the average personal income is often given as the median—the number below which are 50% of personal incomes and above which are 50% of personal incomes—because the mean would be higher by including personal incomes from a few billionaires. For this reason, it is recommended to avoid using the word "average" when discussing measures of central tendency. General properties If all numbers in a list are the same number, then their average is also equal to this number. This property is shared by each of the many types of average. Another universal property is monotonicity: if two lists of numbers ''A'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Water Waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of the wind is known as the ''fetch''. Waves in the oceans can travel thousands of kilometers before reaching land. Wind waves on Earth range in size from small ripples, to waves over high, being limited by wind speed, duration, fetch, and water depth. When directly generated and affected by local wind, a wind wave system is called a wind sea. Wind waves will travel in a great circle route after being generated – curving slightly left in the southern hemisphere and slightly right in the northern hemisphere. After moving out of the area of fetch, wind waves are called '' swells'' and can travel thousands of kilometers. A noteworthy example of this is waves generated south of Tasmania during heavy winds that will travel across the Pacif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The Royal Society A
''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905: * Series A: for papers in physical sciences and mathematics. * Series B: for papers in life sciences. Many landmark scientific discoveries are published in the Proceedings, making it one of the most historically significant science journals. The journal contains several articles written by the most celebrated names in science, such as Paul Dirac, Werner Heisenberg, Ernest Rutherford, Erwin Schrödinger, William Lawrence Bragg, Lord Kelvin, J.J. Thomson, James Clerk Maxwell, Dorothy Hodgkin and Stephen Hawking. In 2004, the Royal Society began ''The Journal of the Royal Society Interface'' for papers at the interface of physical sciences and life sciences. History The journal began in 1831 as a compilation of abstracts of papers in the ''Philosophical Transactions of the Royal Society'', the older Royal Society publication, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Philosophical Transactions Of The Royal Society A
''Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences'' is a fortnightly peer-reviewed scientific journal published by the Royal Society. It publishes original research and review content in a wide range of physical scientific disciplines. Articles can be accessed online a few months prior to the printed journal. All articles become freely accessible two years after their publication date. The current editor-in-chief is John Dainton. Overview ''Philosophical Transactions of the Royal Society A'' publishes themed journal issues on topics of current scientific importance and general interest within the physical, mathematical and engineering sciences, edited by leading authorities and comprising original research, reviews and opinions from prominent researchers. Past issue titles include "Supercritical fluids - green solvents for green chemistry?", "Tsunamis: Bridging science, engineering and society", "Spatial transformations: from f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shock Wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in pressure, temperature, and density of the medium. For the purpose of comparison, in supersonic flows, additional increased expansion may be achieved through an expansion fan, also known as a Prandtl–Meyer expansion fan. The accompanying expansion wave may approach and eventually collide and recombine with the shock wave, creating a process of destructive interference. The sonic boom associated with the passage of a supersonic aircraft is a type of sound wave produced by constructive interference. Unlike solitons (another kind of nonlinear wave), the energy and speed of a shock wave alone dissipates relatively quickly with distance. When a shock wave passes through ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peakon
In the theory of integrable systems, a peakon ("peaked soliton") is a soliton with discontinuous first derivative; the wave profile is shaped like the graph of the function e^. Some examples of non-linear partial differential equations with (multi-)peakon solutions are the Camassa–Holm shallow water wave equation, the Degasperis–Procesi equation and the Fornberg–Whitham equation. Since peakon solutions are only piecewise differentiable, they must be interpreted in a suitable weak sense. The concept was introduced in 1993 by Camassa and Holm in the short but much cited paper where they derived their shallow water equation. A family of equations with peakon solutions The primary example of a PDE which supports peakon solutions is : u_t - u_ + (b+1) u u_x = b u_x u_ + u u_, \, where u(x,t) is the unknown function, and ''b'' is a parameter. In terms of the auxiliary function m(x,t) defined by the relation m = u-u_, the equation takes the simpler form : m_t + m_x u + b m u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nondimensionalization
Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term scaling is used interchangeably with ''nondimensionalization'', in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units. Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic resonance frequency, length, or time constant, nondimensionalization can recover these val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bengt Fornberg
Bengt may refer to: People In arts, entertainment and media Actors * Bengt Djurberg (1898–1941), Swedish actor and singer * Bengt Ekerot (1920–1971), Swedish actor and director * Bengt Eklund (1925–1998), Swedish actor * Bengt Logardt (1914–1994), Swedish actor, screenwriter and film director * Bengt Nilsson (actor) (born 1954), Swedish actor Journalists and writers * Bengt Feldreich (1925-2019), Swedish journalist and teacher * Bengt Frithiofsson (born 1939), Swedish wine writer * Bengt Lidner (1757–1793), Swedish poet * Bengt Linder (1929–1985), Swedish writer and journalist * Bengt Magnusson (born 1950), Swedish journalist and a TV presenter * Bengt Pohjanen (born 1944), Swedish author, translator and priest In music * Bengt Berger (born 1942), Swedish jazz drummer, composer and producer * Bengt Calmeyer, Swedish musician in the band Turbonegro * Bengt Djurberg (1898–1941), Swedish actor and singer * Bengt Forsberg (born 1952), Swedish concert pianist * Bengt H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., f(x)) to its value at zero of its domain (f(0)), or as the weak limit of a sequence of bump functions (e.g., \delta(x) = \lim_ \frace^), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions. The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. Its validity was disputed until Laurent Schwartz developed the theory of distributions where it is defined as a linear form acting on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long Wave
In radio, longwave, long wave or long-wave, and commonly abbreviated LW, refers to parts of the radio spectrum with wavelengths longer than what was originally called the medium-wave broadcasting band. The term is historic, dating from the early 20th century, when the radio spectrum was considered to consist of longwave (LW), medium-wave (MW), and short-wave (SW) radio bands. Most modern radio systems and devices use wavelengths which would then have been considered 'ultra-short'. In contemporary usage, the term ''longwave'' is not defined precisely, and its intended meaning varies. It may be used for radio wavelengths longer than 1,000 m i.e. frequencies up to 300 kilohertz (kHz), including the International Telecommunication Union's (ITU's) low frequency (LF, 30–300 kHz) and very low frequency (VLF, 3–30 kHz) bands. Sometimes the upper limit is taken to be higher than 300 kHz, but not above the start of the medium wave broadcast band at 520&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
