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Nonlinear Tides
Nonlinear tides are generated by hydrodynamic distortions of tides. A tidal wave is said to be nonlinear when its shape deviates from a pure sinusoidal wave. In mathematical terms, the wave owes its nonlinearity due to the nonlinear advection and frictional terms in the governing equations. These become more important in shallow-water regions such as in estuaries. Nonlinear tides are studied in the fields of coastal morphodynamics, coastal engineering and physical oceanography. The nonlinearity of tides has important implications for the transport of sediment. Framework From a mathematical perspective, the nonlinearity of tides originates from the nonlinear terms present in the Navier-Stokes equations. In order to analyse tides, it is more practical to consider the depth-averaged shallow water equations:\frac+\frac D_0 + \eta )u\frac D_0 + \eta )v0,\frac+u\frac+v\frac=-g\frac-\frac, \frac+u\frac+v\frac=-g\frac-\frac. Here, u and v are the zonal (x ) and meridional (y ) flo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diurnal Cycle
A diurnal cycle (or diel cycle) is any pattern that recurs every 24 hours as a result of one full rotation of the planet Earth around its axis. Earth's rotation causes surface temperature fluctuations throughout the day and night, as well as weather changes throughout the year. The diurnal cycle depends mainly on incoming solar radiation. Climate and atmosphere In climatology, the diurnal cycle is one of the most basic forms of climate patterns, including variations in diurnal temperature and rainfall. Diurnal cycles may be approximately sinusoidal or include components of a truncated sinusoid (due to the Sun's rising and setting) and thermal relaxation (Newton cooling) at night. The diurnal cycle also has a great impact on carbon dioxide levels in the atmosphere, due to processes such as photosynthesis and cellular respiration. Biological effects Diurnal cycles of light and temperature can result in similar cycles in biological processes, such as photosynthesis in pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation which is much easier to solve and also valid for inhomogenious media. Introduction The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as Superposition_principle, linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in Scalar field, scalars by scalar functions of a tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temporal frequency, which is defined as the number of wave cycles per unit time (''ordinary frequency'') or radians per unit time (''angular frequency''). In multidimensional systems, the wavenumber is the magnitude of the '' wave vector''. The space of wave vectors is called '' reciprocal space''. Wave numbers and wave vectors play an essential role in optics and the physics of wave scattering, such as X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics. For quantum mechanical waves, the wavenumber multiplied by the reduced Planck's constant is the '' canonical momentum''. Wavenumber can be used to specify quantities other than spatial frequency. For example, in optical spectroscopy, it is of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit time (for example, in rotation) or the rate of change of the phase of a sinusoidal waveform (for example, in oscillations and waves), or as the rate of change of the argument of the sine function. Angular frequency (or angular speed) is the magnitude of the pseudovector quantity angular velocity.(UP1) One turn is equal to 2''π'' radians, hence \omega = \frac = , where: *''ω'' is the angular frequency (unit: radians per second), *''T'' is the period (unit: seconds), *''f'' is the ordinary frequency (unit: hertz) (sometimes ''ν''). Units In SI units, angular frequency is normally presented in radians per second, even when it does not express a rotational value. The unit hertz (Hz) is dimensionally equivalent, but by conventi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. In older texts, the phase of a periodic function is sometimes called the amplitude. Definitions Peak amplitude & semi-amplitude For symmetric periodic waves, like sine waves, square waves or triangle waves ''peak amplitude'' and ''semi amplitude'' are the same. Peak amplitude In audio system measurements, telecommunications and others where the measurand is a signal that swings above and below a reference value but is not sinusoidal, peak amplitude is often used. If the reference is zero, this is the maximum absolute value of the signal; if the reference is a mean value (DC component), the peak amplitude is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nondimensionalization
Nondimensionalization is the partial or full removal of dimensional analysis, physical dimensions from an mathematical equation, equation involving physical quantity, physical quantities by a suitable substitution of variables. This technique can simplify and parametric equation, parameterize problems where Measurement, 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 Wiktionary:intrinsic, intrinsic to the system, rather than units such as International System of Units, SI units. Nondimensionalization is not the same as converting intensive and extensive properties, extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units. Nondimensionalization can also r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Partial Differential Equation
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem. The distinction between a linear and a nonlinear partial differential equation is usually made in terms of the properties of the operator that defines the PDE itself. Methods for studying nonlinear partial differential equations Existence and uniqueness of solutions A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value. Physical interpretation of divergence In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its "outgoingness" – the extent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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