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Spheroidal Harmonic
Spheroidal wave functions are solutions of the Helmholtz equation that are found by writing the equation in spheroidal coordinates and applying the technique of separation of variables, just like the use of spherical coordinates lead to spherical harmonics. They are called ''oblate spheroidal wave functions'' if oblate spheroidal coordinates are used and ''prolate spheroidal wave functions'' if prolate spheroidal coordinates are used. If instead of the Helmholtz equation, the Laplace equation is solved in spheroidal coordinates using the method of separation of variables, the spheroidal wave functions reduce to the spheroidal harmonics. With oblate spheroidal coordinates, the solutions are called ''oblate harmonics'' and with prolate spheroidal coordinates, ''prolate harmonics''. Both type of spheroidal harmonics are expressible in terms of Legendre functions. See also * Oblate spheroidal coordinates Oblate spheroidal coordinates are a three-dimensional orthogonal coordinates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenvalue, and is the (eigen)function. When the equation is applied to waves, is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences. Motivation and uses The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. For example, consider the wave equation \left(\nabla^2-\frac\frac\right) u(\mathbf,t)=0. Separation of variables begins by assumi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Of Variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Ordinary differential equations (ODE) Suppose a differential equation can be written in the form :\frac f(x) = g(x)h(f(x)) which we can write more simply by letting y = f(x): :\frac=g(x)h(y). As long as ''h''(''y'') ≠ 0, we can rearrange terms to obtain: : = g(x) \, dx, so that the two variables ''x'' and ''y'' have been separated. ''dx'' (and ''dy'') can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of ''dx'' as a differential (infinitesimal) is somewhat advanced. Alternative notation Those who dislike Leibniz's notation may prefer to write this as :\frac \frac = g(x), but that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Coordinates
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 article will us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oblate Spheroidal Coordinates
Oblate spheroidal coordinates are a three-dimensional orthogonal coordinates, orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinates, elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius a in the ''x''-''y'' plane. (Rotation about the other axis produces prolate spheroidal coordinates.) Oblate spheroidal coordinates can also be considered as a limiting case (mathematics), limiting case of ellipsoidal coordinates in which the two largest semi-axis, semi-axes are equal in length. Oblate spheroidal coordinates are often useful in solving partial differential equations when the boundary conditions are defined on an oblate spheroid or a hyperboloid, hyperboloid of revolution. For example, they played an important role in the calculation of the Perrin friction factors, which contributed to the awarding of the 1926 Nobel Pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prolate Spheroidal Wave Functions
The prolate spheroidal wave functions are eigenfunctions of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions on certain ellipsoids of revolution (an ellipse rotated around its long axis, “cigar shape“). Related are the oblate spheroidal wave functions (“pancake shaped” ellipsoid). Solutions to the wave equation Solve the Helmholtz equation, \nabla^2 \Phi + k^2 \Phi=0, by the method of separation of variables in prolate spheroidal coordinates, (\xi,\eta,\varphi), with: :\ x=a \sqrt \cos \varphi, :\ y=a \sqrt \sin \varphi, :\ z=a \, \xi \, \eta, and \xi \ge 1, , \eta, \le 1 , and 0 \le \varphi \le 2\pi. Here, 2a > 0 is the interfocal distance of the elliptical cross section of the prolate spheroid. Setting c=ka, the solution \Phi(\xi,\eta,\varphi) can be written as the product of e^, a radial spheroidal wave function R_(c,\xi) and an angular spheroidal wave function S_(c,\eta). The radial wave function R_(c,\xi) satisfies the line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prolate Spheroidal Coordinates
Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length. Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the ''z''-axis. One example is solving for the wavefunction of an electron moving in the electromagnetic field of two positively charged nuclei, as in the hydrogen molecular ion, H2+. Another example is solving for the electric field generated by two small electro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h. This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Functions
In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles. Legendre's differential equation The general Legendre equation reads \left(1 - x^2\right) y'' - 2xy' + \left lambda(\lambda+1) - \frac\righty = 0, where the numbers and may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when is an integer (denoted ), and are the Legendre polynomials ; and when is an integer (denoted ), and is also an integer with are the associated Legendre polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oblate Spheroidal Coordinates
Oblate spheroidal coordinates are a three-dimensional orthogonal coordinates, orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinates, elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius a in the ''x''-''y'' plane. (Rotation about the other axis produces prolate spheroidal coordinates.) Oblate spheroidal coordinates can also be considered as a limiting case (mathematics), limiting case of ellipsoidal coordinates in which the two largest semi-axis, semi-axes are equal in length. Oblate spheroidal coordinates are often useful in solving partial differential equations when the boundary conditions are defined on an oblate spheroid or a hyperboloid, hyperboloid of revolution. For example, they played an important role in the calculation of the Perrin friction factors, which contributed to the awarding of the 1926 Nobel Pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oblate Spheroidal Wave Function
In applied mathematics, oblate spheroidal wave functions (like also prolate spheroidal wave functions and other related functions) are involved in the solution of the Helmholtz equation in oblate spheroidal coordinates. When solving this equation, \Delta \Phi + k^2 \Phi=0, by the method of separation of variables, (\xi,\eta,\varphi), with: :\ z=(d/2) \xi \eta, :\ x=(d/2) \sqrt \cos \varphi, :\ y=(d/2) \sqrt \sin \varphi, :\ \xi \ge 0 \text , \eta, \le 1. the solution \Phi(\xi,\eta,\varphi) can be written as the product of a radial spheroidal wave function R_(-i c,i \xi) and an angular spheroidal wave function S_(-i c,\eta) by e^. Here c=kd/2, with d being the interfocal length of the elliptical cross section of the oblate spheroid. The radial wave function R_(-i c,i \xi) satisfies the linear ordinary differential equation: :\ (\xi^2 +1) \frac + 2\xi \frac -\left(\lambda_(c) -c^2 \xi^2 -\frac\right) = 0 . The angular wave function satisfies the differential equation: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
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 e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
