Helmholtz Equation

The Helmholtz equation, named for Hermann von Helmholtz, is the partial differential equation

where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude.

Read more about Helmholtz Equation:  Motivation and Uses, Solving The Helmholtz Equation Using Separation of Variables, Paraxial Approximation, Inhomogeneous Helmholtz Equation

Other articles related to "helmholtz equation, equation, equations":

Inhomogeneous Helmholtz Equation
... The inhomogeneous Helmholtz equation is the equation where ƒ Rn → C is a given function with compact support, and n = 1, 2, 3 ... This equation is very similar to the screened Poisson equation, and would be identical if the plus sign (in front of the k term) is switched to a minus sign ... In order to solve this equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition uniformly in with ...
Beam Propagation Method - Principles
... BPM is generally formulated as a solution to Helmholtz equation in a time-harmonic case, with the field written as ... the envelope following a slowly varying approximation, Now the solution when replaced into the Helmholtz equation follows, With the aim to calculate the field at all points of space for all times, we only need to ... Since the solution is for the time-harmonic Helmholtz equation, we only need to calculate it over one time period ...
Sommerfeld Radiation Condition
... radiation for a scalar field satisfying the Helmholtz equation as "the sources must be sources, not sinks of energy ... the field." Mathematically, consider the inhomogeneous Helmholtz equation where is the dimension of the space, is a given function with compact support representing a ... A solution to this equation is called radiating if it satisfies the Sommerfeld radiation condition uniformly in all directions (above, is the imaginary unit and is the ...
Prolate Spheroidal Wave Functions
... When solving the Helmholtz equation, by the method of separation of variables in prolate spheroidal coordinates, with and ... satisfies the linear ordinary differential equation The eigenvalue of this Sturm-Liouville differential equation is fixed by the requirement that must be finite for ... The angular wavefunction satisfies the differential equation It is the same differential equation as in the case of the radial wavefunction ...
The Plane Wave Spectrum: The Foundation of Fourier Optics - Solving The Helmholtz Equation: Separation of Variables and Elementary Product Solutions
... Solutions to the Helmholtz equation (2.0) may readily be found in rectangular coordinates via the principle of separation of variables for partial differential equations ... orthogonal coordinates, an elementary product solution to this wave equation may be constructed of the following form i.e ... If this elementary product solution is substituted into the wave equation (2.0), using the scalar Laplacian in rectangular coordinates then the following equation ...

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