PlanetMath: Helmholtz equation

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Helmholtz equation

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The Helmholtz equation is a partial differential equation which, in scalar form is

$\displaystyle \nabla ^2f+k^2f = 0,$

or in vector form is

$\displaystyle \nabla ^2\mathbf{A}+k^2\mathbf{A}= 0,$

where $\nabla ^2$ is the Laplacian. The solutions of this equation represent the solution of the wave equation, which is of great interest in physics.

Consider a wave equation

$\displaystyle \frac{\partial^2\psi}{\partial t^2} = c^2\nabla ^2\psi$

with wave speed

. If we look for time harmonic standing waves of frequency $\omega$ ,

$\displaystyle \psi(\mathbf{x},t) = e^{-j\omega t}\phi(\mathbf{x})$

we find that $\phi(x)$ satisfies the Helmholtz equation:

$\displaystyle (\nabla ^2+k^2)\phi = 0$

where $k=\omega/c$ is the wave number.

Usually the Helmholtz equation is solved by the separation of variables method, in Cartesian, spherical or cylindrical coordinates.

"Helmholtz equation" is owned by Mathprof. [ full author list (2) | owner history (1) ]

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Other names:

Helmholtz differential equation, reduced wave equation

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AMS MSC:	26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)
	35-00 (Partial differential equations :: General reference works )

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