Helmholtz equation

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

$${\nabla }^{2}f+k^{2}f=0,$$

or in vector form is

$${\nabla }^2𝐀+k^2𝐀=0,$$

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

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

with wave speed $c$. If we look for time harmonic standing waves of frequency $\omega$,

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

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

$$({\nabla }^2+k^2)\varphi =0$$

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.