The Helmholtz equation is a partial differential equation which, in scalar form is $$\vnabla^2f+k^2f = 0,$$ or in vector form is $$\vnabla^2\vA+k^2\vA = 0,$$ where $\vnabla^2$ 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 $$\frac{\partial^2\psi}{\partial t^2} = c^2\vnabla^2\psi$$ with wave speed $c$ . If we look for time harmonic standing waves of frequency $\omega$ , $$\psi(\vx,t) = e^{-j\omega t}\phi(\vx)$$ we find that $\phi(x)$ satisfies the Helmholtz equation: $$(\vnabla^2+k^2)\phi = 0$$ 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.