Helmholtz equation

In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: $${\displaystyle \nabla ^{2}f=-k^{2}f,}$$ where ∇² is the Laplace operator, k² is the eigenvalue, and f is the (eigen)function. When the equation is applied to waves, k is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle.

In optics, the Helmholtz equation is the wave equation for the electric field.^[1]

The equation is named after Hermann von Helmholtz, who studied it in 1860.^[2]