Helmholtz equation

In mathematics and physics, the Helmholtz equation, named for Hermann von Helmholtz, is the linear partial differential equation In mathematics and physics, the Helmholtz equation, named for Hermann von Helmholtz, is the linear partial differential equation where ∇ 2 {displaystyle abla ^{2}} is the Laplacian, k {displaystyle k} is the wave number, and A {displaystyle A} is the amplitude. This is also an eigenvalue equation. The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. For example, consider the wave equation Separation of variables begins by assuming that the wave function u ( r , t ) {displaystyle u(mathbf {r} ,t)} is in fact separable: