Chemistry - Helmholtz equation

The Helmholtz equation, named for Hermann von Helmholtz, is the following elliptic partial differential equation:

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

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. For example, consider the wave equation:

$(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial{t}^2})u(\mathbf{x},t)=0$

Applying the technique of separation of variables ( $u=\phi(\mathbf{x})T(t)$ ), we obtain two differential equations:

$\nabla^2\phi + k^2\phi=(\nabla^2 + k^2)\phi=0,\quad \frac{\partial^2{T}}{\partial{t}^2} + k^2c^2T=0,$

where $k$ is the separation constant. We see that we now have Helmholtz's equation for the spatial variable $\mathbf{x}$ and a second-order ordinary differential equation in time. The solution in time will be a linear combination of sine and cosine functions, while the form of the solution in space will depend on the boundary conditions. Alternatively, integral transforms, such as the Laplace or Fourier transform, are often used to transform a hyperbolic PDE into a form of the Helmholtz equation.

Due to its relationship to the wave equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology, and acoustics.

References

Riley, K.F., Hobson, M.P., and Bence, S.J. (2002). Mathematical methods for physics and engineering, Cambridge University Press, ch. 19. ISBN 0-521-89067-5.

External link

Helmholtz Equation at EqWorld: The World of Mathematical Equations.

Categories: Waves | Partial differential equations | Equations