The Helmholtz Equation

Lecture 38

If the Sturm Liouville equation is the most important equation in one dimension, then the Helmholtz equation

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

is the most important, and simplest, eigenvalue equation in two dimensions. The two-dimensional domains we consider are first the Euclidean plane and later the surface of a sphere.

The Helmholtz equation can be written down and then solved relative to any one of many coordinate systems. In three dimensional Euclidean space there are at least eleven such coordinate systems.