After a little work (take the curl of the curl equations, using the identity:

(11.13) |

and using Gauss's source-free Laws) we can easily find that and in free space satisfy the

(11.14) |

(for or ) where

(11.15) |

The wave equation separates^{11.2} for harmonic waves
and we can actually write the following homogeneous PDE for just the
spatial part of
or
:

where the time dependence is implicitly and where .

This is called the *homogeneous Helmholtz equation* (HHE) and we'll
spend a lot of time studying it and its inhomogeneous cousin. Note
that it reduces in the
limit to the familiar homogeneous
Laplace equation, which is basically a special case of this PDE.

Observing that^{11.3}:

(11.16) |

where is a unit vector, we can easily see that the wave equation has (among many, many others)

(11.17) |

where the

(11.18) |

and points in the direction of propagation of this