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

(9.16) |

(9.17) |

(9.18) |

The wave equation separates^{9.2} 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^{9.3}:

(9.19) |

(9.20) |

(9.21) |