After a little work (two curls together, using the identity:

(9.11) |

(9.12) |

(9.13) |

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}:

(9.14) |

(9.15) |

(9.16) |