After a little work (take the curl of the curl equations, using the identity:
The wave equation separates11.2 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.