From vector Helmholtz equation to scalar wave equation

I’m going to simplify the Helmholtz equation further, so that we can have some discussion of the types of solutions we expect.  As a reminder, the vector Helmholtz equation derived in the previous section was:

$latex \displaystyle \nabla^2 \vec{E} – \epsilon \mu \frac{\partial^2 \vec{E} }{\partial t^2}  = 0 $

In rectangular coordinates, the ‘del’ operator is

$latex \displaystyle \nabla = \frac{\partial}{\partial x}\hat{a}_x +\frac{\partial}{\partial y}\hat{a}_y + \frac{\partial}{\partial z}\hat{a}_z $

and $latex \nabla ^2 = \nabla \cdot \nabla $, so

$latex \displaystyle \nabla ^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} $

The vector Helmholtz equation is really a set of three equations, one for each vector component of the electric field. Substituting in $latex \nabla^2$:

 $latex \displaystyle \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) E_x – \epsilon \mu \frac{\partial^2 E_x}{\partial t^2} = 0$

is the equation for the x-component of the electric field $latex E_x$, and the equations for $latex E_y$ and $latex E_z$ are identical.

I’ll describe the plane wave solutions to this equation in more detail later on, including the associated magnetic field, propagation directions and polarization, etc.  For now, let’s suppose we are just interested in electric fields that are varying in the z-direction, and pointing in the x-direction: $latex E_x(z)$.  This seems pretty restrictive, but it simplifies things quite a bit, and the solutions can be generalized pretty easily.  Under these assumptions, we end up with a single equation:

 $latex \displaystyle  \frac{\partial^2 E_x}{\partial z^2} – \epsilon \mu \frac{\partial^2 E_x}{\partial t^2} = 0$

This is a scalar wave equation, as you may have learned in a previous class.   We’ll talk about the solutions to these types of equations in the next section.