5.1.2 Getting agreement with Gauss's law
Substituting the assumed form of the electric field (Equation 7.20) into the empty-space version of Gauss's law (Equation 7.16) gives
The first two partial derivatives are equal to zero because f does not depend on x or y. So we obtain
We are interested in disturbances that propagate in the z-direction, so can ignore the possibility that ∂f/∂z = 0 everywhere. It follows that uz = 0. This means that u is a unit vector perpendicular to the z-direction. With no loss in generality, we can choose u to be equal to ex. It is then appropriate to replace f by Ex, and write Equation 7.20 in the form
A wave of this type, in which the variable of interest oscillates perpendicular to the direction of propagation, is said to be transverse.