The full vectorial wave equation is given by:
Considering into Equation 1, we get:
The term is null.
We can separate Equation 2 into two equations, one for longitudinal terms and another for transversal terms:
for longitudinal terms and
for the transversal terms.
By substituting Equation 3 into the first right hand term of Equation 4, we get:
We can write:
By substituting Equation 6 into the second right hand term of Equation 4, we get:
The third term can be rewritten as:
Now, by using Equation 5 in the left side of Equation 8, we get: