Appendix I

Compatibility:

E-Formulation

The full vectorial wave equation is given by:

BPM - Equation 1

Considering BPM - Equation a into Equation 1, we get:

BPM - Equation 2

The term BPM - Equation is null.

We can separate Equation 2 into two equations, one for longitudinal terms and another for transversal terms:

BPM - Equation 3

for longitudinal terms and

BPM - Equation 4

for the transversal terms.
By substituting Equation 3 into the first right hand term of Equation 4, we get:

BPM - Equation 5

We can write:

BPM - Equation 6

By substituting Equation 6 into the second right hand term of Equation 4, we get:

BPM - Equation 7

The third term can be rewritten as:

BPM - Equation 8

Now, by using Equation 5 in the left side of Equation 8, we get:

BPM - Equation 9