B.2 Calculation of k_ρ, m and/or n as a function of k_∥ and k_⊥

Generally, the initial conditions determine m and k_∥. Then k_⊥ is calculated from the dispersion relation. We need to express k_ρ and n as a function of m, k_∥ and k_⊥. We have

(116)

and

(117)

We get an expression for n

(118)

which is inserted back in (??) to get

(119)

and k_ρ is solution of the equation

(120)

Defining

(121)

and

we have

(124)

such that

(125)

which gives

(126)

To sum up

(127)

where σ_ρ = ±1. We can also express k_ρ as

(128)

The correct solution for k_ρ can be determined from (??).