4.7 Inward propagating ray

The equation (??) for k_ρ has two solutions corresponding to inward and outward propagating waves. The solution is chosen such that the ray initially propagates inward, meaning

(64)

We have

(65)

where

(66)

since N_∥ is independent of k_ρ. For positive energy waves with ∂∕∂ω > 0 the condition (??) thus becomes,

(67)

We have (??)

(68)

In general > such that (??) reduces to

(69)

or equivalently

(70)

where S_⊥ is the energy flow density in the perpendicular direction.