Transverse Wave Functions Amplitude

To solve the recursive formula,

ϕn+1 - Cϕn + ϕn -1 = 0,

(B.9)

one can consider the ansatz ϕ_n = tⁿ and follow similar equation,

2 t - Ct + 1 = 0.

(B.10)

This equation is the generating polynomial of the recursive formula B.9.
The roots of B.10 are

( √ -2----) -C-±----C----4-- t1,2 = 2 .

(B.11)

The general solution of the difference equation is

ϕ = αtn + βtn, n 1 2

(B.12)

since t₁ is a root of the equation, the other root t₂ can be written as: t₂ = t₁^-1.
By substituting those two roots in B.12 one obtains

ϕn = αtn1 + βt-1 n.

(B.13)

Imposing the initial condition ϕ₀ = 0 results in

α + β = 0, α = - β,

(B.14)

and from the B.13,

ϕn = α(tn1 - t-1n).

(B.15)

We obtain

α = √--ϕ1----, β = - √---ϕ1---. C2 - 4 C2 - 4

(B.16)

By substituting B.11 and B.16 in B.15, one obtains

( √ -------)n ( √ -------)n ---ϕ1---- C-+---C2----4- ---ϕ1---- C-----C2----4- ϕn = √ -2----- 2 - √ -2----- 2 . C - 4 C - 4

(B.17)

B.17 can be rewritten as

( √ --2----)n ( √ --2----)n C-+---C----4-- - C-----C-----4- -------2---------------------2---------- ϕn = ∘ --2---- ϕ1. C - 4

(B.18)