B.2 Transverse Wave Functions Amplitude

To solve the recursive formula,

ϕn+1 -  Cϕn + ϕn -1 = 0,
(B.9)

one can consider the ansatz ϕn = 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)