Hydromagnetic Resonators - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 106

and the conditions of bi-orthonormality

l z

Q n G m = Q n ( z ) Q m ( z )d z = δ mn (5.11)

0

are fulfilled between them.

5.3 Explicit Eigenmodes

Let us obtain explicit expressions for Q n ( z )and k n . The solution of (5.9) is
Q ( z )= sin kz
k

c
ωX −

+ i cos kz (5.12)

It satisfies the boundary condition (5.10) at z = 0. Substituting this function
into the boundary condition at z = l z , we obtain
1+( πq ) 2 α − α + sin πq
πq

+ i ( α + + α − )cos πq =0 , (5.13)

where

q = kl z
π

c
ωl z X ∓

, .

=

∓

Consider two limiting cases of high and low conductivities Σ P . For the
perfectly conducting ionospheres, i.e. Σ P →∞ , the parameters α ± = 0 . Then

(5.13) has only real roots

k n = nπ q n = n, , n =1 , 2 , 3 ,....

l z

The eigenfunctions of (5.12) reduce to
Q n ( z )= 2

π sin( k n z ) .

(5.14)

At small but finite α ± the first-order correction on α + + α − can be found by
substituting q n = n + δq n ( k n = k n + δk n ) into (5.13). Expanding it into a
series by δq n and α ± and leaving only linear terms, we obtain

nπ
l z −

in π

l z ( α + + α − ) ,n =1 , 2 , 3 ,.... (5.15)
In the opposite case of small conductivities Σ P , we find that (5.13) in the
first approximation by 1 /α ± for q n becomes

δq n ≈− in ( α + + α − ) ,k n ≈

i 1
α −

.

1
α +

πq tan( πq )= − +