Hydromagnetic Resonators - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 120

Assume there are no plasma displacements normal to boundaries x =0
('equatorial ionosphere') and x = l x ('magnetopause'), i.e.

ξ x | x =0; l x =0 . (5.46)

For ideal plasma the latter condition is equivalent to the requirement that the
tangential electric components vanish:

E y | x =0; l x =0 .

The impedance boundary conditions are satisfied on the lower and upper
boundaries of the box:

∂ξ ⊥
∂z ∓

X c ∂ξ ⊥

∂t

=0 . (5.47)

z =0; l z

At the initial moment t = 0 plasma displacements and velocities are given by

t =0

∂ξ ⊥
∂t

ξ ⊥ | t =0 = ξ ⊥ 0 ( x, z ) , = ξ ⊥ 1 ( x, z ) . (5.48)

Laplace Transform

The Laplace transform from displacement ξ ⊥ ( x, z, t )and b ( x, z, t ) is deter-
mined in the usual way. Write it in the likewise Fourier form :

∞

ξ ⊥ω ( x, z )= ξ ⊥ ( x, z, t )exp( iωt )d t, (5.49)

0

where ω is complex in general. Applying this transform to (5.7a)-(5.8),
boundary and initial conditions (5.46), (5.47) and (5.48) yields a system of
non-uniform equations:

L ω ξ yω = ik y b ω
B 0

+ f yω , (5.50)

b ω
B 0

d
d x

= L ω ξ xω + f xω , (5.51)

b ω
B 0 −

d
d x ξ xω =

− ik y ξ yω ,

(5.52)

with boundary conditions

ξ x | x =0; l x =0 , (5.53)

d ξ ⊥ ω
d z c ξ ⊥ω

ω

± iX ∓ ·

=0 .

(5.54)

z =0; l z