Propagation of MHD-Beams - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 247

8.4 Large Distances

Sommerfeld-Watson Transformation

For investigating spatial distributions at large distances from the beam axis
as well as electric type wave propagation, let us return to exact expressions
(7.108) and (7.105) for matrices R ( k )and T ( k ) and to the integral presen-
tation (8.6). It is convenient to study the amplitude and phase dependencies
with the field expanded by normal modes propagating along the layers. We
shall use the Sommerfeld-Watson transformation [5]. Deform the initial con-
tour of integration Γ 1 in the complex plane of horizontal wavenumbers k to
the contour Γ m (see Fig. 8.1). If the ground is modeled by a half-space with a
finite conductivity σ g , the integration contour must be supplemented by the
contour Γ g .
The transition from Γ 1 to Γ m requires knowledge of the analytical proper-
ties of R ( k )and T ( k ) on the entire complex plane k (see Section 7.6). The
poles of R ( k )and T ( k ) determine the wavenumbers of normal waveguide
modes propagating in the ground-lower ionosphere waveguide. They can be
found from

∆ SK ∆ A =0 . (8.35)
Two types of waveguide modes, magnetic and electric, prove to be coupled
with one another because of the anisotropy of the ionosphere, and normal
modes are of mixed character. However, the separation of the waves into
two types remains useful. Divide the roots of (8.35) into two groups. The
wavenumbers of the first group are determined by

∆ SK ( k )= ζ 4

κ S
k 0

ζ 3 −

X K −

=0 ,

(8.36)

and of the other group by

X
sin 2 I

= ζ 1 ∆ A ( k ) / | sin I |

ζ 2 −

=0 , (8.37)

where X K = X + Y 2 / X , ζ i ( i =1

− 4) see (7.39) and (7.45).

0, when the magnetic and electric modes do not couple, find from
(8.36) the wavenumbers of magnetic-type normal modes and from (8.37) those
for the electric type. In the general case, Σ H

At Σ H →

= 0, we preserve the designations
of magnetic and electric modes for the branches turning by continuity into the
respective modes at Σ H →

0.
Make a general note concerning the behavior of ∆ A and ∆ SK : one can
separate two regions on the complex plane k with essentially different depen-
dencies ∆ A ( k )and ∆ SK ( k ) . One of them is far from the contours Γ m and Γ g
where ∆ A ≈−X/ (see (7.120)) and the other one is close to Γ m and

Γ g . Here a special study is necessary.

| sin I |