MHD-Waves in Layered Media - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 209

Im k

Imκ g =0

Im κ S =0

k 3 =− k 0 ε g 1/2

k 2 =− k 0 ε m 1/2 k 1 =− k 0 ε m 1/2

Re k

k 4 =− k 0 ε g 1/2

Imκ s =0

Im κ g =0

Fig. 7.3. Lo ca tion of cuts Im κ g =Im κ s = 0 and branch points k 1 , 2 = ±k 0 √ ε m ,
k 3 , 4 = ±k 0 √ ε g on the complex plane of wave numbers k

In the case of the finite conductivity layer underlain by a perfect conductor,
only two sheets remain:

• the physical sheet: Im κ s > 0and

• the non-physical sheet: Im κ s < 0.

Apart from branch points, the poles of R and T can appear on the
complex plane, and their location is determined by the roots of ∆ ( k ) (7.101):

∆ ( k )=0 .

(7.106)

The Role of a Magnetic Inclination

Before proceeding to the investigation of R ( k )and T ( k ) in various wavenum-
ber ranges, note that the character of the reflection of MHD-waves does not

Table 7.2. Numeration of the sheets of the Riemann surface

Physical sheet Non-physical sheets

1. Im κ g > 0andIm κ s > 0 2. Im κ s < 0, Im κ g > 0

3. Im κ s < 0, Im κ g < 0
4. Im κ s > 0, Im κ g < 0