Propagation of MHD-Beams - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 266

Fig. 8.A.1. (a) A contour Γ D for w k =2 ik + b − b exp − 2 k . (b) Hodograph of

w k when turning over Γ D

Let us show that there are no roots on the physical sheet for τ K <π/ 2 . The
left part of the first equation of (8.A.4) is negative, and is positive for x< 0 .
The second equation of (8.A.4) has roots on the physical sheet in the 1-st
quadrant, if they exist, are just for x> 0and π ( n +1 / 2) <y<π ( n +1) .
Substituting these values of y into the 1-st equation of (8.A.4), we found that
a root can be just for τ K >π/ 2 . Considering that (8.A.5) is obtained from
(8.A.4) by replacing of

k by - k, there are no roots on the whole physical

sheet.
Let us prove that roots of (8.A.4) exist in the 3-rd and 4-th quadrants

Re ¯ k< 0 . Consider a function
w k = u + iv =2 ik + τ K −

τ K exp − 2 k .

According to the argument principle, the number of zeroes of an analytical
function without singularities in an area D is equal to an argument increment
of this function over 2 π obtained when the complete circuit of D is traversed
along the contour Γ D containing D. Adopt a contour Γ D shown in Fig. 8.A.1a

for the function w k . When moving along Γ D 1 and Γ D 3 , the values of u and
v are

u = τ K + π

2 −

2 nπ on Γ D 1 ,

3 π
2 −

u = τ K − 2 nπ on Γ D 3 ,

2 x ) n Γ D 1 and Γ D 3 .
Choose large x 1 and x 2 , then w k ≈−

v =2 x + τ K exp ( −

τ K exp − 2 k on Γ D 4 and w k =2 ik

on Γ D 2 . Hodograph of the function w k , when turning over Γ D , is shown
in Fig. 8.A.1b. It follows from the given expressions that the hodograph one
time envelopes the origin of the coordinate system, i.e. (8.38) has one root on
the nonphysical sheet in the band