Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

assuming that the TM magnetic source field j.s due to a uniforrn
sheet current at height z = -h, h > 0. T1lj.s assun~ptj-on, however,
is immaterial for the following.
In the sequel all field quantities are split into a normal and
anoinalous part, denoted by the subscripts "nu and "a" , respectivel-y.
The normal part refers to a one-dimensional conductivity structure.
Let
u(y,zl = un(z) + U,(Y,Z) (3.7)
H(y,z) = Hn(z) + Ha(y,z)
E and H are defined as solutions of the equations
n n
(3.3b)
e
AEn = k2 E + j (3.10a)
n 11
d l d
-(- - 1-1 ) = H z > Or H (0) =
dz ,?2 dz n n ' - n
vanishing for z + m.
In virtue of (3.5a,b), (3.9arb), and (3.103,b) Ea and H satisfy
a
1 d 1. 1 dIln
div (- gradH ) = H a - - - , z > 0
k
a
k * k2 dz -
n
If the anomalous domain is of fin2te extent, E has to vanish uni-
a
forinly at infinity. Under the same condition Ha has to vanish uni-
formly in the lower half-space. At z=o H is zero.
a
If the anomalous domain is of infi.nite extent in horizontal direc-
tion, we can demand only that Ear H +O for z + m.
a
For a numerical solution of (3.11 a) the following three clioices of
a basic dornain are possible (boundaries hatched).
In approach A, (3.11 a) is solved by Finite differences subject to
the boundary condition Ea=O or better subject to an inpedance
boundary collclition (below). In approach B (3.11a) is solved by
finite differences only in the anomal.ous slab. At the ho~izontal
boundaries boundary conditions involicity the normal structure
above and below the slab are applied. I approach C (3.11~~) is re-
duced to an integral equation over the anon~alous domain. These
approaches will now be discussed in details.