Electromagnetic Effects Resulted from Natural Disasters - Ultra and Extremely Low Frequency Electromagnetic Fields - page 408

The directional cosines are transformed to the view

x D sin cos 0 cos .' ' 0 / cos sin 0 ; (10.67)

y D sin sin .' ' 0 /;

(10.68)

z D sin 0 sin cos .' ' 0 / C cos 0 cos :

(10.69)

Substituting Eq. ( 10.63 )for u i and Eqs. ( 10.67 )-( 10.69 )for i into the set of
Eqs. ( 10.64 )-( 10.66 ), and rearranging, we obtain

8r 2 C t 3 w 2 f 1 1 w 2 f 2 ;

1

u r D (10.70)

sin 2 0 cos 2 .' ' 0 /

sin 2 .M zz M xx /
8r 2 C l

u D

cos 2 cos .' ' 0 / cos 2 0 ;

sin 2 0
2

C (10.71)

sin 0 sin .' ' 0 /.M xx M zz / z
4r 2 C l

u ' D : (10.72)

As is seen from Eqs. ( 10.70 )-( 10.72 ), the displacement components depend on the
random angles 0 and ' 0 as well as on the fixed/deterministic angles and ',
which define the direction from the cracked zone to the ground-recording station.
Assuming for the moment that there is an equal probability for the crack plane
orientation, the probability density distribution does not depend on azimuthal angle
' 0 . In such a case the averaging of Eqs. ( 10.70 )-( 10.72 ) with respect to ' 0 yields

( .M xx M zz / " sin 2 sin 2 0

2

3 w 2

1
8r 2 C t

h u r iD

!

1 # C w 2 .M xx C M zz / ) ;

C cos 2 cos 2 0

(10.73)

sin 2 0 ! ; ǝ u ' Ǜ D 0:

sin 2 .M zz M xx /
8r 2 C l

2 cos 2 0

h u iD (10.74)