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r 2 @ r r 2 @ r ıB C
m
1
sin @ .sin @ ıB /
@ t ıB D
sin 2 C 2@ ıB r
ıB
B 0 sin
r
C
@ r .rV/;
(7.33)
m
r f @ r .rıB / @ ıB r g :
E 0 D
(7.34)
where m is the magnetic diffusion coefficient, and @ t and @ r denote partial
derivatives with respect to time and radius, respectively. The mass velocity, V D
V .r;t/, is supposed to be a given function.
The set of Eqs. ( 7.32 )-( 7.34 ) should be supplemented by the proper initial and
boundary conditions at the infinity and at the origin of the coordinate system. At the
initial moment t D 0 there is a uniform constant magnetic field B 0 and therefore
the initial conditions are ıB r .r;0/ D ıB .r;0/ D 0 and E 0 .r;0/ D 0.All
the functions have to tend to zero as r !1 , besides they have to be finite as r ! 0.
A rigorous solution of this problem is found in Appendix G. The components of
the electromagnetic perturbations are given by
Z
r
r 02 g 1 r 0 ;t dr 0 ;
B 0 cos
r 3
ıB r D
(7.35)
0
0
1
Z
r
r 02 g 1 r 0 ;t dr 0 g 1 .r;t/
B 0 sin
2
1
r 3
@
A ;
ıB D
(7.36)
0
m B 0 sin
2
E 0 D
@ r g 1 .r;t/;
(7.37)
where the function g 1 can be written as
( exp
!
Z
Z
t
.r C r 0 / 2
4 m .t t 0 /
dt 0
.t t 0 / 1=2
1
. m / 1=2 r
g 1 .r;t/ D
0
0
exp
!)
.r r 0 / 2
4 m .t t 0 /
r 0 @ r 0 V C 2V dr 0 :
(7.38)
This solution can be applied for an arbitrary function V D V .r;t/.
Large-scale seismic sources such as crust and deep-focus EQs and underground
explosions differ essentially from one another by the duration, spectrum, and etc.
During powerful underground explosions the most part of the seismic wave energy
is radiated by a crushing wave moving at supersonic velocity in the ground (e.g.,
 
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