Electromagnetic Effects Resulted from Explosions - Ultra and Extremely Low Frequency Electromagnetic Fields - page 435

influence of the gravity, presence of large-scale inhomogeneity of the medium and
other causes. So, the actual distribution of the shock polarization is not entirely
spherically symmetrical. For instance, we assume the cylindrical symmetry of such
a distribution (Surkov 1986 ). The origin of coordinate system is placed in the center
of symmetry of the spherical SW. The radial polarization of the medium is described
by the following equation:

… 1 .r;;t t a / D .1 C LJ cos /….r;t t a / O r ; (11.19)

where LJ is the small parameter of asymmetry, O r is the unit vector and the parameter
t a denotes the moment of the SW arrival at the point with spherical coordinates r
and . Here the polar angle is measured from the axis of symmetry z .
As has already been stated in Sect. 9.1 , the SW in a solid carries the electric
charge. Let R 0 be the initial radius where the primary charge of the SW is formed.
By assuming that the SW velocity, U , is constant, we obtain that t a D .r R 0 /=U.
The pressure magnitude in the spherical SW changes with distance by a power law,
i.e. P m D P .R 0 =r/ n , where P denotes the pressure magnitude at the radius r D
R 0 . For the ground the value n 1:6 is usually accepted though the exponent n can
vary depending on distance (e.g., see Rodionov et al. 1971 ). The actual values of
the characteristic times f and r can be of the order of the SW duration. Since the
width of shock wavefront in the ground can reach several meters, these parameters
can be as large as a few ms or even more. In the model by Grigor'iev et al. ( 1979 )
the SW rise time is inversely proportional to the pressure magnitude and hence it
follows that f D 0 .r=R 0 / n where 0 is the constant.
The vector of the dipole electric moment d of the polarized matter is directed
along the axis of symmetry z . In order to find the absolute value of d , one should
integrate the projection of … 1 on z -axis over the volume V occupied by the SW, i.e.

over the volume restricted by the radius R f D R 0 C Ut

Z

d .t/ D … 1 .r;;t t a / cos dV: (11.20)

V

Here we consider the time interval when the SW has not yet reached the ground
surface.
The space charges due to the shock polarization of the medium are mainly
situated in the vicinity of the shock wavefront in the layer with depth on the order

of U r C f . These charges have a certain sign depending on the properties of
the rock. The opposite charges are situated behind the SW front at the distance
which is equal to the charge relaxation length or they are concentrated around the
underground cavity. Certainly, in all cases the total electric charge contained in
the rock is equal to zero.

Below we assume an inequality U r C f R f . Since the integral sum in

Eq. ( 11.20 ) is mainly accumulated within a short length U r C f , the functions