Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 285

To summarize, we note that all the electromagnetic amplitudes given by
Eqs. ( 7.25 )-( 7.27 ) and ( 7.29 )-( 7.31 ) are proportional to the amplitude of the
medium mass velocity V 0 . Hence we conclude that the attenuation of the GMP
in the seismic zone is the same as that of the mass velocity. For example, once the
amplitude of the longitudinal elastic wave radiated by the spherical seismic source
decreases inversely proportional to the distance r, the amplitudes of the magnetic
and electric perturbations originated from the seismic wave propagation decrease in
a similar manner, i.e., as r 1 . This conclusion is not valid for non-perfect elastic or
dissipative media, which require a special consideration (Dunin and Surkov 1992 ).

7.2.6 Spherically Symmetric Seismic Wave

Far away from the source a seismic longitudinal wave can be considered as
approximately spherically symmetric one at least as a first approximation. This
approach is applied to longitudinal waves radiated from expansion of the spherical
pore in the ground and from an underground explosion at least up to the moment of
the wave reflection from the boundary with atmosphere. The EQ focal zone, as is
often the case, is equivalent to a dislocation/shear crack, whose seismic radiation is
nonsymmetric (Aki and Richards 2002 ). In the analysis that follows, we consider,
however, the spherically symmetric wave propagating in a homogeneous medium
in order to analyze a rigorous solution of the problem, which are valid for both the
diffusion and acoustic zones.
So, let us consider an infinite homogeneous conductive medium immersed in
the constant magnetic field with induction B 0 . A spherically symmetric source
begins to radiate a longitudinal acoustic/seismic wave at the moment t D 0.
The perturbations, ı B , of the external magnetic field caused by the acoustic wave
propagation are supposed to be small, so that j ı B jj B 0 j as before. In such a case
the magnetic perturbation ı B and the electric field E in an earth-fixed reference
frame are defined by Eqs. ( 7.11 ) and ( 7.12 ). The origin of the coordinate system
is chosen to be in the center of symmetry of the acoustic source. We use the
coordinate system whose z axis is directed along the external field B 0 . We apply
spherical coordinates r, , and , where the polar angle is measured from the z
axis and the azimuthal angle is measured from the positive direction of x axis.
The mass velocity of the medium is radially directed, i.e., V D V .r;t/ O r , where O r
is the radially directed unit vector. In such a case all functions are independent of
the azimuthal angle . Taking into account the symmetry of the problem, only three
components of the electromagnetic perturbations are nonzero, namely ıB r , ıB ,
and E . Using the spherical coordinates, Eqs. ( 7.11 ) and ( 7.12 ) can be thus rewritten
as (Landau and Lifshits 1982 )

@ r r 2 @ r ıB r 2ıB r

m
r 2

@ t ıB r D

sin @ Œsin .@ ıB r 2ıB /

1 2VB 0 cos

r

C ; (7.32)