Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 306

into Eq. ( 7.98 ) and rearranging, we arrive at the conventional wave equation for the
function f

@ t f D C l @ r f:

(7.99)

The solution of Eq. ( 7.99 ) in the form of outgoing wave is an arbitrary twice
differentiable function depending on the variable t 1 D t .r R 0 /=C l , i.e.,
f D f .t 1 /. So, the solution of the problem has the form
u r D R 0 @ r f .t 1 /
r

@ t f .t 1 / C

r f .t 1 / :

R 0
rC l

C l

D (7.100)

In rearranging this equation we have used that @ t f D C l @ r f . The mass velocity,
V .r;t/ D @ t u r .r;t/ O r , can be expressed through Eq. ( 7.100 ) in the form given by
Eq. ( 7.40 ).
The dimensionless function f .t 1 / is usually referred to as the so-called normal-
ized potential of the elastic displacement. The normalized potential can be extracted
from seismic recordings. In situ measurements the instruments are arranged far
away from the seismic source so that the distances are much greater than the
characteristic seismic wavelength, i.e., r a . In such a region, referred to as a
wave zone, the last term on the right-hand side of Eq. ( 7.100 ) can be neglected in
comparison with the first term. This implies that the recording of the displacement is
proportional to the time-derivative of the normalized potential, so that the function
f .t/ can be found through the data processing of the seismic recordings.
In a theory the analytical form of the normalized potential can be found
from Eq. ( 7.100 ) and the initial and boundary conditions. For example, as the
displacement at the surface of the source is a given function u .t/, taking Eq. ( 7.100 )
at the boundary r D R 0 we come to the following differential equation for the
function f .t/

u .t/ D R 0 R 0

C l d t f .t/ C f .t/ ;

(7.101)

where the symbol d t stands for time-derivative. The solution of Eq. ( 7.101 )isgiven
by Eq. ( 7.40 ).
In the theory of underground explosion the crushing zone can serve as an
effective source of seismic waves (Rodionov et al. 1971 ). In such a case the radial
component of the stress tensor s rr at the crushing zone boundary is considered as a
given function. For example, the radial stress can be approximated by Eq. ( 7.42 ).
In accordance with Hooke law the radial component of the stress tensor can be
expressed through the radial displacement (Landau and Lifshits 1987 )

s rr D m C l Œ.1 /@ r u r C 2 u r =r; (7.102)

where m is medium density and is Poisson 's coefficient which defines the ratio
of the transverse and longitudinal components of the strain tensor. The Poisson 's