Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 287

see Chadwick et al. 1964 ; Rodionov et al. 1971 ). Due to the large mechanical stress
behind the front of the crushing wave, the rock fails in shock compression and
is broken to pieces and fragments. The speed of such a wave decreases gradually
owing to dissipation of the mechanical energy, and the crushing wave stops abruptly
as the stress at the front falls off below the crushing strength of the rock. For nuclear
underground explosions the final radius of the crushing wave or the so-called radius
of the crushing zone is of the order of several tens or hundreds meters that may
characterize a size of seismic source associated with the underground explosion.
A peculiarity of EQs is that the shear waves make larger contribution to the
total seismic energy as compared to the underground explosions. In the case of
earthquakes the effective seismic radiator is situated at the depth of several tens
or hundred kilometers and its radius is determined by the typical scale of a focal
zone. For the EQs with magnitudes M>6the typical size of the seismic source
exceeds 10 km (Scholz 1990 ).
We next derive an order-of-magnitude estimate of the mass velocity for the case
of a large-scale seismic source. The acoustic wave equation for a uniform elastic
medium can be written as (Landau and Lifshits 1987 )

@ t u D C l r . r u / C t r . r u /;

(7.39)

where u is the vector of medium displacement, and C l and C t are the longitudinal
and transverse sound velocities. Let us consider a spherically symmetric acoustic
source with a radius R 0 . The source radiates the longitudinal acoustic wave in which
the displacement vector u has only radial component u r D u r .r;t/.
A general solution of this problem in the form of outgoing wave is found in
Appendix G. This solution can be expressed through the so-called normalized
potential of the elastic displacement f D f .t 1 / which depends on the variable
t 1 D t .r R 0 /=C l . The mass velocity, V .r;t/ D @ t u r .r;t/ O r , is represented by
the potential as follows:

@ t f .t 1 / C

r @ t f .t 1 / :

R 0 O r
rC l

C l

V D (7.40)

To find the explicit expression of the normalized potential we require the
boundary and initial conditions. At first consider the conventional formulation of the
problem. The radial displacement of the medium at the radius r D R 0 is supposed
to be a given function, i.e., u r .R 0 ;t/ D u .t/. At the initial moment u r .r;0/ D 0
everywhere and u .0/ D 0 and @ t u .0/ D 0. In such a case the normalized potential
is given by

u t 0 exp C l

R 0 t 0 t dt 0 :

Z

t

C l
R 0

f .t/ D (7.41)

0