Earthquakes - Quantitative Plate Tectonics - page 280

seismogenic source. The fundamental problem
of the earthquake source theory is determining
the radiation pattern of displacements u D u ( r , t )
from a source located at the origin, assuming
elastic properties of the transmission medium. In
this context, the building block of analog models
that are representative of real seismic sources is
simply a body force (per unit volume), f , applied
to a point r D r 0 . In principle, the objective of
determining the radiation pattern associated with
this simple source can be accomplished solving
the non-homogeneous version ( 7.50 ) of Cauchy's
momentum equation with f ( r , t ) D g ( t )•( r - r 0 ):

f C .œ C 2/ r . r u / rr u D 0
(10.15)

where we have used the identity:

2 u

rr u Dr . r u / r (10.16)

In this problem, the body force field f D f ( r )
is concentrated at the origin, so that it must be
expressed in terms of Dirac's delta function. By
Gauss' theorem, this function can be written as
the Laplacian of a scalar field:

2 1
r

1
4 r

•.r/ D (10.17)

¡ @ 2 u i

@£ ij
@x j C g i .t/•.r r 0 /

@t 2 D

(10.12)

Therefore following Lay and Wallace ( 1995 )
we can write: where •( r - r 0 ) is the Dirac delta function, which

is defined by the following functional relations
(e.g., Panofsky and Phillips 2005 ):
8
<

2 n
4 r

f .r/ D n•.r/ Dr

n
4 r

r n
4 r

•.r/ D 0 for r ¤ 0

Dr Crr

Z

• r r 0 dx 0 dy 0 d z 0 D 1 for any region R

(10.18)

R

such that r 2 R

where n is a unit vector representing the direction
of f . If we insert this force into the static equilib-
rium Eq. ( 10.15 ) we obtain:

Z

:

f r 0 • r r 0 dx 0 dy 0 d z 0 D f.r/ for any

R

n
4 r

region R such that r 2 R

(10.13)

r n
4 r

r Crr

We also note that the second of these proper-
ties is a consequence of the third, more general,
property. To determine the radiation pattern as-
sociated with f , it is useful to start from the static
displacement field generated by the application of
aforce f at the origin of the reference frame. In
equilibrium conditions, the displacement is zero
far from the origin, and assuming a homogeneous
medium we have:

C .œ C 2/ r . r u / rr u D 0
(10.19)

Therefore,

n h

4 r C .œ C 2/ u io

n r r

n

4 r u

Crr D 0 (10.20)

@x i C @ 2 u i

0 D .œ C / @

We search a solution u D u ( r ) having the form:
u Dr r A p r . r A s / (10.21)

@x j C f i

(10.14)

where have simply rewritten (8.11) without the
acceleration term and including the body force
contribution. In vector notation, this equation
assumes the form:

To this purpose, we note that for an arbitrary
vector field u D u ( r ) we can always determine a
vector field A D A ( r ) such that: