Geology Reference
In-Depth Information
P
(a)
l
(b)
F
F
P
P
l
+
Δ
l
longitudinal stress
F/A
longitudinal strain
volume stress
P
volume strain
E
=
K
=
Δ
l
/
l
Δ
v/v
τ
(c)
(d)
l
θ
F
F
l
+
Δ
l
shear stress
τ
shear strain tan
θ
longitudinal stress
F/A
longitudinal strain
Fig. 3.2
The elastic moduli. (a)Young's
modulus
E
. (b) Bulk modulus
K
. (c) Shear
modulus
m
. (d) Axial modulus
y
.
μ
=
ψ
=
Δ
l
/
l
(no lateral strain)
which they pass.There are two groups of seismic waves,
body waves
and
surface waves
.
12
y
r
=
È
Í
˘
˙
v
p
3.3.1 Body waves
Body waves can propagate through the internal volume
of an elastic solid and may be of two types.
Compressional
waves
(the longitudinal, primary or
P
-
waves
of earth-
quake seismology) propagate by compressional and dila-
tional uniaxial strains in the direction of wave travel.
Particle motion associated with the passage of a com-
pressional wave involves oscillation, about a fixed point,
in the direction of wave propagation (Fig. 3.3(a)).
Shear
waves
(the transverse, secondary or
S
-
waves
of earth-
quake seismology) propagate by a pure shear strain in a
direction perpendicular to the direction of wave travel.
Individual particle motions involve oscillation, about a
fixed point, in a plane at right angles to the direction of
wave propagation (Fig. 3.3(b)). If all the particle oscilla-
tions are confined to a plane, the shear wave is said to be
plane-polarized.
The velocity of propagation of any body wave in any
homogeneous, isotropic material is given by:
4
3
or, since
y
=
K
+
m
,by
12
K
+
4
3
m
È
Í
˘
˙
v
=
p
r
and the velocity
v
s
of a shear body wave, which involves a
pure shear strain, is given by
12
m
r
=
È
Í
˘
˙
v
s
It will be seen from these equations that compres-
sional waves always travel faster than shear waves in the
same medium. The ratio
v
p
/
v
s
in any material is deter-
mined solely by the value of Poisson's ratio (
s
) for that
material
12
21
12
-
(
s
s
)
È
Í
˘
˙
v
ps
=
(
-
)
12
appropriate elastic ulus of material
of material
mod
=
È
Í
˘
˙
and since Poisson's ratio for consolidated rocks is typi-
cally about 0.25,
v
p
~
1.7
v
s
. While knowledge of the P-
wave velocity is useful, it is a function of three separate
properties of the rock and is only a very ambiguous
indicator of rock lithology. The
v
p
/
v
s
ratio, however, is
v
density
r
Hence the velocity
v
p
of a compressional body wave,
which involves a uniaxial compressional strain, is given by
