Geoscience Reference
In-Depth Information
vibrate about an equilibrium position, in the same direction as the direction in
which the P-wave is travelling. In contrast, the particle motion of S-waves is
transverse, that is, perpendicular to the direction of motion of the S-wave. The
S-wave motion can be split into a horizontally polarized motion termed SH and
avertically polarized motion termed SV.
The
wave equation
is derived and discussed in Appendix 2. The three-
dimensional
compressional wave equation
(Eq. (A2.48)) is
∂
2
φ
∂
t
2
2
2
=
α
∇
φ
(4.1)
where
ø
is the
scalar displacement potential
and
the speed at which dilatational
waves travel. The three-dimensional
rotational wave equation
(Eq. (A2.49)) is
∂
α
∂
t
2
2
2
2
=
β
∇
(4.2)
where
the speed at which rota-
tional waves travel. The
displacement
u
of the medium by any wave is given by
Eq. (A2.45):
is the
vector displacement potential
and
β
u
=∇
φ
+∇∧
(4.3)
The speeds at which dilatational and rotational waves travel,
, are termed
the
P-wave
and
S-wave seismic velocities
. Often the symbols
v
p
and
v
s
are used
instead of
α
and
β
.
The P- and S-wave velocities depend on the physical properties of the material
through which the wave travels (Eqs. (A2.31), (A2.39) and (A2.44)):
α
and
β
K
+
4
3
µ
α
=
(4.4)
ρ
µ
ρ
β
=
(4.5)
where
K
is the bulk modulus or incompressibility,
µ
the shear modulus or rigidity
and
the density.
The bulk modulus
K
,which is defined as the ratio of the increase in pressure
to the resulting fractional change in volume, is a measure of the force per unit
area required to compress material. The shear modulus
ρ
is a measure of the
force per unit area needed to change the shape of a material. Since P-waves
involve change of volume and shape,
µ
α
is a function of
K
and
µ
,whereas
β
is
only a function of
because S-waves involve no change in volume. Since the
bulk modulus
K
must be positive, Eqs. (4.4) and (4.5) show that
µ
α
is always
greater than
β
,or, in other words, P-waves always travel faster than S-waves. The
rigidity
for a liquid is zero, a liquid has no rigidity and cannot sustain shear;
Eq. (4.5) therefore indicates that S-waves cannot be propagated through liquids.
Thus, S-waves cannot be transmitted through the Earth's liquid outer core.
The dependence of
µ
on density is not immediately obvious, but, in
general, denser rocks have higher seismic velocities, contrary to what one would
α
and
β
