Geoscience Reference
In-Depth Information
First, if we differentiate the
u
,
v
and
w
equations with respect to
x
,
y
and
z
, respectively,
and add the results, we obtain
ρ
∂
t
2
∂
2
∂
2
u
x
+
∂
v
y
+
∂
w
)
∂
2
u
=
(
λ
+
µ
+
µ
∇
∂
∂
∂
∂
z
∂
x
2
∂
x
2
∂
2
)
∂
v
+
(
λ
+
µ
+
µ
∇
∂
y
2
∂
y
2
∂
2
)
∂
w
+
(
λ
+
µ
+
µ
∇
(A2.37)
∂
z
2
∂
z
or
2
ρ
∂
2
2
=
(
λ
+
µ
)
∇
+
µ
∇
∂
t
2
2
(A2.38)
=
(
λ
+
2
µ
)
∇
This is a
wave equation
for a dilatational disturbance transmitted through the material with a
speed
λ
+
2
µ
ρ
(A2.39)
α
=
In seismology, as discussed in Chapter 4, this type of wave involves only dilatation and no
rotation and is termed the
primary wave
or
P-wave
.
Second, we can differentiate Eq. (A2.36a) with respect to
y
and Eq. (A2.36b) with respect
to
x
:
t
2
∂
2
∂
ρ
∂
2
u
)
∂
2
u
=
(
λ
+
µ
y
+
µ
∇
(A2.40)
∂
∂
y
∂
x
∂
∂
y
and
t
2
∂
2
∂
2
2
ρ
∂
v
)
∂
v
=
(
λ
+
µ
y
+
µ
∇
(A2.41)
∂
∂
x
∂
x
∂
∂
x
Subtracting Eq. (A2.41) from (A2.40)gives
ρ
∂
t
2
∂
2
∂
2
u
y
−
∂
v
u
y
−
∂
v
=
µ
∇
(A2.42a)
∂
∂
∂
x
∂
∂
x
By differentiating and subtracting derivatives, we obtain the other two equations:
ρ
∂
t
2
∂
2
∂
2
u
z
−
∂
w
u
z
−
∂
w
∂
=
µ
∇
(A2.42b)
∂
∂
∂
x
∂
x
t
2
∂
2
∂
2
ρ
∂
v
z
−
∂
w
v
z
−
∂
w
∂
=
µ
∇
(A2.42c)
∂
∂
∂
y
∂
y
However, since
∂
u
/∂
y
−
∂
v
/∂
x
and so on are the components of curl
u
(or
∇
∧
u
; see
Appendix 1), these three equations can be written
ρ
∂
2
2
(curl
u
)
(A2.43)
t
2
(curl
u
)
=
µ
∇
∂
This is a vector wave equation for a rotational disturbance transmitted through the material
with a speed
µ
ρ
β
=
(A2.44)
In seismology, as discussed in Chapter 4, this type of wave involves only rotation and no
change in volume and is called the
secondary wave
or
S-wave
.
