Geoscience Reference
In-Depth Information
Displacement potentials
We can use the method of Helmholtz to express the displacement
u
as the sum of the gradient
of a scalar potential
φ
and the curl of a vector potential
. The divergence of the vector
potential must be zero:
∇·
=
0. The displacement is then expressed as
u
=
∇
φ
+
∇
∧
(A2.45)
The two potentials
are called the
displacement potentials
. Substituting Eq. (A2.45)
into Eqs. (A2.38) and (A2.43) and using the vector identities
φ
and
∇·
(
∇
∧
V
)
=
0,
∇
∧
(
∇
S
)
=
2
V
,where
S
is a scalar and
V
avector, gives
0 and
∇
∧
(
∇
∧
V
)
=
∇
(
∇·
V
)
−
∇
λ
+
2
∂
2
µ
2
2
(
2
(A2.46)
t
2
(
∇
φ
)
=
∇
∇
φ
)
∂
ρ
and
2
∂
t
2
(
∇
∂
µ
ρ
∇
2
4
(A2.47)
)
=
The potentials therefore satisfy the wave equations
∂
λ
+
2
φ
2
µ
2
(A2.48)
=
∇
φ
∂
t
2
ρ
and
2
∂
∂
t
2
µ
ρ
∇
2
(A2.49)
=
Equation (A2.48)isthus an alternative expression of Eq. (A2.38), the wave equation for
P-waves, and Eq. (A2.49)isanalternative expression of Eq. (A2.43), the wave equation for
S-waves.
Plane waves
Consider the case in which
φ
is a function of
x
and
t
only. Then Eq. (A2.48) simplifies to
∂
2
2
φ
∂
t
2
λ
+
2
µ
ρ
∂
φ
λ
x
2
=
2
∂
φ
2
=
α
(A2.50)
∂
x
2
Any function of
x
±
at
,
φ
=
φ
(
x
±
at
)isasolution to Eq. (A2.50), provided that
∂φ/∂
x
,
2
x
2
,
2
t
2
∂
φ/∂
∂φ/∂
t
and
∂
φ/∂
are continuous. The simplest harmonic solution to Eq. (A2.50)
is
(A2.51)
φ
=
cos[
κ
(
x
−
α
t
)]
where
κ
is a constant termed the
wave number
. Equation (A2.51) describes a plane wave
travelling in the
x
direction with velocity
α
. The displacement of the medium due to the
passage of this wave is given by Eq. (A2.45):
u
=
∇
φ
∂φ
∂
x
,
∂φ
y
,
∂φ
=
∂
∂
z
(A2.52)
=
(
−
κ
sin[
κ
(
x
−
at
)]
,
0
,
0)
