64 Basic Seismological Theory
P–SV waves have no effect on the SH waves, and vice versa,
so there is no coupling between P–SV waves and SH waves.
However, P waves and SV waves are coupled, because both
affect the same components of displacement and traction. Thus
at interfaces, P waves convert to SV waves, and vice versa,
whereas SH waves do not convert to either P or SV waves.
When treating the earth as a horizontally layered medium,
we assume that P– SV and SH waves propagating between
any two points are decoupled and can be treated separately.
The situation is more complicated when dipping interfaces are
present. P–SV and SH are coupled at a dipping interface if its
normal is not in the plane of propagation, the vertical plane
containing the source and the receiver. Thus, for dipping interfaces, the waves will be coupled for most pairs of source and
receiver positions.
As a result, in most applications we treat the P–SV system of
propagating waves as distinct from SH. In the last section, we
saw that P waves are described by the scalar potential that
satisfies the scalar wave equation (Eqn 2.4.37), whereas the S
waves are described by the vector potential ϒ
ϒ ϒ
ϒ ϒ satisfying the
vector wave equation (Eqn 2.4.38). To see that the SV and
SH potentials each satisfy the vector wave equation separately,
we substitute Eqn 1 into it:
∇
∇ ∇
∇ ∇ 2 [Ψ Ψ Ψ
Ψ Ψ(x, z, t) + ∇
∇ ∇
∇ ∇ × χ
χ χ
χ χ (x, z, t)] =
1
2
2
2
β
∂
∂t
[Ψ Ψ Ψ
Ψ Ψ(x, z, t)
+ ∇
∇ ∇
∇ ∇ × χ
χ χ
χ χ(x, z, t)],
(8)
and regroup the terms:
∇
∇ ∇
∇ ∇
2
Ψ
Ψ Ψ
Ψ Ψ(x, z, t) −
1
2
2
2
β
∂
∂
Ψ Ψ( , , )
x z t
t
= −∇ ∇ ∇
∇ ∇ 2 [∇ ∇ ∇
∇ ∇ × χ
χ χ
χ χ(x, z, t)] +
1
2
2
2
β
∂
∂t
[∇ ∇ ∇
∇ ∇ × χ
χ χ
χ χ(x, z, t)],
(9)
so the two potentials can be treated separately. Thus the P–SV
system is described by
∇
=
2
2
2
2
1
φ
α
φ
( , , )
( , , ) ,
x z t
x z t
t
∂
∂
∇
=
2
2
2
2
1
ψ
β
ψ
( , , )
( , , ) .
x z t
x z t
t
∂
∂
(10)
Both of these are scalar wave equations, because ψ is the scalar
function forming the y component of the SV vector potential
(Eqn 3).
For SH waves we have two choices. Interchanging the curl
and the other derivatives in the right side of Eqn 9 shows that
the scalar function χ, the y component of the SH vector potential, satisfies a scalar wave equation. Alternatively, we can take
the curl and recognize that by Eqns 4 and 5
u y = ∇
∇ ∇
∇ ∇ × ∇
∇ ∇
∇ ∇ × χ
χ χ
χ χ (x, z, t),
(11)
(P)
∇ ∇φ
φ
φ
( , , )
( , , ) , ,
( , , )
x z t
x z t
x
x z t
z
=
⎛
⎝
⎜
⎞
⎠
⎟
∂
∂
∂
∂
0
(SV)
∇ ∇
( , , )
( , , ) , ,
( , , )
×
=
⎛
⎝
⎜
⎞
⎠
⎟
ψ ψ x z t
x z t
z
x z t
x
−∂
∂
∂
∂
ψ
ψ
0
(SH) ∇ ∇ ∇ ∇
( , , )
,
( , , )
( , , ) , .
× ×
=
−
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎛
⎝
⎜
⎞
⎠
⎟
χ χ x z t
x z t
x
x z t
z
0
0
∂
∂
∂
∂
2
2
2
2
χ
χ
(4)
Thus P and SV contribute to the x and z components of displacement, whereas SH contributes only to the y component.
The divergences ∇
∇ ∇
∇ ∇ · Ψ and ∇
∇ ∇
∇ ∇ · χ equal zero because only their
y components are nonzero, and ∂/ ∂y of these components is
zero. Hence, as expected, neither SH nor SV gives rise to a
volume change.
The components of the displacement vector are found by
grouping the components from Eqn 4:
u x z t
x z t
x
x z t
z
x ( , , )
( , , )
( , , )
=
−
∂
∂
∂
∂
φ
ψ
u x z t
x z t
z
x z t
x
z ( , , )
( , , )
( , , )
=
+
∂
∂
∂
∂
φ
ψ
u x z t
x z t
x
x z t
z
x z t
y ( , , )
( , , )
( , , )
( , , )
= −
+
⎛
⎝
⎜
⎞
⎠
⎟ = −∇
∂
∂
∂
∂
2
2
2
2
χ
χ
χ
2
. (5)
These equations demonstrate that P–SV waves are independent
of SH waves. The x and z components of displacement depend
on both the P-wave potential φ and the SV-potential ψ. Thus
for waves propagating in the x–z plane, the P and SV waves
form a coupled system, which gives rise to two components
of displacement. Neither the P nor the SV potentials contribute
to the y component of displacement. Hence SH waves, which
alone contribute to the y component of displacement, are
decoupled from P and SV waves.
This coupling and decoupling persists when these waves
interact with a horizontal interface parallel to the x–y plane.
The boundary conditions at the interface constrain the displacements and tractions (Section 2.3.10). Because the normal
to the interface has only a z component,
4 = (0, 0, 1), n j = δ j3 ,
(6)
the tractions on the interface are given by
T i = σ ij n j = σ i3 = (σ xz , σ yz , σ zz ).
(7)
The P–SV system gives rise to nonzero components of displacement u x and u z , and hence tractions σ xz and σ zz . For these
waves, both u y = 0 and σ yz = 0. By contrast, the SH waves
contribute only a y component of displacement, and their only
nonzero traction component is σ yz . Thus, at the interface, the
P–SV waves have no effect on the SH waves, and vice versa,
so there is no coupling between P–SV waves and SH waves.
However, P waves and SV waves are coupled, because both
affect the same components of displacement and traction. Thus
at interfaces, P waves convert to SV waves, and vice versa,
whereas SH waves do not convert to either P or SV waves.
When treating the earth as a horizontally layered medium,
we assume that P– SV and SH waves propagating between
any two points are decoupled and can be treated separately.
The situation is more complicated when dipping interfaces are
present. P–SV and SH are coupled at a dipping interface if its
normal is not in the plane of propagation, the vertical plane
containing the source and the receiver. Thus, for dipping interfaces, the waves will be coupled for most pairs of source and
receiver positions.
As a result, in most applications we treat the P–SV system of
propagating waves as distinct from SH. In the last section, we
saw that P waves are described by the scalar potential that
satisfies the scalar wave equation (Eqn 2.4.37), whereas the S
waves are described by the vector potential ϒ
ϒ ϒ
ϒ ϒ satisfying the
vector wave equation (Eqn 2.4.38). To see that the SV and
SH potentials each satisfy the vector wave equation separately,
we substitute Eqn 1 into it:
∇
∇ ∇
∇ ∇ 2 [Ψ Ψ Ψ
Ψ Ψ(x, z, t) + ∇
∇ ∇
∇ ∇ × χ
χ χ
χ χ (x, z, t)] =
1
2
2
2
β
∂
∂t
[Ψ Ψ Ψ
Ψ Ψ(x, z, t)
+ ∇
∇ ∇
∇ ∇ × χ
χ χ
χ χ(x, z, t)],
(8)
and regroup the terms:
∇
∇ ∇
∇ ∇
2
Ψ
Ψ Ψ
Ψ Ψ(x, z, t) −
1
2
2
2
β
∂
∂
Ψ Ψ( , , )
x z t
t
= −∇ ∇ ∇
∇ ∇ 2 [∇ ∇ ∇
∇ ∇ × χ
χ χ
χ χ(x, z, t)] +
1
2
2
2
β
∂
∂t
[∇ ∇ ∇
∇ ∇ × χ
χ χ
χ χ(x, z, t)],
(9)
so the two potentials can be treated separately. Thus the P–SV
system is described by
∇
=
2
2
2
2
1
φ
α
φ
( , , )
( , , ) ,
x z t
x z t
t
∂
∂
∇
=
2
2
2
2
1
ψ
β
ψ
( , , )
( , , ) .
x z t
x z t
t
∂
∂
(10)
Both of these are scalar wave equations, because ψ is the scalar
function forming the y component of the SV vector potential
(Eqn 3).
For SH waves we have two choices. Interchanging the curl
and the other derivatives in the right side of Eqn 9 shows that
the scalar function χ, the y component of the SH vector potential, satisfies a scalar wave equation. Alternatively, we can take
the curl and recognize that by Eqns 4 and 5
u y = ∇
∇ ∇
∇ ∇ × ∇
∇ ∇
∇ ∇ × χ
χ χ
χ χ (x, z, t),
(11)
(P)
∇ ∇φ
φ
φ
( , , )
( , , ) , ,
( , , )
x z t
x z t
x
x z t
z
=
⎛
⎝
⎜
⎞
⎠
⎟
∂
∂
∂
∂
0
(SV)
∇ ∇
( , , )
( , , ) , ,
( , , )
×
=
⎛
⎝
⎜
⎞
⎠
⎟
ψ ψ x z t
x z t
z
x z t
x
−∂
∂
∂
∂
ψ
ψ
0
(SH) ∇ ∇ ∇ ∇
( , , )
,
( , , )
( , , ) , .
× ×
=
−
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎛
⎝
⎜
⎞
⎠
⎟
χ χ x z t
x z t
x
x z t
z
0
0
∂
∂
∂
∂
2
2
2
2
χ
χ
(4)
Thus P and SV contribute to the x and z components of displacement, whereas SH contributes only to the y component.
The divergences ∇
∇ ∇
∇ ∇ · Ψ and ∇
∇ ∇
∇ ∇ · χ equal zero because only their
y components are nonzero, and ∂/ ∂y of these components is
zero. Hence, as expected, neither SH nor SV gives rise to a
volume change.
The components of the displacement vector are found by
grouping the components from Eqn 4:
u x z t
x z t
x
x z t
z
x ( , , )
( , , )
( , , )
=
−
∂
∂
∂
∂
φ
ψ
u x z t
x z t
z
x z t
x
z ( , , )
( , , )
( , , )
=
+
∂
∂
∂
∂
φ
ψ
u x z t
x z t
x
x z t
z
x z t
y ( , , )
( , , )
( , , )
( , , )
= −
+
⎛
⎝
⎜
⎞
⎠
⎟ = −∇
∂
∂
∂
∂
2
2
2
2
χ
χ
χ
2
. (5)
These equations demonstrate that P–SV waves are independent
of SH waves. The x and z components of displacement depend
on both the P-wave potential φ and the SV-potential ψ. Thus
for waves propagating in the x–z plane, the P and SV waves
form a coupled system, which gives rise to two components
of displacement. Neither the P nor the SV potentials contribute
to the y component of displacement. Hence SH waves, which
alone contribute to the y component of displacement, are
decoupled from P and SV waves.
This coupling and decoupling persists when these waves
interact with a horizontal interface parallel to the x–y plane.
The boundary conditions at the interface constrain the displacements and tractions (Section 2.3.10). Because the normal
to the interface has only a z component,
4 = (0, 0, 1), n j = δ j3 ,
(6)
the tractions on the interface are given by
T i = σ ij n j = σ i3 = (σ xz , σ yz , σ zz ).
(7)
The P–SV system gives rise to nonzero components of displacement u x and u z , and hence tractions σ xz and σ zz . For these
waves, both u y = 0 and σ yz = 0. By contrast, the SH waves
contribute only a y component of displacement, and their only
nonzero traction component is σ yz . Thus, at the interface, the