2.4 Seismic waves
2.4.1 The seismic wave equation
The ideas of elasticity in the last section let us show that the
equation of motion has solutions that describe the two types
of propagating seismic (or elastic) waves, compressional and
shear waves. We will see that these wave types propagate
differently, with velocities that depend in different ways on
the elastic properties of the material. Our approach to showing that the equations of elasticity have propagating wave
solutions is conceptually similar to the way we showed (Section 2.2) that the physics of a string gives rise to traveling
waves. In that analysis, we first demonstrated that waves
occur on a uniform string, and then considered how waves
propagate between strings of differing properties. That analysis considered propagating waves without regard to how
they were generated.
Following that approach, we consider a homogeneous 1 region, one of uniform properties, within an elastic material. We
assume that the region contains no source of seismic waves,
which requires a body force. Once the waves propagate away
from the source, the relation between the stresses and displacements is given by the homogeneous equation of motion,
which includes no body force term, so F = ma becomes
σ
ρ
ij j
i
t
u
t
t
, ( , )
( , ) .
x
x
=
∂
∂
2
2
(1)
Before solving the equation, two points are worth noting.
The equation of motion can be written and solved entirely in
terms of displacements, because the stress is related to the
strain, which is formed from derivatives of the displacement.
The stress and strain are related by the constitutive relation,
which characterizes the material. Thus, although the equation of motion does not depend on the elastic constants, the
solution does. Second, the equation of motion relates spatial
derivatives of the stress tensor to a time derivative of the displacement vector. The resulting solutions give the displacement
vector and hence the strain and stress tensors as functions of
both space and time. Often, for simplicity, these dependences
are not explicitly written.
We solve Eqn 1 in Cartesian (x, y, z) coordinates, beginning
with the x component,
∂
∂
∂
∂
∂
∂
∂
∂
2
2
σ
σ
σ
ρ
xx
xy
xz
x
t
x
t
y
t
z
u
t
t
( , )
( , )
( , )
( , ) .
x
x
x
x
+
+
=
(2)
To express this in terms of displacements, we use the constitutive law for an isotropic elastic medium (Eqn 2.3.70),
1 Unfortunately, this word is used for two different concepts: a homogeneous
medium has properties that do not vary with position, whereas a homogeneous equation has no forcing function or source term.
σ ij = λθδ ij + 2µe ij ,
(3)
and write the strains in terms of displacements, which yields
σ xx = λθ + 2µe xx = λθ + 2µ
∂
∂
u
x
x
σ xy = 2µe xy = µ
∂
∂
∂
∂
u
y
u
x
x
y
+
⎛
⎝
⎜
⎞
⎠
⎟
σ xz = 2µe xz =
+
⎛
⎝
⎜
⎞
⎠
⎟ .
µ
∂
∂
∂
∂
u
z
u
x
x
z
(4)
We then take derivatives of the stress components
∂
∂
∂
∂
∂
∂
2
σ
λ
θ
µ
xx
x
x
x
u
x
=
+2
2
∂
∂
∂
∂
∂
∂ ∂
2
2
σ
µ
xy
x
y
y
u
y
u
y x
=
+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
2
∂
∂
∂
∂
∂
∂ ∂
2
2
σ
µ
xz
x
z
z
u
z
u
z x
=
+
⎛
⎝
⎜
⎞
⎠
⎟
2
(5)
using the fact that for a homogeneous material the elastic
constants do not vary with position. Finally, substituting the
derivatives into the equation of motion and using the definitions of the dilatation
θ = ⋅ =
+
+
∇ ∇ u
∂
∂
∂
∂
∂
∂
u
x
u
y
u
z
x
y
z
(6)
and of the Laplacian (Section A.6.5)
∇
=
+
+
2 ( )
u
u
x
u
y
u
z
x
x
x
x
∂
∂
∂
∂
∂
∂
2
2
2
2
2
2
(7)
yields
(
)
( )
λ µ
θ µ
ρ
+
+ ∇
=
∂
∂
∂
∂
2
2
x
u
u
t
x
x
2
(8)
for the x component of the equation of motion (1).
Similar equations can be obtained for the y and z components of displacement. The three equations can be combined,
using the vector Laplacian of the displacement field
∇
∇ ∇
∇ ∇
2 u = (∇
2
u x , ∇
2
u y , ∇
2 u z ),
(9)
into a single vector equation:
(λ + µ)∇ ∇ ∇
∇ ∇(∇ ∇ ∇
∇ ∇ · u(x, t)) + µ∇ ∇ ∇
∇ ∇ 2 u(x, t) =
( , ) .
ρ
∂
∂
2
2
u x t
t
(10)
2.4 Seismic waves 53