38 Basic Seismological Theory
The normal mode solution (Eqn 45) gives insight into the
relation between the medium in which waves propagate and
the source that generates them. The waves are expressed as
the sum of eigenfunctions weighted by amplitudes that depend
on the source. The physical properties of the string control its
velocity and thus its eigenfrequencies and spatial eigenfunctions.
The displacement due to any particular source corresponds to a
different weighting of the eigenfunctions. By analogy, we use
the eigenfrequencies of the earth’s normal modes to study the
properties of the medium (earth structure), and the displacement (the specific weighting of eigenfunctions) to study the
source (generally an earthquake) that excited them.
The normal mode solution generates all the incident, reflected,
and transmitted waves, although they do not appear individually as they do in a traveling wave solution like Eqn 8. The
mode solution is thus less intuitive, and individual modes are
not physically meaningful, although their sum is. For example,
each mode mathematically starts vibrating along the entire
string at time zero, even though no waves have reached the
string ends. When the modes are summed, the resulting waves
propagate at the correct velocity.
The solution also illustrates an important relation between
the positions of the source and the receiver. The fact that
Eqn 45 depends in the same way on the positions of the source
(x s ) and the receiver (x) illustrates the principle of reciprocity,
which states that under appropriate conditions the same displacement occurs if the positions of the source and the receiver
are interchanged. This principle is important for studying earth
structure because it is often convenient to place the source or
the receiver at a particular site. We can do this knowing that
the same ray paths and thus waves result. 6 Equation 45 also
illustrates an important point about the relation of the source
position to the waves generated: namely, a source at a point
where a particular mode has no displacement will not excite
that mode. For example, in Fig. 2.2-8, modes with numbers
that are multiples of five give zero displacement because the
source term sin (nπx s /20) is zero. Analogously, in the earth,
surface waves whose displacements are largest near the surface
are not excited well by deep earthquakes.
Finally, although we have discussed the normal modes of
a uniform string, we could generalize these ideas to find the
modes of a non-uniform string. One way to do this is to extend
the method used to find the reflection and transmission coefficients (Section 2.2.3). We treat the string as a set of uniform
pieces, use the harmonic wave solution in each piece, and
impose displacement and traction boundary conditions at the
junctions. We then numerically find eigenfrequencies that
satisfy the fixed boundary condition at the string’s end. The
normal modes of the non-uniform string are then summed to
give the traveling waves. The waves on the non-uniform string
in Fig. 2.2-6 were calculated in this way.
2.3 Stress and strain
2.3.1 Introduction
By applying Newton’s second law of motion, F = ma, to a string,
we found that deforming the string gave rise to propagating
waves. Similarly, deforming the solid earth produces seismic
waves. We study these waves using concepts from continuum
mechanics, which describes the behavior of a continuous
deformable material made up of particles packed so closely
together that density, force, and displacement can be thought
of as continuous and differentiable functions. This approximation breaks down on an atomic distance scale, but is adequate
for most seismological problems.
For these applications, we write Newton’s second law in
terms of the force per unit volume and the density, the mass per
unit volume. If the density does not change with time, the force
per unit volume f(x, t) equals the inertial term, the product
of the density ρ and the second derivative of the displacement
vector u(x, t) with time. Thus F = ma becomes
f x
u x
( , )
( , ) .
t
t
t
= ρ
∂
∂
2
2
(1)
This vector equation can be written as a set of three equations,
one for each component of the force and displacement vectors 1
f
t
u
t
t
i
i
( , )
( , ) .
x
x
= ρ
∂
∂
2
2
(2)
In seismic wave propagation, both the displacement and the
force vectors can vary in space and time. Although this dependence is generally not written explicitly, we will sometimes do
so to remind ourselves that the solutions depend on space and
time.
The goal of this section is to use Newton’s second law to
characterize a continuous medium and its response to applied
forces. We first introduce the stress tensor that describes the
forces acting on a deformable continuous medium. We then
formulate the equation of motion, the version of Newton’s law
appropriate for a continuous medium, which relates the stress
to the displacement. The variation in displacement within the
material, described by the strain tensor, gives rise to internal
deformation. This deformation is related to the stress via the
constitutive equation that characterizes the properties of the
material. Our brief discussion covers some basic results of
continuum mechanics necessary for introductory seismology.
The suggested reading listed at the end of the chapter provides
further treatment of these and related topics.
6 A familiar version for light waves, seen on the back of large trucks, warns other
drivers that “If you can’t see my mirrors, I can’t see you.”
1 The three equations are written as one using index notation (Section A.3.5) in
which the index i ranges from 1 to 3 over the coordinate axes. Index notation makes
cumbersome vector equations shorter, clearer, and often easier to solve. These equations are often made even more compact using a dot superscript to indicate differentiation with respect to time, so the acceleration is ü i .
The normal mode solution (Eqn 45) gives insight into the
relation between the medium in which waves propagate and
the source that generates them. The waves are expressed as
the sum of eigenfunctions weighted by amplitudes that depend
on the source. The physical properties of the string control its
velocity and thus its eigenfrequencies and spatial eigenfunctions.
The displacement due to any particular source corresponds to a
different weighting of the eigenfunctions. By analogy, we use
the eigenfrequencies of the earth’s normal modes to study the
properties of the medium (earth structure), and the displacement (the specific weighting of eigenfunctions) to study the
source (generally an earthquake) that excited them.
The normal mode solution generates all the incident, reflected,
and transmitted waves, although they do not appear individually as they do in a traveling wave solution like Eqn 8. The
mode solution is thus less intuitive, and individual modes are
not physically meaningful, although their sum is. For example,
each mode mathematically starts vibrating along the entire
string at time zero, even though no waves have reached the
string ends. When the modes are summed, the resulting waves
propagate at the correct velocity.
The solution also illustrates an important relation between
the positions of the source and the receiver. The fact that
Eqn 45 depends in the same way on the positions of the source
(x s ) and the receiver (x) illustrates the principle of reciprocity,
which states that under appropriate conditions the same displacement occurs if the positions of the source and the receiver
are interchanged. This principle is important for studying earth
structure because it is often convenient to place the source or
the receiver at a particular site. We can do this knowing that
the same ray paths and thus waves result. 6 Equation 45 also
illustrates an important point about the relation of the source
position to the waves generated: namely, a source at a point
where a particular mode has no displacement will not excite
that mode. For example, in Fig. 2.2-8, modes with numbers
that are multiples of five give zero displacement because the
source term sin (nπx s /20) is zero. Analogously, in the earth,
surface waves whose displacements are largest near the surface
are not excited well by deep earthquakes.
Finally, although we have discussed the normal modes of
a uniform string, we could generalize these ideas to find the
modes of a non-uniform string. One way to do this is to extend
the method used to find the reflection and transmission coefficients (Section 2.2.3). We treat the string as a set of uniform
pieces, use the harmonic wave solution in each piece, and
impose displacement and traction boundary conditions at the
junctions. We then numerically find eigenfrequencies that
satisfy the fixed boundary condition at the string’s end. The
normal modes of the non-uniform string are then summed to
give the traveling waves. The waves on the non-uniform string
in Fig. 2.2-6 were calculated in this way.
2.3 Stress and strain
2.3.1 Introduction
By applying Newton’s second law of motion, F = ma, to a string,
we found that deforming the string gave rise to propagating
waves. Similarly, deforming the solid earth produces seismic
waves. We study these waves using concepts from continuum
mechanics, which describes the behavior of a continuous
deformable material made up of particles packed so closely
together that density, force, and displacement can be thought
of as continuous and differentiable functions. This approximation breaks down on an atomic distance scale, but is adequate
for most seismological problems.
For these applications, we write Newton’s second law in
terms of the force per unit volume and the density, the mass per
unit volume. If the density does not change with time, the force
per unit volume f(x, t) equals the inertial term, the product
of the density ρ and the second derivative of the displacement
vector u(x, t) with time. Thus F = ma becomes
f x
u x
( , )
( , ) .
t
t
t
= ρ
∂
∂
2
2
(1)
This vector equation can be written as a set of three equations,
one for each component of the force and displacement vectors 1
f
t
u
t
t
i
i
( , )
( , ) .
x
x
= ρ
∂
∂
2
2
(2)
In seismic wave propagation, both the displacement and the
force vectors can vary in space and time. Although this dependence is generally not written explicitly, we will sometimes do
so to remind ourselves that the solutions depend on space and
time.
The goal of this section is to use Newton’s second law to
characterize a continuous medium and its response to applied
forces. We first introduce the stress tensor that describes the
forces acting on a deformable continuous medium. We then
formulate the equation of motion, the version of Newton’s law
appropriate for a continuous medium, which relates the stress
to the displacement. The variation in displacement within the
material, described by the strain tensor, gives rise to internal
deformation. This deformation is related to the stress via the
constitutive equation that characterizes the properties of the
material. Our brief discussion covers some basic results of
continuum mechanics necessary for introductory seismology.
The suggested reading listed at the end of the chapter provides
further treatment of these and related topics.
6 A familiar version for light waves, seen on the back of large trucks, warns other
drivers that “If you can’t see my mirrors, I can’t see you.”
1 The three equations are written as one using index notation (Section A.3.5) in
which the index i ranges from 1 to 3 over the coordinate axes. Index notation makes
cumbersome vector equations shorter, clearer, and often easier to solve. These equations are often made even more compact using a dot superscript to indicate differentiation with respect to time, so the acceleration is ü i .