We then introduce concepts of wave propagation in the
earth, with emphasis on how waves behave when they encounter changes in physical properties. These ideas give us the tools
for Chapter 3, which discusses how seismic waves are used
to study the interior of the earth, and Chapter 4, where we discuss how seismic waves are used to study earthquakes.
Although we focus on seismic waves, many of the concepts
are similar to ones for other types of waves, so we will sometimes draw analogies to familar behavior of light, water, and
sound waves.
2.2 Waves on a string
2.2.1 Theory
We consider an idealized mathematical string that extends in
the x direction. Initially the string is straight in response to
a tension force τ exerted along it, so u, the displacement from
the equilibrium position in the y direction, is zero everywhere.
After the string is plucked, portions of the string are displaced
from their equilibrium positions and disturbances move along
the string.
Our goal is to describe the displacement u(x, t) as a function
of both position along the string and of time. To do this, we
apply Newton’s second law of motion, F = ma, which states that
the force vector equals the mass times the acceleration vector, 1
to a segment dx of the string. Once the string segment is displaced, the string is stretched and the tension directed along the
2 Basic Seismological Theory
A very interesting example of sound waves in a solid, both longitudinal and transverse, are waves in the solid earth. Inside the earth,
from time to time, there are earthquakes so sound waves travel around in the earth. Therefore if we place a seismograph at some location and watch the way the thing jiggles after there has been an earthquake somewhere else, we might get a jiggling, and a quieting
down, and another jiggling . . . By using a large number of observations of many earthquakes at different places, we know what is
inside the earth.
Richard Feynman, The Feynman Lectures on Physics, 1963
1 Bold face is commonly used to denote vectors; see Section A.3.1.
2.1 Introduction
We begin the study of seismic waves in the earth by addressing
two basic questions. First, what in the physics of the solid earth
allows waves to propagate through it? Second, how does the
propagation of seismic waves depend on the nature of the
material within the earth?
We will see that seismic waves propagate through the earth
because the material within it, though solid, can undergo
internal deformation. As a result, earthquakes and other disturbances generate seismic waves, which give information
about both the source of the waves and the material they pass
through.
To motivate these ideas, we first discuss a stretched string, a
simple physical system that gives rise to waves analogous to
seismic waves in the earth. As for the solid earth, deforming the
string causes displacements that are functions of space and time
satisfying the wave equation. The velocity of the propagating
waves depends on the physical properties of the string in a way
similar to that for waves in the earth, and the waves respond to
changes in the physical properties of the string in ways
analogous to what occurs for waves in the earth.
After discussing the string, we develop basic ideas about the
mechanics of the solid earth. We introduce the stress tensor,
which describes the forces acting within a deformable solid
material, and the strain tensor, which describes the deformation. We then explore the relation between these tensors,
and show that the displacements within the material can be
described as functions of position and time satisfying the wave
equation. Specifically, we will see how two types of seismic
waves, P and S, propagate.
earth, with emphasis on how waves behave when they encounter changes in physical properties. These ideas give us the tools
for Chapter 3, which discusses how seismic waves are used
to study the interior of the earth, and Chapter 4, where we discuss how seismic waves are used to study earthquakes.
Although we focus on seismic waves, many of the concepts
are similar to ones for other types of waves, so we will sometimes draw analogies to familar behavior of light, water, and
sound waves.
2.2 Waves on a string
2.2.1 Theory
We consider an idealized mathematical string that extends in
the x direction. Initially the string is straight in response to
a tension force τ exerted along it, so u, the displacement from
the equilibrium position in the y direction, is zero everywhere.
After the string is plucked, portions of the string are displaced
from their equilibrium positions and disturbances move along
the string.
Our goal is to describe the displacement u(x, t) as a function
of both position along the string and of time. To do this, we
apply Newton’s second law of motion, F = ma, which states that
the force vector equals the mass times the acceleration vector, 1
to a segment dx of the string. Once the string segment is displaced, the string is stretched and the tension directed along the
2 Basic Seismological Theory
A very interesting example of sound waves in a solid, both longitudinal and transverse, are waves in the solid earth. Inside the earth,
from time to time, there are earthquakes so sound waves travel around in the earth. Therefore if we place a seismograph at some location and watch the way the thing jiggles after there has been an earthquake somewhere else, we might get a jiggling, and a quieting
down, and another jiggling . . . By using a large number of observations of many earthquakes at different places, we know what is
inside the earth.
Richard Feynman, The Feynman Lectures on Physics, 1963
1 Bold face is commonly used to denote vectors; see Section A.3.1.
2.1 Introduction
We begin the study of seismic waves in the earth by addressing
two basic questions. First, what in the physics of the solid earth
allows waves to propagate through it? Second, how does the
propagation of seismic waves depend on the nature of the
material within the earth?
We will see that seismic waves propagate through the earth
because the material within it, though solid, can undergo
internal deformation. As a result, earthquakes and other disturbances generate seismic waves, which give information
about both the source of the waves and the material they pass
through.
To motivate these ideas, we first discuss a stretched string, a
simple physical system that gives rise to waves analogous to
seismic waves in the earth. As for the solid earth, deforming the
string causes displacements that are functions of space and time
satisfying the wave equation. The velocity of the propagating
waves depends on the physical properties of the string in a way
similar to that for waves in the earth, and the waves respond to
changes in the physical properties of the string in ways
analogous to what occurs for waves in the earth.
After discussing the string, we develop basic ideas about the
mechanics of the solid earth. We introduce the stress tensor,
which describes the forces acting within a deformable solid
material, and the strain tensor, which describes the deformation. We then explore the relation between these tensors,
and show that the displacements within the material can be
described as functions of position and time satisfying the wave
equation. Specifically, we will see how two types of seismic
waves, P and S, propagate.