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

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.

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 t 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 string gives rise to forces (Fig. 2.2-1) in the y direction of t sin ?2 and -t sin ?1 at the ends of the segment. The net force in the y direction equals the inertial term, which is the acceleration (second time derivative of the displacement) times the mass, where the mass is the product of the density ? and dx. Hence, the vector equation F = ma becomes the scalar equation

Fig. 2.2-1 Geometry of a segment of a string subject to a tension t. A slight difference in the angles ?1 and ?2 provides a net force in the y direction of F = t sin ?2 - t sin ?1, which accelerates the string.

(1)

If the angles ? are small, sin ? ~ ? ~ tan ? can be approximated by the slope, so

(2)

which can be expanded by forming a Taylor series and discarding the higher-order terms:

(3)

yielding the wave equation:

(4)

where v = (t/?)1/2.

This equation gives the relationship between the time and space derivatives of the displacement u(x, t) along the string. We will see that the coupling between the two partial derivatives gives rise to waves propagating along the string with a velocity v. Because (4) describes the propagation of the scalar quantity u(x, t) in one space dimension, it is called the onedimensional scalar wave equation.

Fig. 2.2-2 "Snapshots" of a string showing a pulse f(x - 2t) traveling to the right in the +x direction. Because the velocity is 2, the pulse moves two distance units during each time unit. This pulse is one of many forms a traveling wave can take.

The wave equation is easily solved, because any function with the form u(x, t) = f(x ± vt) is a solution. To show this, note that the partial derivatives are

(5)

where f? is the second derivative of f with respect to its argument. Thus, although we often think of solutions to the wave equation as sines and cosines, any function whose argument is (x ± vt) is a solution.

To see that a function f(x - vt) describes a propagating wave, consider how it varies in space and time. As time increases by an increment dt, the argument stays constant provided that the distance increases by vdt. Because the function's value stays the same when its argument is constant, f(x - vt) describes a wave of constant shape propagating with velocity v in the positive x direction (Fig. 2.2-2). Similarly, because (x + vt) is constant if x decreases as time increases, f(x + vt) describes a wave propagating with velocity v in the -x direction. The sign relating the x and t terms thus shows which way the wave travels. We follow seismological convention and use the vector term "velocity" for v, although it is a scalar and thus better termed a "speed."

The velocity v = (t/?)1/2 at which the waves propagate depends on two physical properties of the string: the tension with which it is stretched and its density. Equation 1 shows how these properties interact. Because the tension provides the force that tends to restore any displacement to the equilibrium position, greater tension gives higher acceleration and thus faster wave propagation. In contrast, because the density appears in the inertial term, higher density gives lower acceleration and slower wave propagation.

The fact that the velocity depends on the density illustrates one of the reasons why the string is a useful analogy for seismic waves in the earth. One goal of seismology is to study the composition of the earth. For this purpose, we measure the time that waves take to travel between sources and receivers, find the velocity at which the waves propagated, and thus learn about the properties of the earth.

2.2.2 Harmonic wave solution

Any function of the form f(x ± vt) describes a propagating wave as a function of time and distance. A particularly useful form is a harmonic or sinusoidal wave2

(6)

A harmonic wave is characterized by its amplitude A and two parameters, ? and k, which we will discuss shortly. Substituting into the wave equation (4) and canceling the exponential and constant show that the wave velocity is the ratio

(7)

Although the exponential function u(x, t) in Eqn 6 is complex, the physical displacement must be real. We thus describe the displacement as the real part of u(x, t). The complex exponential form can be used for most purposes, because when a complex exponential appears in the solution of a physical problem, its conjugate also appears, so their sum yields a real displacement.

To understand the harmonic wave solution, consider the wave given by the real part of u(x, t), which is A cos (?t - kx). Figure 2.2-3 shows how this function varies with both distance and time. The value of u is constant when the phase (?t - kx) remains constant, as for a crest or a trough. Such lines of constant phase...