Oscillations and Waves

249 Pages·2013·2.435 MB·English

by  Richard Fitzpatrick

Oscillations and Waves
Richard Fitzpatrick
Professor of Physics
The University of Texas at Austin
Contents
1 Introduction 5
2 Simple Harmonic Oscillation 7
2.1 Mass on a Spring . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Simple Harmonic Oscillator Equation . . . . . . . . . . . . . 12
2.3 LC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Simple Pendulum . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Damped and Driven Harmonic Oscillation 23
3.1 Damped Harmonic Oscillation . . . . . . . . . . . . . . . . . 23
3.2 Quality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 LCR Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Driven Damped Harmonic Oscillation . . . . . . . . . . . . . 29
3.5 Driven LCR Circuit . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Transient Oscillator Response . . . . . . . . . . . . . . . . . 36
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Coupled Oscillations 43
4.1 Two Spring-Coupled Masses . . . . . . . . . . . . . . . . . . 43
4.2 Two Coupled LC Circuits . . . . . . . . . . . . . . . . . . . . 48
4.3 Three Spring Coupled Masses . . . . . . . . . . . . . . . . . 51
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2 OSCILLATIONSANDWAVES
5 Transverse Standing Waves 55
5.1 Normal Modes of a Beaded String . . . . . . . . . . . . . . . 55
5.2 Normal Modes of a Uniform String . . . . . . . . . . . . . . 63
5.3 General Time Evolution of a Uniform String . . . . . . . . . 69
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6 Longitudinal Standing Waves 81
6.1 Spring Coupled Masses . . . . . . . . . . . . . . . . . . . . . 81
6.2 Sound Waves in an Elastic Solid . . . . . . . . . . . . . . . . 86
6.3 Sound Waves in an Ideal Gas . . . . . . . . . . . . . . . . . . 91
6.4 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Traveling Waves 103
7.1 Standing Waves in a Finite Continuous Medium . . . . . . . 103
7.2 Traveling Waves in an Infinite Continuous Medium . . . . . 104
7.3 Wave Interference . . . . . . . . . . . . . . . . . . . . . . . . 107
7.4 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . 109
7.5 Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . 113
7.6 Reflection and Transmission at Boundaries . . . . . . . . . . 115
7.7 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . 122
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8 Wave Pulses 131
8.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . 131
8.2 General Solution of the Wave Equation . . . . . . . . . . . . 137
8.3 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
9 Dispersive Waves 151
9.1 Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . 151
9.2 Electromagnetic Wave Propagation in Plasmas . . . . . . . . 154
9.3 Electromagnetic Wave Propagation in Conductors . . . . . . 161
9.4 Surface Wave Propagation in Water . . . . . . . . . . . . . . 165
9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10 Multi-Dimensional Waves 173
10.1 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.2 Three-Dimensional Wave Equation . . . . . . . . . . . . . . 175
10.3 Laws of Geometric Optics . . . . . . . . . . . . . . . . . . . . 175
CONTENTS 3
10.4 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
10.5 Cylindrical Waves . . . . . . . . . . . . . . . . . . . . . . . . 180
10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
11 Wave Optics 183
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
11.2 Two-Slit Interference . . . . . . . . . . . . . . . . . . . . . . 183
11.3 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
11.4 Multi-Slit Interference . . . . . . . . . . . . . . . . . . . . . 197
11.5 Fourier Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 202
11.6 Single-Slit Diffraction . . . . . . . . . . . . . . . . . . . . . . 203
11.7 Multi-Slit Diffraction . . . . . . . . . . . . . . . . . . . . . . 206
11.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
12 Wave Mechanics 213
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
12.2 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . 214
12.3 Electron Diffraction . . . . . . . . . . . . . . . . . . . . . . . 216
12.4 Representation of Waves via Complex Numbers . . . . . . . 216
12.5 Schro¨dinger’s Equation . . . . . . . . . . . . . . . . . . . . . 219
12.6 Probability Interpretation of the Wavefunction . . . . . . . . 220
12.7 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.8 Heisenberg’s Uncertainty Principle . . . . . . . . . . . . . . . 227
12.9 Collapse of the Wavefunction . . . . . . . . . . . . . . . . . 228
12.10 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . 230
12.11 Three-Dimensional Wave Mechanics . . . . . . . . . . . . . . 234
12.12 Particle in a Finite Potential Well . . . . . . . . . . . . . . . . 237
12.13 Square Potential Barrier . . . . . . . . . . . . . . . . . . . . 241
12.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
A Useful Information 247
A.1 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . 247
A.2 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . 247
A.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
A.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
4 OSCILLATIONSANDWAVES
Introduction 5
1 Introduction
Oscillations and waves are ubiquitous phenomena that are encountered in
many different areas of physics. An oscillation is a disturbance in a physical
systemthatisrepetitiveintime. Awaveisadisturbanceinanextendedphys-
ical system that is both repetitive in time and periodic in space. In general,
an oscillation involves a continuous back and forth flow of energy between
twodifferentenergytypes: e.g.,kineticandpotentialenergy,inthecaseofa
pendulum. Awaveinvolvessimilarrepetitiveenergyflowstoanoscillation,
but, in addition, is capable of transmitting energy and information from
place to place. Now, although sound waves and electromagnetic waves,
for example, rely on quite distinct physical mechanisms, they, nevertheless,
sharemanycommonproperties. Thesameistrueofdifferenttypesofoscil-
lation. It turns out that the common factor linking various types of wave is
that they are all described by the same mathematical equations. Again, the
same is true of various types of oscillation.
The aim of this course is to develop a unified mathematical theory of
oscillations and waves in physical systems. Examples will be drawn from
the dynamics of discrete mechanical systems; continuous gases, fluids, and
elastic solids; electronic circuits; electromagnetic waves; and quantum me-
chanical systems.
This course assumes a basic familiarity with the laws of physics, such
as might be obtained from a two-semester introductory college-level survey
course. Students are also assumed to be familiar with standard mathemat-
ics, up to and including trigonometry, linear algebra, differential calculus,
integral calculus, ordinary differential equations, partial differential equa-
tions, and Fourier series.
Thetextbookswhichwereconsultedmostoftenduringthedevelopment
of the course material are:
Waves, Berkeley Physics Course, Vol. 3, F.S. Crawford, Jr. (McGraw-Hill,
New York NY, 1968).
Vibrations and Waves, A.P. French (W.W. Norton & Co., New York NY,
1971).
IntroductiontoWavePhenomena,A.Hirose,andK.E.Lonngren(JohnWiley
& Sons, New York NY, 1985).
6 OSCILLATIONSANDWAVES
The Physics of Vibrations and Waves, 5th Edition, H.J. Pain (John Wiley &
Sons, Chichester UK, 1999).
SimpleHarmonicOscillation 7
2 Simple Harmonic Oscillation
2.1 Mass on a Spring
Consider a compact mass m which slides over a frictionless horizontal sur-
face. Suppose that the mass is attached to one end of a light horizontal
spring whose other end is anchored in an immovable wall. See Figure 2.1.
At time t, let x(t) be the extension of the spring: i.e., the difference be-
tween the spring’s actual length and its unstretched length. Obviously, x(t)
can also be used as a coordinate to determine the instantaneous horizontal
displacement of the mass.
Theequilibriumstateofthesystemcorrespondstothesituationinwhich
the mass is at rest, and the spring is unextended (i.e., x = x˙ = 0, where
˙ d/dt). In this state, zero horizontal force acts on the mass, and so there
≡
is no reason for it to start to move. However, if the system is perturbed
from its equilibrium state (i.e., if the mass is displaced, so that the spring
becomes extended) then the mass experiences a horizontal restoring force
given by Hooke’s law:
f(x) = −kx. (2.1)
Here, k > 0 is the so-called force constant of the spring. The negative sign
indicates that f(x) is indeed a restoring force (i.e., if the displacement is
positive then the force is negative, and vice versa). Note that the magnitude
oftherestoringforceisdirectlyproportionaltothedisplacementofthemass
from its equilibrium position (i.e., |f| x). Of course, Hooke’s law only
∝
holds for relatively small spring extensions. Hence, the displacement of the
mass cannot be made too large. Incidentally, the motion of this particular
dynamical system is representative of the motion of a wide variety of me-
chanical systems when they are slightly disturbed from a stable equilibrium
state (see Section 2.4).
Newton’s second law of motion gives following time evolution equation
for the system:
mx¨ = −kx, (2.2)
where¨ d2/dt2. Thisdifferentialequationisknownasthesimpleharmonic
≡
oscillator equation, and its solution has been known for centuries. In fact,
the solution is
x(t) = a cos(ωt−φ), (2.3)
8 OSCILLATIONSANDWAVES
x
m
x = 0
Figure 2.1: Mass on a spring
where a > 0, ω > 0, and φ are constants. We can demonstrate that Equa-
tion(2.3)isindeedasolutionofEquation(2.2)bydirectsubstitution. Plug-
ging the right-hand side of (2.3) into Equation (2.2), and recalling from
standard calculus that d(cosθ)/dθ = −sinθ and d(sinθ)/dθ = cosθ, so
that x˙ = −ωa sin(ωt − φ) and x¨ = −ω2a cos(ωt − φ), where use has
been made of the chain rule, we obtain
−mω2a cos(ωt−φ) = −ka cos(ωt−φ). (2.4)
It follows that Equation (2.3) is the correct solution provided
k
ω = . (2.5)
sm
Figure 2.2 shows a graph of x versus t obtained from Equation (2.3).
Thetypeofbehaviorshownhereiscalledsimpleharmonicoscillation. Itcan
be seen that the displacement x oscillates between x = −a and x = +a.
Here, a is termed the amplitude of the oscillation. Moreover, the motion
is repetitive in time (i.e., it repeats exactly after a certain time period has
elapsed). In fact, the repetition period is
2π
T = . (2.6)
ω
This result is easily obtained from Equation (2.3) by noting that cosθ is a
periodic function of θ with period 2π: i.e., cos(θ+2π) cosθ. It follows
≡
that the motion repeats every time ωt increases by 2π: i.e., every time t
SimpleHarmonicOscillation 9
Figure 2.2: Simple harmonic oscillation.
increases by 2π/ω. The frequency of the motion (i.e., the number of oscilla-
tions completed per second) is
1 ω
f = = . (2.7)
T 2π
It can be seen that ω is the motion’s angular frequency; i.e., the frequency
f converted into radians per second. Of course, f is measured in Hertz—
otherwiseknownascyclespersecond. Finally,thephaseangle,φ,determines
thetimesatwhichtheoscillationattainsitsmaximumdisplacement,x = a.
Infact,since themaximaofcosθoccuratθ = n2π, wherenisanarbitrary
integer, the times of maximum displacement are
φ
t = T n+ . (2.8)
max
2π
(cid:18) (cid:19)
Clearly,varyingthephaseanglesimplyshiftsthepatternofoscillationback-
ward and forward in time. See Figure 2.3.
Table 2.1 lists the displacement, velocity, and acceleration of the mass
at various different phases of the simple harmonic oscillation cycle. The in-
formationcontainedinthistablecaneasilybederivedfromEquation(2.3).
Note that all of the non-zero values shown in this table represent either the
10 OSCILLATIONSANDWAVES
Figure 2.3: Simple harmonic oscillation. The solid, short-dashed, and long
dashed-curves correspond to φ = 0, +π/4, and −π/4, respectively.
maximum or the minimum value taken by the quantity in question during
the oscillation cycle.
We have seen that when a mass on a spring is disturbed it executes sim-
ple harmonic oscillation about its equilibrium position. In physical terms, if
the mass’s initial displacement is positive (x > 0) then the restoring force is
negative, and pulls the mass toward the equilibrium point (x = 0). How-
ever, when the mass reaches this point it is moving, and its inertia thus
carries it onward, so that it acquires a negative displacement (x < 0). The
restoring force then becomes positive, and again pulls the mass toward the
equilibrium point. However, inertia again carries it past this point, and the
mass acquires a positive displacement. The motion subsequently repeats it-
ωt−φ 0 π/2 π 3π/2
x +a 0 −a 0
x˙ 0 −ωa 0 +ωa
x¨ −ω2a 0 +ω2a 0
Table 2.1: Simple harmonic oscillation.