the physics of vibrations and waves

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00000___0c5a055e41241f6e52e0d152fde439dc.pdf

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00001___69475d49ae48ab8fccae1e2d39cee8ab.pdfTHE PHYSICS OF VIBRATIONS

AND WAVES

Sixth Edition

H. J. PainFormerly of Department of Physics,

Imperial College of Science and Technology, London, UK

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00002___da723f9a5c372a060216e1ce80daeb30.pdf

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The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00007___015be0b69b013f60a1777f8454cc8b01.pdfContents

Introduction to First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Introduction to Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Introduction to Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Introduction to Fourth Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Introduction to Fifth Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Introduction to Sixth Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Simple Harmonic Motion 1

Displacement in Simple Harmonic Motion 4

Velocity and Acceleration in Simple Harmonic Motion 6

Energy of a Simple Harmonic Oscillator 8

Simple Harmonic Oscillations in an Electrical System 10

Superposition of Two Simple Harmonic Vibrations in One Dimension 12

Superposition of Two Perpendicular Simple Harmonic Vibrations 15Polarization 17Superposition of a Large Number n of Simple Harmonic Vibrations of

Equal Amplitude a and Equal Successive Phase Difference d 20Superposition of n Equal SHM Vectors of Length a with Random Phase 22Some Useful Mathematics 25

2 Damped Simple Harmonic Motion 37

Methods of Describing the Damping of an Oscillator 43

3 The Forced Oscillator 53

The Operation of i upon a Vector 53

Vector form of Ohms Law 54

The Impedance of a Mechanical Circuit 56

Behaviour of a Forced Oscillator 57

v

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00008___f93574a5dffda46a9f5bad7a9a52ddaa.pdfBehaviour of Velocity vv in Magnitude and Phase versus Driving Force Frequency x 60

Behaviour of Displacement versus Driving Force Frequency x 62

Problem on Vibration Insulation 64

Significance of the Two Components of the Displacement Curve 66

Power Supplied to Oscillator by the Driving Force 68

Variation of P av with x. Absorption Resonance Curve 69

The Q-Value in Terms of the Resonance Absorption Bandwidth 70

The Q-Value as an Amplification Factor 71

The Effect of the Transient Term 74

4 Coupled Oscillations 79

Stiffness (or Capacitance) Coupled Oscillators 79

Normal Coordinates, Degrees of Freedom and Normal Modes of Vibration 81

The General Method for Finding Normal Mode Frequencies, Matrices,

Eigenvectors and Eigenvalues 86

Mass or Inductance Coupling 87

Coupled Oscillations of a Loaded String 90

The Wave Equation 95

5 Transverse Wave Motion 107

Partial Differentiation 107

Waves 108

Velocities in Wave Motion 109

The Wave Equation 110

Solution of the Wave Equation 112

Characteristic Impedance of a String (the string as a forced oscillator) 115

Reflection and Transmission of Waves on a String at a Boundary 117

Reflection and Transmission of Energy 120

The Reflected and Transmitted Intensity Coefficients 120

The Matching of Impedances 121

Standing Waves on a String of Fixed Length 124

Energy of a Vibrating String 126

Energy in Each Normal Mode of a Vibrating String 127

Standing Wave Ratio 128

Wave Groups and Group Velocity 128

Wave Group of Many Components. The Bandwidth Theorem 132

Transverse Waves in a Periodic Structure 135

Linear Array of Two Kinds of Atoms in an Ionic Crystal 138

Absorption of Infrared Radiation by Ionic Crystals 140

Doppler Effect 141

6 Longitudinal Waves 151

Sound Waves in Gases 151

vi Contents

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00009___3914921e2696f976c0d504aecb0be119.pdfEnergy Distribution in Sound Waves 155

Intensity of Sound Waves 157

Longitudinal Waves in a Solid 159

Application to Earthquakes 161

Longitudinal Waves in a Periodic Structure 162

Reflection and Transmission of Sound Waves at Boundaries 163

Reflection and Transmission of Sound Intensity 164

7 Waves on Transmission Lines 171

Ideal or Lossless Transmission Line 173

Coaxial Cables 174

Characteristic Impedance of a Transmission Line 175

Reflections from the End of a Transmission Line 177

Short Circuited Transmission Line ZL 0 178The Transmission Line as a Filter 179

Effect of Resistance in a Transmission Line 183

Characteristic Impedance of a Transmission Line with Resistance 186

The Diffusion Equation and Energy Absorption in Waves 187

Wave Equation with Diffusion Effects 190

Appendix 191

8 Electromagnetic Waves 199

Maxwells Equations 199

Electromagnetic Waves in a Medium having Finite Permeability l and

Permittivity e but with Conductivity r 0 202The Wave Equation for Electromagnetic Waves 204

Illustration of Poynting Vector 206

Impedance of a Dielectric to Electromagnetic Waves 207

Electromagnetic Waves in a Medium of Properties l, e and r (where r 6 0) 208Skin Depth 211

Electromagnetic Wave Velocity in a Conductor and Anomalous Dispersion 211

When is a Medium a Conductor or a Dielectric? 212

Why will an Electromagnetic Wave not Propagate into a Conductor? 214

Impedance of a Conducting Medium to Electromagnetic Waves 215

Reflection and Transmission of Electromagnetic Waves at a Boundary 217

Reflection from a Conductor (Normal Incidence) 222

Electromagnetic Waves in a Plasma 223

Electromagnetic Waves in the Ionosphere 227

9 Waves in More than One Dimension 239

Plane Wave Representation in Two and Three Dimensions 239

Wave Equation in Two Dimensions 240

Contents vii

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00010___69c96f48e5e0938750cee8dd4d0a186e.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00011___c887b1abc0459cadfc59a971a3c96749.pdf12 Interference and Diffraction 333

Interference 333

Division of Amplitude 334

Newtons Rings 337

Michelsons Spectral Interferometer 338

The Structure of Spectral Lines 340

Fabry -- Perot Interferometer 341

Resolving Power of the Fabry -- Perot Interferometer 343

Division of Wavefront 355

Interference from Two Equal Sources of Separation f 357

Interference from Linear Array of N Equal Sources 363

Diffraction 366

Scale of the Intensity Distribution 369

Intensity Distribution for Interference with Diffraction from N Identical Slits 370

Fraunhofer Diffraction for Two Equal Slits N 2 372Transmission Diffraction Grating (N Large) 373

Resolving Power of Diffraction Grating 374

Resolving Power in Terms of the Bandwidth Theorem 376

Fraunhofer Diffraction from a Rectangular Aperture 377

Fraunhofer Diffraction from a Circular Aperture 379

Fraunhofer Far Field Diffraction 383

The Michelson Stellar Interferometer 386

The Convolution Array Theorem 388

The Optical Transfer Function 391

Fresnel Diffraction 395

Holography 403

13 Wave Mechanics 411

Origins of Modern Quantum Theory 411

Heisenbergs Uncertainty Principle 414

Schrodingers Wave Equation 417

One-dimensional Infinite Potential Well 419

Significance of the Amplitude w nx of the Wave Function 422Particle in a Three-dimensional Box 424

Number of Energy States in Interval E to E dE 425The Potential Step 426

The Square Potential Well 434

The Harmonic Oscillator 438

Electron Waves in a Solid 441

Phonons 450

14 Non-linear Oscillations and Chaos 459

Free Vibrations of an Anharmonic Oscillator -- Large Amplitude Motion of

a Simple Pendulum 459

Contents ix

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00012___e9a59a54186dc6a7e1204d326bfc2029.pdfForced Oscillations Non-linear Restoring Force 460

Thermal Expansion of a Crystal 463

Non-linear Effects in Electrical Devices 465

Electrical Relaxation Oscillators 467

Chaos in Population Biology 469

Chaos in a Non-linear Electrical Oscillator 477

Phase Space 481

Repellor and Limit Cycle 485

The Torus in Three-dimensional _xx; x; t) Phase Space 485Chaotic Response of a Forced Non-linear Mechanical Oscillator 487

A Brief Review 488

Chaos in Fluids 494

Recommended Further Reading 504

References 504

15 Non-linear Waves, Shocks and Solitons 505

Non-linear Effects in Acoustic Waves 505

Shock Front Thickness 508

Equations of Conservation 509

Mach Number 510

Ratios of Gas Properties Across a Shock Front 511

Strong Shocks 512

Solitons 513

Bibliography 531

References 531

Appendix 1: Normal Modes, Phase Space and Statistical Physics 533

Mathematical Derivation of the Statistical Distributions 542

Appendix 2: Kirchhoffs Integral Theorem 547

Appendix 3: Non-Linear Schrodinger Equation 551

Index 553

x Contents

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Simple Harmonic Motion

At first sight the eight physical systems in Figure 1.1 appear to have little in common.

1.1(a) is a simple pendulum, a mass m swinging at the end of a light rigid rod of length l.

1.1(b) is a flat disc supported by a rigid wire through its centre and oscillating through

small angles in the plane of its circumference.

1.1(c) is a mass fixed to a wall via a spring of stiffness s sliding to and fro in the x

direction on a frictionless plane.

1.1(d) is a mass m at the centre of a light string of length 2l fixed at both ends under a

constant tension T. The mass vibrates in the plane of the paper.

1.1(e) is a frictionless U-tube of constant cross-sectional area containing a length l of

liquid, density , oscillating about its equilibrium position of equal levels in eachlimb.

1.1(f ) is an open flask of volume V and a neck of length l and constant cross-sectional

area A in which the air of density vibrates as sound passes across the neck.1.1(g) is a hydrometer, a body of mass m floating in a liquid of density with a neck of

constant cross-sectional area cutting the liquid surface. When depressed slightly

from its equilibrium position it performs small vertical oscillations.

1.1(h) is an electrical circuit, an inductance L connected across a capacitance C carrying

a charge q.

All of these systems are simple harmonic oscillators which, when slightly disturbed from

their equilibrium or rest postion, will oscillate with simple harmonic motion. This is the

most fundamental vibration of a single particle or one-dimensional system. A small

displacement x from its equilibrium position sets up a restoring force which is proportional

to x acting in a direction towards the equilibrium position.

Thus, this restoring force F may be written

F sx

where s, the constant of proportionality, is called the stiffness and the negative sign shows

that the force is acting against the direction of increasing displacement and back towards

The Physics of Vibrations and Waves, 6th Edition H. J. Pain# 2005 John Wiley & Sons, Ltd

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Damped Simple Harmonic Motion

Initially we discussed the case of ideal simple harmonic motion where the total energy

remained constant and the displacement followed a sine curve, apparently for an infinite

time. In practice some energy is always dissipated by a resistive or viscous process; for

example, the amplitude of a freely swinging pendulum will always decay with time as

energy is lost. The presence of resistance to motion means that another force is active,

which is taken as being proportional to the velocity. The frictional force acts in the

direction opposite to that of the velocity (see Figure 2.1) and so Newtons Second law

becomes

mxx sx r _xxwhere r is the constant of proportionality and has the dimensions of force per unit of

velocity. The presence of such a term will always result in energy loss.

The problem now is to find the behaviour of the displacement x from the equation

mxx r _xx sx 0 2:1where the coefficients m, r and s are constant.

When these coefficients are constant a solution of the form x C et can always befound. Obviously, since an exponential term is always nondimensional, C has the

dimensions of x (a length, say) and has the dimensions of inverse time, T 1. We shallsee that there are three possible forms of this solution, each describing a different

behaviour of the displacement x with time. In two of these solutions C appears explicitly as

a constant length, but in the third case it takes the form

C A Bt


The number of constants allowed in the general solution of a differential equation is always equalto the order (that is, the highest differential coefficient) of the equation. The two values A and B areallowed because equation (2.1) is second order. The values of the constants are adjusted to satisfy theinitial conditions.

37

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The Forced Oscillator

The Operation of i upon a Vector

We have already seen that a harmonic oscillation can be conveniently represented by the

form ei!t. In addition to its mathematical convenience i can also be used as a vector

operator of physical significance. We say that when i precedes or operates on a vector the

direction of that vector is turned through a positive angle (anticlockwise) of =2, i.e. iacting as an operator advances the phase of a vector by 90. The operator i rotates thevector clockwise by =2 and retards its phase by 90. The mathematics of i as an operatordiffers in no way from its use as

ffiffiffiffiffiffiffi1p and from now on it will play both roles.The vector r a ib is shown in Figure 3.1, where the direction of b is perpendicular to

that of a because it is preceded by i. The magnitude or modulus or r is written

r jrj a2 b21=2

and

r 2 a2 b2 a iba ib rr;

where a ib r is defined as the complex conjugate of a ib; that is, the sign of i ischanged.

The vector r a ib is also shown in Figure 3.1.The vector r can be written as a product of its magnitude r (scalar quantity) and its phase

or direction in the form (Figure 3.1)

r r ei rcos i sin a ib

showing that a r cos and b r sin.


53

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00076___d7764732f49c7c18bd0ccf9143033547.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00077___7ed532de48684623d1c05dac73db599a.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00078___0991d3e5c7da517c88f2f45e6efa4f90.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00079___5609ff3d82734d0107928b7bf6f068e3.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00080___f51ed4fdf13426486962d332d4b6fdba.pdfThe complete solution for x in the equation of motion consists of two terms:

(1) a transient term which dies away with time and is, in fact, the solution to the equation

mxx r _xx sx 0 discussed in Chapter 2. This contributes the term

x C ert=2m eis=mr 2=4m 2 1=2 t

which decays with ert=2m. The second term(2) is called the steady state term, and describes the behaviour of the oscillator after the

transient term has died away.

Both terms contribute to the solution initially, but for the moment we shall concentrate

on the steady state term which describes the ultimate behaviour of the oscillator.

To do this we shall rewrite the force equation in vector form and represent cos!t by e i!t

as follows:

mxx r _xx sx F0 e i!t 3:1Solving for the vector x will give both its magnitude and phase with respect to the drivingforce F0 e

i!t. Initially, let us try the solution x A ei!t, where A may be complex, so that itmay have components in and out of phase with the driving force.

The velocity

_xx i!A ei!t i!xso that

xx i 2!2x !2x

and equation (3.1) becomes

A!2m i!Ar As ei!t F0 e i!t

which is true for all t when

A F0i!r s !2m

or, after multiplying numerator and denominator by i

A iF0!r i!m s=!

iF0!Zm

Hence

x A e i!t iF0 ei!t

!Zm iF0 e

i!t

!Zm ei

iF0 ei!t

!Zm

58 The Forced Oscillator

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00081___89bdbcb36f75db3688f20b4ad1454533.pdfwhere

Zm r 2 !m s=!21=2

This vector form of the steady state behaviour of x gives three pieces of information and

completely defines the magnitude of the displacement x and its phase with respect to the

driving force after the transient term dies away. It tells us

1. That the phase difference exists between x and the force because of the reactive part!m s=! of the mechanical impedance.

2. That an extra difference is introduced by the factor i and even if were zero thedisplacement x would lag the force F0 cos!t by 90

.

3. That the maximum amplitude of the displacement x is F0=!Zm. We see that this isdimensionally correct because the velocity x=t has dimensions F0=Zm.

Having used F0 ei!t to represent its real part F0 cos!t, we now take the real part of the

solution

x iF0 ei!t

!Zm

to obtain the actual value of x. (If the force had been F0 sin!t, we would now take that partof x preceded by i.)Now

x iF0!Zm

ei!t

iF0!Zm

cos !t i sin !t

iF0!Zm

cos !t F0!Zm

sin !t

The value of x resulting from F0 cos!t is therefore

x F0!Zm

sin !t

[the value of x resulting from F0 sin!t would be F0 cos !t =!Zm.Note that both of these solutions satisfy the requirement that the total phase difference

between displacement and force is plus the =2 term introduced by the i factor. When 0 the displacement x F0 sin!t=!Zm lags the force F0 cos!t by exactly 90.

Behaviour of a Forced Oscillator 59

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00082___04c954280d34aea1eb1c5ca9ca1b9780.pdfTo find the velocity of the forced oscillation in the steady state we write

v _xx i! iF0!Zm

ei!t

F0Zm

ei!t

We see immediately that

1. There is no preceding i factor so that the velocity v and the force differ in phase onlyby , and when 0 the velocity and force are in phase.

2. The amplitude of the velocity is F0=Zm, which we expect from the definition ofmechanical impedance Zm F=v.

Again we take the real part of the vector expression for the velocity, which will

correspond to the real part of the force F0 ei!t. This is

v F0Zm

cos !t

Thus, the velocity is always exactly 90 ahead of the displacement in phase and differsfrom the force only by a phase angle , where

tan !m s=!r

Xmr

so that a force F0 cos!t gives a displacement

x F0!Zm

sin !t

and a velocity

v F0Zm

cos !t

(Problems 3.1, 3.2, 3.3, 3.4)

Behaviour of Velocity vv in Magnitude and Phase versus DrivingForce Frequency x

The velocity amplitude is

F0

Zm F0

r 2 !m s=!21=2

so that the magnitude of the velocity will vary with the frequency ! because Zm isfrequency dependent.


The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00083___e911b75898afe762b3bac528b194d9bb.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00084___39809c953a5b71785b40b9e0ed05382a.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00085___5851d8e448eb427c2ec8e7e872ead0df.pdfAt high frequencies Zm ! !m, so that x F0=!2m, which tends to zero as ! becomesvery large. At very high frequencies, therefore, the displacement amplitude is almost zero

because of the mass-controlled or inertial effect.

The velocity resonance occurs at !20 s=m, where the denominator Zm of the velocityamplitude is a minimum, but the displacement resonance will occur, since x F0=!Zmsin !t , when the denominator !Zm is a minimum. This takes place when

d

d!!Zm d

d!!r 2 !m s=!21=2 0

i.e. when

2!r 2 4!m!2m s 0or

2!r 2 2m!2m s 0so that either

! 0or

!2 sm r

2

2m2 !20

r 2

2m2

Thus the displacement resonance occurs at a frequency slightly less than !0, thefrequency of velocity resonance. For a small damping constant r or a large mass m these

two resonances, for all practical purposes, occur at the frequency !0.Denoting the displacement resonance frequency by

! r sm r

2

2m2

1=2

we can write the maximum displacement as

xmax F0! rZm

The value of ! rZm at ! r is easily shown to be equal to !0r where

! 02 sm r

2

4m2 !20

r 2

4m2

The value of x at displacement resonance is therefore given by

xmax F0! 0r

where

! 0 !20 r 2

4m2

1=2

Behaviour of Displacement versus Driving Force Frequency x 63

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00086___b0241472b1e68c9060d7a5848366a4bd.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00087___eaf977c3636d7efff32055c08d367567.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00088___99d9ef284e5824e99cd259e5a508cb65.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00089___b08b7020d1086fa4cf06de4cee8eef98.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00090___b61d4ca0d52a77409942f4d963db31e1.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00091___6a1674d9f1af7fc562a5144adf33cda5.pdfbecause T0

cos!t sin!t dt 0

and

1

T

T0

cos2 !t dt 12

The power supplied by the driving force is not stored in the system, but dissipated as

work expended in moving the system against the frictional force r _xx.The rate of working (instantaneous power) by the frictional force is

r _xx _xx r _xx2 r F20

Z 2mcos2!t

and the average value of this over one period of oscillation

1

2

rF 20Z 2m

12

F 20Zm

cos forr

Zm cos

This proves the initial statement that the power supplied equals the power dissipated.

In an electrical circuit the power is given by VI cos, where V and I are the instantaneousr.m.s. values of voltage and current and cos is known as the power factor.

VI cos V2

Zecos V

20

2Zecos

since

V V0ffiffiffi2

p

(Problem 3.11)

Variation of Pav with x. Absorption Resonance Curve

Returning to the mechanical case, we see that the average power supplied

P av F 20=2Zm cos

is a maximum when cos 1; that is, when 0 and !m s=! 0 or !20 s=m. Theforce and the velocity are then in phase and Zm has its minimum value of r. Thus

P av(maximum) F 20=2r

Variation of Pav with !. Absorption Resonance Curve 69

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00092___f41e20f1001d0badcb8331931395e2a4.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00093___c21f05589268afac35a0aa61d084db31.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00094___d294f11414e11facdedddf308b66319a.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00095___5d56bf558435ac7f1b701d70edb7e185.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00096___11128a62a2b46280f591401b43175327.pdfQ-values are of the order of a few hundred; at higher radio frequencies resonant copper

cavities have Q-values of about 30 000 and piezo-electric crystals can produce Q-values of

500 000. Optical absorption in crystals and nuclear magnetic resonances are often

described in terms of Q-values. The Mossbauer effect in nuclear physics involves Q-values

of 1010.

The Effect of the Transient Term

Throughout this chapter we have considered only the steady state behaviour without

accounting for the transient term mentioned on p. 58. This term makes an initial

contribution to the total displacement but decays with time as ert=2m. Its effect is bestdisplayed by considering the vector sum of the transient and steady state components.

The steady state term may be represented by a vector of constant length rotating

anticlockwise at the angular velocity ! of the driving force. The vector tip traces a circle.Upon this is superposed the transient term vector of diminishing length which rotates anti

clockwise with angular velocity ! 0 s=m r 2=4m21=2. Its tip traces a contracting spiral.The locus of the magnitude of the vector sum of these terms is the envelope of the

varying amplitudes of the oscillator. This envelope modulates the steady state oscillations

of frequency ! at a frequency which depends upon ! 0 and the relative phase between !tand ! 0t.Thus, in Figure 3.12(a) where the total oscillator displacement is zero at time t 0 we

have the steady state and transient vectors equal and opposite in Figure 3.12(b) but because

! 6 ! 0 the relative phase between the vectors will change as the transient term decays.The vector tip of the transient term is shown as the dotted spiral and the total amplitude

assumes the varying lengths OA1, OA2, OA3, OA4, etc.

(Problems 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.18)

Problem 3.1Show, if F0 e

i!t represents F0 sin!t in the vector form of the equation of motion for the forcedoscillator that

x F0!Zm

cos !t

and the velocity

v F0Zm

sin !t

Problem 3.2The displacement of a forced oscillator is zero at time t 0 and its rate of growth is governed by therate of decay of the transient term. If this term decays to ek of its original value in a time t showthat, for small damping, the average rate of growth of the oscillations is given by x 0=t F0=2km!0where x 0 is the maximum steady state displacement, F0 is the force amplitude and !

20 s=m.


The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00097___01a2f895444add32a463b56228860d19.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00098___1e7f1993e39e39f1af1b82c344589ad0.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00099___3f3b580656717b271b9ba5dc7ad37ba7.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00100___5b4e954b460f755f55134d5c8b9e38a5.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00101___53bc9693425fc8e5fddf35c375abc0f1.pdf4

Coupled Oscillations

The preceding chapters have shown in some detail how a single vibrating system will

behave. Oscillators, however, rarely exist in complete isolation; wave motion owes its

existence to neighbouring vibrating systems which are able to transmit their energy to each

other.

Such energy transfer takes place, in general, because two oscillators share a common

component, capacitance or stiffness, inductance or mass, or resistance. Resistance coupling

inevitably brings energy loss and a rapid decay in the vibration, but coupling by either of

the other two parameters consumes no power, and continuous energy transfer over many

oscillators is possible. This is the basis of wave motion.

We shall investigate first a mechanical example of stiffness coupling between two

pendulums. Two atoms set in a crystal lattice experience a mutual coupling force and

would be amenable to a similar treatment. Then we investigate an example of mass, or

inductive, coupling, and finally we consider the coupled motion of an extended array of

oscillators which leads us naturally into a discussion on wave motion.

Stiffness (or Capacitance) Coupled Oscillators

Figure 4.1 shows two identical pendulums, each having a mass m suspended on a light rigid

rod of length l. The masses are connected by a light spring of stiffness s whose natural

length equals the distance between the masses when neither is displaced from equilibrium.

The small oscillations we discuss are restricted to the plane of the paper.

If x and y are the respective displacements of the masses, then the equations of

motion are

mxx mg xl sx y

and

myy mg yl sx y


79

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00102___426f9b68a439defb734d87bb429e0a65.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00103___1b81cd9ff380686a3876211abb1b23f8.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00104___3f248f08b181cf14efda14ac6e09c97d.pdfdifferent ways in which the system can take up energy defines its number of degrees of

freedom and its number of normal coordinates. Each harmonic oscillator has two

degrees of freedom, it may take up both potential energy (normal coordinate X) and

kinetic energy (normal coordinate _XX). In our two normal modes the energies may bewritten

EX a _XX 2 bX 2 4:3aand

EY c _YY 2 dY 2 4:3bwhere a, b, c and d are constant.

Our system of two coupled pendulums has, then, four degrees of freedom and four

normal coordinates.

Any configuration of our coupled system may be represented by the super-position of the

two normal modes

X x y X0 cos !1t 1

and

Y x y Y0 cos !2t 2

where X0 and Y0 are the normal mode amplitudes, whilst !21 g=l and !22 g=l 2s=m

are the normal mode frequencies. To simplify the discussion let us choose

X0 Y0 2a

and put

1 2 0

The pendulum displacements are then given by

x 12X Y a cos!1t a cos!2t

and

y 12X Y a cos!1t a cos!2t

with velocities

_xx a!1 sin!1t a!2 sin!2tand

_yy a!1 sin!1t a!2 sin!2t

82 Coupled Oscillations

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00105___42ce2c44794f0a17c7740ef9af2d2686.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00106___1368659bdcb38eb2c5993247d8621bd9.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00107___2e406c439b0ffdabbea51496a89fde2d.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00108___f77f8df673266cee13dbb121e9ef19dd.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00109___2229985d7ba20f8f80df7e2983846126.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00110___efd3904b44555f86469a974a46c3ff26.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00111___7a80f2b4d4a70a7650bb1eda7c647ccb.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00112___aaabacdce83915b1ece4d291eb5d0b01.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00113___9e36ec5cea5f548db31df62c2adba28c.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00114___92d356e773c85b9ee163582a5108c76e.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00115___56256db4d5fa40fda20901a49fcaaf6d.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00116___cdc1599803fbfd8d6a87fae3b74d414b.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00117___42040183d0e6e85cebbc09f3ca4d2c09.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00118___f23be76af9d195ef4e400dcb98a762c5.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00119___bfedf365b0c486762719e3a946e28230.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00120___1824dd51e69eaa649bbc68c37b5fa18c.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00121___072f2eedfcbe95b66a308553b96e3fb0.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00122___b8f73f6c5e649c8571235a3654c77da6.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00123___21695feab7ec9e8a1c6d111953886127.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00124___3b1eb3fc1ed2f494879394e225c4f141.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00125___3777ab9a8ba634694fdd266a883fe5de.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00126___5601352b3031aa1a4cffa788af1ffa97.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00127___62f618b83bfc6cae6775eb065fc56c22.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00128___691f585fe33ffb4c75efa6bdbcf2479c.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00129___5fdb07ea4132f5904f35889cc5486574.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00130___8b915b3b6f52f855cf26f3fbdcde194a.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00131___70560023ae87ec9974ef3364f8b50715.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00132___2d2a5aaeae3ac745281a2ba1e1c7e44c.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00133___52d67ce94f565680275512ad6ad636f0.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00134___e58922ee35e45e90e13e1f7920e5e0f0.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00135___8adc8857af8139182507c994cef435d2.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00136___2d7d9d93d8073972e6f70fe8c838b464.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00137___de95cbc33cb01359184d3a3b922437d5.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00138___88aa818638d546e51ce207912a8c643e.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00139___121cc2201ddb413dda1777e7855660f8.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00140___e8c447e47c46768896521ae1b131f458.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00141___33c5018fc273bd7de35eaa4649a6636c.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00142___1257e576746892aa3ef915b8708bd660.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00143___19daedb49040a1073b2817a8c6a9004a.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00144___c3a1062bd949d9a23074051066ae9ad0.pdfThe incident, reflected and transmitted displacements at the junctions x 0 and x l areshown in Figure 5.9 and we seek to make the ratio

Transmitted energy

Incident energy Z3A

23

Z1A21

equal to unity.

The boundary conditions are that y and [email protected][email protected] are continuous across the junctionsx 0 and x l.Between Z1 and Z2 the continuity of y gives

A1 ei!tk 1x B1 e i!tk 1x A2 e i!tk 2x B2 e i!tk 2x

or

A1 B1 A2 B2 at x 0 5:3

Similarly the continuity of [email protected][email protected] at x 0 gives

Tik1A1 ik1B1 Tik2A2 ik2B2

Dividing this equation by ! and remembering that Tk=! T=c c Z we haveZ1A1 B1 Z2A2 B2 5:4

Similarly at x l, the continuity of y gives

A2 eik 2l B2 e ik 2l A3 5:5

and the continuity of [email protected][email protected] gives

Z2A2 eik 2l B2 e ik 2l Z3A3 5:6

From the four boundary equations (5.3), (5.4), (5.5) and (5.6) we require the ratio A3=A1.We use equations (5.3) and (5.4) to eliminate B1 and obtain A1 in terms of A2 and B2. We

then use equations (5.5) and (5.6) to obtain both A2 and B2 in terms of A3. Equations (5.3)

and (5.4) give

Z1A1 A2 B2 A1 Z2A2 B2or

A1 A2r12 1 B2r12 12r12

5:7

where

r12 Z1Z2

122 Transverse Wave Motion

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00145___297e514144c94f43cda72fbe3dbcf6e5.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00146___70694bed2087e77c11593c63ce074662.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00147___c16c9f501e04c481cc6fd5d9e8dec51e.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00148___e2789b2626a21c961d4bac6d7cf91a5f.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00149___042cf602d9d51bcf2d9b48644b43057f.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00150___b46ed66c4cf4678f7bceba7fc114724f.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00151___91ce232d00ca03451e91435411132195.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00152___af768eb3df565f941fa54780d730bdb8.pdfThe beats of maximum intensity fluctuations thus have a frequency equal to the

difference 1 2 of the components. In the example here where the components haveequal amplitudes a, superposition will produce an amplitude which varies between 2a and

0; this is called complete or 100% modulation.

More generally an amplitude modulated wave may be represented by

y A cos !t kx

where the modulated amplitude

A a b cos! 0tThis gives

y a cos !t kx b2fcos ! ! 0t kx cos ! ! 0t kxg

so that here amplitude modulation has introduced two new frequencies ! ! 0, known ascombination tones or sidebands. Amplitude modulation of a carrier frequency is a common

form of radio transmission, but its generation of sidebands has led to the crowding of radio

frequencies and interference between stations.

Wave Groups and Group Velocity

Suppose now that the two frequency components of the last section have different phase

velocities so that !1=k1 6 !2=k2. The velocity of the maximum amplitude of the group;that is, the group velocity

!1 !2k1 k2

!

k

is now different from each of these velocities; the superposition of the two waves will no

longer remain constant and the group profile will change with time.

A medium in which the phase velocity is frequency dependent !=k not constant) isknown as a dispersive medium and a dispersion relation expresses the variation of ! as afunction of k. If a group contains a number of components of frequencies which are nearly

equal the original expression for the group velocity is written

!

k d!

dk

The group velocity is that of the maximum amplitude of the group so that it is the velocity

with which the energy in the group is transmitted. Since ! kv, where v is the phasevelocity, the group velocity

v g d!dk

ddk

kv v k dvdk

v dvd


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wavelength of 5:5 107 m. Show that reflections at this wavelength are avoided by a coating ofrefractive index 1.22 and thickness 1:12 107 m.

Problem 5.11Prove that the displacement yn of the standing wave expression in equation (5.10) satisfies the time

independent form of the wave equation

@ 2y

@x2 k 2y 0:

Problem 5.12The total energy En of a normal mode may be found by an alternative method. Each section dx of the

string is a simple harmonic oscillator with total energy equal to the maximum kinetic energy of

oscillation

k:e:max 12 dx _yy2nmax 12 dx!2ny 2nmaxNow the value of y2nmax at a point x on the string is given by

y2nmax A2n B2n sin2!nx

c

Show that the sum of the energies of the oscillators along the string; that is, the integral

12!2n

l0

y2nmax dx

gives the expected result.

Problem 5.13The displacement of a wave on a string which is fixed at both ends is given by

yx; t A cos !t kx rA cos !t kx

where r is the coefficient of amplitude reflection. Show that this may be expressed as the

superposition of standing waves

yx; t A1 r cos!t cos kx A1 r sin!t sin kx:

Problem 5.14A wave group consists of two wavelengths and where = is very small.Show that the number of wavelengths contained between two successive zeros of the modulating

envelope is =.

Problem 5.15The phase velocity v of transverse waves in a crystal of atomic separation a is given by

v c sin ka=2ka=2


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Longitudinal Waves

In deriving the wave equation

@ 2y

@x2 1

c2@ 2y

@t 2

in Chapter 5, we used the example of a transverse wave and continued to discuss waves of

this type on a vibrating string. In this chapter we consider longitudinal waves, waves in

which the particle or oscillator motion is in the same direction as the wave propagation.

Longitudinal waves propagate as sound waves in all phases of matter, plasmas, gases,

liquids and solids, but we shall concentrate on gases and solids. In the case of gases,

limitations of thermodynamic interest are imposed; in solids the propagation will depend

on the dimensions of the medium. Neither a gas nor a liquid can sustain the transverse

shear necessary for transverse waves, but a solid can maintain both longitudinal and

transverse oscillations.

Sound Waves in Gases

Let us consider a fixed mass of gas, which at a pressure P0 occupies a volume V0 with a

density 0. These values define the equilibrium state of the gas which is disturbed, ordeformed, by the compressions and rarefactions of the sound waves. Under the influence of

the sound waves

the pressure P0 becomes P P0 pthe volume V0 becomes V V0 v

and

the density 0 becomes 0 d:

The excess pressure pm is the maximum pressure amplitude of the sound wave and p is an

alternating component superimposed on the equilibrium gas pressure P0.


151

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water or steel on the other leads to an extreme mismatch of impedances when the

transmission of acoustic energy between these media is attempted.

There is an almost total reflection of sound wave energy at an air-water interface,

independent of the side from which the wave approaches the boundary. Only 14% of

acoustic energy can be transmitted at a steel-water interface, a limitation which has severe

implications for underwater transmission and detection devices which rely on acoustics.

(Problems 6.10, 6.11, 6.12, 6.13, 6.14, 6.15, 6.16, 6.17)

Problem 6.1Show that in a gas at temperature T the average thermal velocity of a molecule is approximatley

equal to the velocity of sound.

Problem 6.2The velocity of sound in air of density 1.29 kg m3 may be taken as 330 m s1. Show that theacoustic pressure for the painful sound of 10 W m2 6:5 104 of an atmosphere.

Problem 6.3Show that the displacement amplitude of an air molecule at a painful sound level of 10 W m2 at500 Hz 6:9 105 m.

Problem 6.4Barely audible sound in air has an intensity of 1010 I0. Show that the displacement amplitude of anair molecule for sound at this level at 500 Hz is 1010 m; that is, about the size of the moleculardiameter.

Problem 6.5Hi-fi equipment is played very loudly at an intensity of 100 I 0 in a small room of cross section

3 m 3 m. Show that this audio output is about 10 W.

Problem 6.6Two sound waves, one in water and one in air, have the same intensity. Show that the ratio of their

pressure amplitudes (p water/p air) is about 60. When the pressure amplitudes are equal show that

the intensity ratio is 3 102.

Problem 6.7A spring of mass m, stiffness s and length L is stretched to a length L l. When longitudinal wavespropagate along the spring the equation of motion of a length dx may be written

dx@ 2

@t 2 @F

@xdx

where is the mass per unit length of the spring, is the longitudinal displacement and F is therestoring force. Derive the wave equation to show that the wave velocity v is given by

v 2 sL l=

Reflection and Transmission of Sound Intensity 165

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Waves on Transmission Lines

In the wave motion discussed so far four major points have emerged. They are

1. Individual particles in the medium oscillate about their equilibrium positions with

simple harmonic motion but do not propagate through the medium.

2. Crests and troughs and all planes of equal phase are transmitted through the medium to

give the wave motion.

3. The wave or phase velocity is governed by the product of the inertia of the medium and

its capacity to store potential energy; that is, its elasticity.

4. The impedance of the medium to this wave motion is governed by the ratio of the

inertia to the elasticity (see table on p. 546).

In this chapter we wish to investigate the wave propagation of voltages and currents and

we shall see that the same physical features are predominant. Voltage and current waves are

usually sent along a geometrical configuration of wires and cables known as transmission

lines. The physical scale or order of magnitude of these lines can vary from that of an

oscilloscope cable on a laboratory bench to the electric power distribution lines supported

on pylons over hundreds of miles or the submarine telecommunication cables lying on an

ocean bed.

Any transmission line can be simply represented by a pair of parallel wires into one end

of which power is fed by an a.c. generator. Figure 7.1a shows such a line at the instant

when the generator terminal A is positive with respect to terminal B, with current flowing

out of the terminal A and into terminal B as the generator is doing work. A half cycle later

the position is reversed and B is the positive terminal, the net result being that along each of

the two wires there will be a distribution of charge as shown, reversing in sign at each half

cycle due to the oscillatory simple harmonic motion of the charge carriers (Figure 7.1b).

These carriers move a distance equal to a fraction of a wavelength on either side of their

equilibrium positions. As the charge moves current flows, having a maximum value where

the product of charge density and velocity is greatest.


171

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travelling along the cable is

V

I Z0

ffiffiffiffiffiffiL0

C0

r

where Z0 defines the impedance seen by the waves moving down an infinitely long cable. It

is called the Characteristic Impedance.

We write " "r "0 where "r is the relative permittivity (dielectric constant) of a materialand r0, where r is the relative permeability. Polythene, which commonly fills thespace between r1 and r2, has "r 10 and r 1.Hence

Z0 ffiffiffiffiffiffiL0

C0

r 1

2

ffiffiffi

"

rloge

r2

r1 1

2

1ffiffiffi"

pr

loger2

r1

ffiffiffiffiffi0"0

r

where

ffiffiffiffiffi0"0

r 376:6

Typically, the ratio r2=r1 varies between 2 and 102 and for a laboratory cable using

polythene Z0 5075 with a signal speed c=3 where c is the speed of light.Coaxial cables can be made to a very high degree of precision and the time for an

electrical signal to travel a given length can be accurately calculated because the velocity is

known.

Such a cable can be used as a delay line in order to separate the arrival of signals at a

given point by very small intervals of time.

Characteristic Impedance of a Transmission Line

The solutions to equations (7.3) and (7.4) are, of course,

V V0 sin 2vt x

and

I I0 sin 2vt x

where V0 and I0 are the maximum values and where the subscript + refers to a wave

moving in the positive x-direction. Equation (7.1), @[email protected] L0 @[email protected], therefore givesV 0 vL0I 0, where the superscript refers to differentiation with respect to the bracketvt x.Integration of this equation gives

V vL0I

Characteristic Impedance of a Transmission Line 175

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00198___1d5bb5243b96be03e747775e27f56bf2.pdfwhere the constant of integration has no significance because we are considering only

oscillatory values of voltage and current whilst the constant will change merely the d.c.

level.

The ratio

VI

vL0 ffiffiffiffiffiffiL0

C0

r

and the value offfiffiffiffiffiffiffiffiffiffiffiffiffiL0=C0

p, written as Z0, is a constant for a transmission line of given

properties and is called the characteristic impedance. Note that it is a pure resistance

(no dimensions of length are involved) and it is the impedance seen by the wave

system propagating along an infinitely long line, just as an acoustic wave experiences a

specific acoustic impedance c. The physical correspondence between c andL0v

ffiffiffiffiffiffiffiffiffiffiffiffiffiL0=C0

p Z0 is immediately evident.The value of Z0 for the coaxial cable considered earlier can be shown to be

Z0 12

ffiffiffi

"

rloge

r2

r1

Electromagnetic waves in free space experience an impedance Z0 ffiffiffiffiffiffiffiffiffiffiffiffi0="0

p 376:6 .So far we have considered waves travelling only in the x-direction. Waves which travel

in the negative x-direction will be represented (from solving the wave equation) by

V V0 sin 2vt x

and

I I0 sin 2vt x

where the negative subscript denotes the negative x-direction of propagation.

Equation (7.1) then yields the results that

VI

vL0 Z0

so that, in common with the specific acoustic impedance, a negative sign is introduced into

the ratio when the waves are travelling in the negative x-direction.

When waves are travelling in both directions along the transmission line the total voltage

and current at any point will be given by

V V Vand

I I IWhen a transmission line has waves only in the positive direction the voltage and current

waves are always in phase, energy is propagated and power is being fed into the line by the

generator at all times. This situation is destroyed when waves travel in both directions;

176 Waves on Transmission Lines

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Zi iffiffiffiffiffiffiL 0

C 0

rtan

2l

and by sketching the variation of the ratio Zi=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiL 0=C0

pwith l, show that for l just greater than

2n 1=4, Zi is capacitative, and for l just greater than n=2 it is inductive. (This provides apositive or negative reactance to match another line.)

Problem 7.9Show that a line of characteristic impedance Z 0 may be matched to a load ZL by a loss-free quarter

wavelength line of characteristic impedance Zm if Z2m Z 0ZL.

(Hintcalculate the input impedance at the Z 0Zm junction.)

Problem 7.10Show that a short-circuited quarter wavelength loss-free line has an infinite impedance and that if it

is bridged across another transmission line it will not affect the fundamental wavelength but will

short-circuit any undesirable second harmonic.

Problem 7.11Show that a loss-free line of characteristic impedance Z 0 and length n=2 may be used to couple twohigh frequency circuits without affecting other impedances.

Problem 7.12A transmission line has Z1 i!L and Z2 i!C1. If, for a range of frequencies !, the phase shiftper section is very small show that k the wave number and that the phase velocity isindependent of the frequency.

Problem 7.13In a transmission line with losses where R0=!L 0 and G 0=!C 0 are both small quantities expand theexpression for the propagation constant

R0 i!L 0G0 i!C 0 1=2

to show that the attenuation constant

R02

ffiffiffiffiffiffiC0

L 0

r G 0

2

ffiffiffiffiffiffiL 0

C0

r

and the wave number

k !ffiffiffiffiffiffiffiffiffiffiffiL 0C 0

p !

v

Show that for G 0 0 the Q value of such a line is given by k=2.

194 Waves on Transmission Lines

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Electromagnetic Waves

Earlier chapters have shown that the velocity of waves through a medium is determined by

the inertia and the elasticity of the medium. These two properties are capable of storing

wave energy in the medium, and in the absence of energy dissipation they also determine

the impedance presented by the medium to the waves. In addition, when there is no loss

mechanism a pure wave equation with a sine or cosine solution will always be obtained, but

this equation will be modified by any resistive or loss term to give an oscillatory solution

which decays with time or distance.

These physical processes describe exactly the propagation of electromagnetic waves

through a medium. The magnetic inertia of the medium, as in the case of the transmission

line, is provided by the inductive property of the medium, i.e. the permeability , which hasthe units of henries per metre. The elasticity or capacitive property of the medium is

provided by the permittivity ", with units of farads per metre. The storage of magneticenergy arises through the permeability ; the potential or electric field energy is storedthrough the permittivity ".If the material is defined as a dielectric, only and " are effective and a pure wave

equation for both the magnetic field vector H and the electric field vector E will result. If

the medium is a conductor, having conductivity (the inverse of resistivity) withdimensions of siemens per metre or (ohms m)1, in addition to and ", then some of thewave energy will be dissipated and absorption will take place.

In this chapter we will consider first the propagation of electromagnetic waves in a

medium characterized by and " only, and then treat the general case of a medium having, " and properties.

Maxwells Equations

Electromagnetic waves arise whenever an electric charge changes its velocity. Electrons

moving from a higher to a lower energy level in an atom will radiate a wave of a particular

frequency and wavelength. A very hot ionized gas consisting of charged particles will

radiate waves over a continuous spectrum as the paths of individual particles are curved in


199

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positive charge neex on the other side (where n i ne).This charge separation generates an electric field E in the plasma of magnitude

E n eex"0

which produces an electric force n ee2x="0 acting on each electron in the direction of itsequilibrium position.

The equation of motion of each electron is therefore

m exx n ee2x

"0 0

and the electron motion is simple harmonic with an angular frequency !p where

!2p nee

2

"0m e

The angular frequency !p is called the electron plasma frequency and plays a significantrole in the propagation of electromagnetic waves in the plasma.

In the expression for the refractive index

" r n2 1!2p!20

8:8

n is real for all values of ! and waves of that frequency will propagate. However, when

" r n2 1!2p!2

8:9

waves will propagate only when ! > !pWhen !2p=!

2 > 1

n2 c2

v 2 c

2k 2

!2 1 !

2p

!2

is negative and the wave number k is considered to be complex with

k k0 i:

In this case, electromagnetic waves incident on the plasma will be attenuated within the

plasma, or if is large enough, will be reflected at the plasma surface.The electric field of the wave E E0 e i!tkz becomes E E0ez ei!tk z and is

reduced to E0e1 when z 1= the penetration depth. When k0, the penetration

Electromagnetic Waves in a Plasma 225

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00248___f26735b47ee4694cc94c2ef123fe4c82.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00249___3a6c46d86749db6a2cc888767569803b.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00250___2fb712ed85011d361c7492bd776af1fe.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00251___d0853bfda65b36e5d11fa5cdb76c453c.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00252___4d1680e55ca1aec6618bc2e71ef2492d.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00253___7a6cb4487f476fffa592b8fe100930fb.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00254___9b38aecebd05247e8b54b54c6070e874.pdfProblem 8.1The solutions to the e.m. wave equations are given in Figure 8.3 as

Ex E 0 sin 2vt z

and

Hy H 0 sin 2vt z

Use equations (8.1a) and (8.2a) to prove that they have the same wavelength and phase as shown in

figure.

Problem 8.2Show that the concept of B2=2 (magnetic energy per unit volume) as a magnetic pressure accountsfor the fact that two parallel wires carrying currents in the same direction are forced together and that

reversing one current will force them apart. (Consider a point midway between the two wires.) Show

that it also explains the motion of a conductor carrying a current which is situated in a steady

externally applied magnetic field.

Problem 8.3At a distance r from a charge e on a particle of mass m the electric field value is E e=4" 0r 2.Show by integrating the electrostatic energy density over the spherical volume of radius a to infinity

and equating it to the value mc 2 that the classical radius of the electron is given by

a 1:41 1015 m

Problem 8.4The rate of generation of heat in a long cylindrical wire carrying a current I is I 2R, where R is the

resistance of the wire. Show that this joule heating can be described in terms of the flow of energy

into the wire from surrounding space and is equal to the product of the Poynting vector and the

surface area of the wire.

Problem 8.5Show that when a current is increasing in a long uniformly wound solenoid of coil radius r the total

inward energy flow rate over a length l (the Poynting vector times the surface area 2rl) gives thetime rate of change of the magnetic energy stored in that length of the solenoid.

Problem 8.6The plane polarized electromagnetic wave (Ex, Hy) of this chapter travels in free space. Show that its

Poynting vector (energy flow in watts per squaremetre) is given by

S ExHy c12 " 0E 2x 12 0H 2y c" 0E 2x

232 Electromagnetic Waves

The_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00255___11a65fceed45034104d30c06f55b62c2.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00256___7ffca158b46ae7b7f5899bfc034a1d7d.pdfThe_Physics_of_Vibrations_and_Waves__6th_Edition/047001296X/files/00257___b57e810844b57e81b947d29047109390.pdfProblem 8.16Show that when a plane electromagnetic wave travelling in air is reflected normally from a plane

conducting surface the transmitted magnetic field value Ht 2Hi, and that a magnetic standingwave exists in air with a very large standing wave ratio. If the wave is travelling in a conductor and is

reflected normally from a plane conductorair interface, show that Et 2Ei. Show that these twocases are respectively analogous to a short-circuited and an open-circuited transmission line.

Problem 8.17Show that in a conductor the average value of the Poynting vector is given by

S av 12E0H0 cos 45 1

2H 20 real part of ZcWm2

where E0 and H0 are the peak field values. A plane 1000 MHz wave travelling in air with E0 1 V m1 is incident normally on a large copper sheet. Show firstly that the real part of the conductorimpedance is 8.2103 and then (remembering from Problem 8.16 that H0 doubles in theconductor) show that the average power absorbed by the copper per square metre is 1.6107 W.

Problem 8.18For a good conductor " r r 1. Show that when an electromagnetic wave is reflected normallyfrom such a conducting surface its fractional loss of energy (1reflection coefficient I r) is

ffiffiffiffiffiffiffiffiffiffiffiffiffi8!"=p . Note that the ratio of the displacement current density to the conduction current densityis therefore a direct measure of the reflectivity of the surface.

Problem 8.19Using the value of the Poynting vector in the conductor from Problem 8.17, show that the ratio of

this value to the value of the Poynting vector in air is ffiffiffiffiffiffiffiffiffiffiffiffiffi8!"=p , as expected from Problem 8.18.Problem 8.20The electromagnetic wave of Problems 8.18 and 8.19 has electric and magnetic field magnitudes in

the conductor given by

Ex A ekz e i!tkzand

Hy A !

1=2ekz e i!tkz ei=4

where k !=2 1=2.Show that the average value of the Poynting vector in the conductor is given by

S av 12A2

2!

1=2e2kz Wm2

This is the power absorbed per unit area by the conductor. We know, however, that the wavepropagates only a distance of the order of the skin depth, so that this power is rapidly transformed.The rate at which it changes with distance is given by @S [email protected], which gives the energy transformedper unit volume in unit time. Show that this quantity is equal to the conductivity times the squareof the mean value of the electric field vector E, that is, the joule heating from currents flowing in thesurface of the conductor down to a depth of the order of the skin depth.

Electromagnetic Waves in the Ionosphere 235

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Waves in More thanOne Dimension

Plane Wave Representation in Two and Three Dimensions

Figure 9.1 shows that in two dimensions waves of velocity c may be represented by lines of

constant phase propagating in a direction k which is normal to each line, where themagnitude of k is the wave number k 2=.The direction cosines of k are given by

l k1k; m k2

kwhere k 2 k 21 k 22

and any point rx; y on the line of constant phase satisfies the equationlx my p ct

where p is the perpendicular distance from the line to the origin. The displacements at all

points rx; y on a given line are in phase and the phase difference between the origin anda given line is

2(path difference) 2

p k r k1x k2y kp

Hence, the bracket !t !t kx used in a one dimensional wave is replaced by!t k r in waves of more than one dimension, e.g. we shall use the exponentialexpression

ei!tkr

In three dimensions all points rx; y; z in a given wavefront will lie on planes of constantphase satisfying the equation

lx my nz p ct


239

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A ei!tkr for rx; y; z:

Wave Guides

Reflection from walls y 0; y b in a two-dimensional wave guide for a wave offrequency ! and vector direction kk1; k2 gives normal modes in the y direction withk2 n=b and propagation in the x direction with phase velocity

v p !k1>!

k v

and group velocity

v g @[email protected]

such that v pv g v 2

Cut-off frequency

Only frequencies ! nv=b will propagate where n is mode number.

Normal Modes in Three Dimensions

Wave equation separates into three equations (one for each variable x, y, z) to give solution

A sincos

k1xsin

cosk2y

sin

cosk3z

sin

cos!t

(Boundary conditions determine final form of solution.)

For waves of velocity c, the number of normal modes per unit volume of an enclosure in

the frequency range to d

the physics of vibrations and waves

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