The Physics of Vibrations and Waves E-Book

H. John Pain

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 708

Veröffentlichungsjahr: 2013

Beschreibung

The main theme of this highly successful book is that the transmission of energy by wave propogation is fundamental to almost every branch of physics. Therefore, besides giving students a thorough grounding in the theory of waves and vibrations, the book also demonstrates the pattern and unity of a large part of physics.

This new edition has been thoroughly revised and has been redeisgned to meet the best contemporary standards. It includes new material on electron waves in solids using the Kronig-Penney model to show how their allowed energies are limited to Brillouin zones, The role of phonons is also discussed. An Optical Transform is used to demonstrate the modern method of lens testing. In the last two chapters the sections on chaos and solitons have been reduced but their essential contents remain.

As with earlier editions, the book has a large number of problems together with hints on how to solve them. The Physics of Vibrations and Waves, 6th Edition will prove invaluable for students taking a first full course in the subject across a variety of disciplines particularly physics, engineering and mathematics.

Leseprobe

Contents

Introduction to First Edition

Introduction to Second Edition

Introduction to Third Edition

Introduction to Fourth Edition

Introduction to Fifth Edition

Introduction to Sixth Edition

1 Simple Harmonic Motion

Displacement in Simple Harmonic Motion

Velocity and Acceleration in Simple Harmonic Motion

Energy of a Simple Harmonic Oscillator

Simple Harmonic Oscillations in an Electrical System

Superposition of Two Simple Harmonic Vibrations in One Dimension

Superposition of Two Perpendicular Simple Harmonic Vibrations

* Polarization

Superposition of a Large Number n of Simple Harmonic Vibrations of Equal Amplitude a and Equal Successive Phase Difference δ

* Superposition of n Equal SHM Vectors of Length a with Random Phase

Some Useful Mathematics

2 Damped Simple Harmonic Motion

Methods of Describing the Damping of an Oscillator

3 The Forced Oscillator

The Operation of i upon a Vector

Vector form of Ohm’s Law

The Impedance of a Mechanical Circuit

Behaviour of a Forced Oscillator

Behaviour of Velocity v in Magnitude and Phase versus Driving Force Frequency ω

Behaviour of Displacement versus Driving Force Frequency ω

Problem on Vibration Insulation

Significance of the Two Components of the Displacement Curve

Power Supplied to Oscillator by the Driving Force

Variation of Pav with ω. Absorption Resonance Curve

The Q-Value in Terms of the Resonance Absorption Bandwidth

The Q-Value as an Amplification Factor

The Effect of the Transient Term

4 Coupled Oscillations

Stiffness (or Capacitance) Coupled Oscillators

Normal Coordinates, Degrees of Freedom and Normal Modes of Vibration

The General Method for Finding Normal Mode Frequencies, Matrices, Eigenvectors and Eigenvalues

Mass or Inductance Coupling

Coupled Oscillations of a Loaded String

The Wave Equation

5 Transverse Wave Motion

Partial Differentiation

Waves

Velocities in Wave Motion

The Wave Equation

Solution of the Wave Equation

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

Reflection and Transmission of Waves on a String at a Boundary

Reflection and Transmission of Energy

The Reflected and Transmitted Intensity Coefficients

The Matching of Impedances

Standing Waves on a String of Fixed Length

Energy of a Vibrating String

Energy in Each Normal Mode of a Vibrating String

Standing Wave Ratio

Wave Groups and Group Velocity

Wave Group of Many Components. The Bandwidth Theorem

Transverse Waves in a Periodic Structure

Linear Array of Two Kinds of Atoms in an Ionic Crystal

Absorption of Infrared Radiation by Ionic Crystals

Doppler Effect

6 Longitudinal Waves

Sound Waves in Gases

Energy Distribution in Sound Waves

Intensity of Sound Waves

Longitudinal Waves in a Solid

Application to Earthquakes

Longitudinal Waves in a Periodic Structure

Reflection and Transmission of Sound Waves at Boundaries

Reflection and Transmission of Sound Intensity

7 Waves on Transmission Lines

Ideal or Lossless Transmission Line

Coaxial Cables

Characteristic Impedance of a Transmission Line

Reflections from the End of a Transmission Line

The Transmission Line as a Filter

Effect of Resistance in a Transmission Line

Characteristic Impedance of a Transmission Line with Resistance

The Diffusion Equation and Energy Absorption in Waves

Wave Equation with Diffusion Effects

Appendix

8 Electromagnetic Waves

Maxwell’s Equations

The Wave Equation for Electromagnetic Waves

Illustration of Poynting Vector

Impedance of a Dielectric to Electromagnetic Waves

Electromagnetic Waves in a Medium of Properties μ, ε and σ (where σ ≠ 0)

Skin Depth

Electromagnetic Wave Velocity in a Conductor and Anomalous Dispersion

When is a Medium a Conductor or a Dielectric?

Why will an Electromagnetic Wave not Propagate into a Conductor?

Impedance of a Conducting Medium to Electromagnetic Waves

Reflection and Transmission of Electromagnetic Waves at a Boundary

Reflection from a Conductor (Normal Incidence)

Electromagnetic Waves in a Plasma

Electromagnetic Waves in the Ionosphere

9 Waves in More than One Dimension

Plane Wave Representation in Two and Three Dimensions

Wave Equation in Two Dimensions

Wave Guides

Normal Modes and the Method of Separation of Variables

Two-Dimensional Case

Three-Dimensional Case

Normal Modes in Two Dimensions on a Rectangular Membrane

Normal Modes in Three Dimensions

Frequency Distribution of Energy Radiated from a Hot Body. Planck’s Law

Debye Theory of Specific Heats

Reflection and Transmission of a Three-Dimensional Wave at a Plane Boundary

Total Internal Reflection and Evanescent Waves

10 Fourier Methods

Fourier Series

Application of Fourier Sine Series to a Triangular Function

Application to the Energy in the Normal Modes of a Vibrating String

Fourier Series Analysis of a Rectangular Velocity Pulse on a String

The Spectrum of a Fourier Series

Fourier Integral

Fourier Transforms

Examples of Fourier Transforms

The Slit Function

The Fourier Transform Applied to Optical Diffraction from a Single Slit

The Gaussian Curve

The Dirac Delta Function, its Sifting Property and its Fourier Transform

Convolution

The Convolution Theorem

11 Waves in Optical Systems

Light. Waves or Rays?

Fermat’s Principle

The Laws of Reflection

The Law of Refraction

Rays and Wavefronts

Ray Optics and Optical Systems

Power of a Spherical Surface

Magnification by the Spherical Surface

Power of Two Optically Refracting Surfaces

Power of a Thin Lens in Air (Figure 11.12)

Principal Planes and Newton’s Equation

Optical Helmholtz Equation for a Conjugate Plane at Infinity

The Deviation Method for (a) Two Lenses and (b) a Thick Lens

The Matrix Method

12 Interference and Diffraction

Interference

Division of Amplitude

Newton’s Rings

Michelson’s Spectral Interferometer

The Structure of Spectral Lines

Fabry – Perot Interferometer

Resolving Power of the Fabry – Perot Interferometer

Division of Wavefront

Interference from Two Equal Sources of Separation f

Interference from Linear Array of N Equal Sources

Diffraction

Scale of the Intensity Distribution

Intensity Distribution for Interference with Diffraction from N Identical Slits

Transmission Diffraction Grating (N Large)

Resolving Power of Diffraction Grating

Resolving Power in Terms of the Bandwidth Theorem

Fraunhofer Diffraction from a Rectangular Aperture

Fraunhofer Diffraction from a Circular Aperture

Fraunhofer Far Field Diffraction

The Michelson Stellar Interferometer

The Convolution Array Theorem

The Optical Transfer Function

Fresnel Diffraction

Holography

13 Wave Mechanics

Origins of Modern Quantum Theory

Heisenberg’s Uncertainty Principle

Schrödinger’s Wave Equation

One-dimensional Infinite Potential Well

Significance of the Amplitude ψn(x) of the Wave Function

Particle in a Three-dimensional Box

Number of Energy States in Interval E to E + dE

The Potential Step

The Square Potential Well

The Harmonic Oscillator

Electron Waves in a Solid

Phonons

14 Non-linear Oscillations and Chaos

Free Vibrations of an Anharmonic Oscillator – Large Amplitude Motion of a Simple Pendulum

Forced Oscillations – Non-linear Restoring Force

Thermal Expansion of a Crystal

Non-linear Effects in Electrical Devices

Electrical Relaxation Oscillators

Chaos in Population Biology

Chaos in a Non-linear Electrical Oscillator

Phase Space

Repellor and Limit Cycle

The Torus in Three-dimensional (, x, t) Phase Space

Chaotic Response of a Forced Non-linear Mechanical Oscillator

A Brief Review

Chaos in Fluids

Recommended Further Reading

References

15 Non-linear Waves, Shocks and Solitons

Non-linear Effects in Acoustic Waves

Shock Front Thickness

Equations of Conservation

Mach Number

Ratios of Gas Properties Across a Shock Front

Strong Shocks

Solitons

Bibliography

References

Appendix 1: Normal Modes, Phase Space and Statistical Physics

Mathematical Derivation of the Statistical Distributions

Appendix 2: Kirchhoff’s Integral Theorem

Appendix 3: Non-Linear Schrödinger Equation

Index

Copyright © 2005 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England

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Introduction to First Edition

The opening session of the physics degree course at Imperial College includes an introduction to vibrations and waves where the stress is laid on the underlying unity of concepts which are studied separately and in more detail at later stages. The origin of this short textbook lies in that lecture course which the author has given for a number of years. Sections on Fourier transforms and non-linear oscillations have been added to extend the range of interest and application.

At the beginning no more than school-leaving mathematics is assumed and more advanced techniques are outlined as they arise. This involves explaining the use of exponential series, the notation of complex numbers and partial differentiation and putting trial solutions into differential equations. Only plane waves are considered and, with two exceptions, Cartesian coordinates are used throughout. Vector methods are avoided except for the scalar product and, on one occasion, the vector product.

Opinion canvassed amongst many undergraduates has argued for a ‘working’ as much as for a ‘reading’ book; the result is a concise text amplified by many problems over a wide range of content and sophistication. Hints for solution are freely given on the principle that an undergraduates gains more from being guided to a result of physical significance than from carrying out a limited arithmetical exercise.

The main theme of the book is that a medium through which energy is transmitted via wave propagation behaves essentially as a continuum of coupled oscillators. A simple oscillator is characterized by three parameters, two of which are capable of storing and exchanging energy, whilst the third is energy dissipating. This is equally true of any medium.

The product of the energy storing parameters determines the velocity of wave propagation through the medium and, in the absence of the third parameter, their ratio governs the impedance which the medium presents to the waves. The energy dissipating parameter introduces a loss term into the impedance; energy is absorbed from the wave system and it attenuates.

This viewpoint allows a discussion of simple harmonic, damped, forced and coupled oscillators which leads naturally to the behaviour of transverse waves on a string, longitudinal waves in a gas and a solid, voltage and current waves on a transmission line and electromagnetic waves in a dielectric and a conductor. All are amenable to this common treatment, and it is the wide validity of relatively few physical principles which this book seeks to demonstrate.

H. J. PAINMay 1968

Introduction to Second Edition

The main theme of the book remains unchanged but an extra chapter on Wave Mechanics illustrates the application of classical principles to modern physics.

Any revision has been towards a simpler approach especially in the early chapters and additional problems. Reference to a problem in the course of a chapter indicates its relevance to the preceding text. Each chapter ends with a summary of its important results.

Constructive criticism of the first edition has come from many quarters, not least from successive generations of physics and engineering students who have used the book; a second edition which incorporates so much of this advice is the best acknowledgement of its value.

H. J. PAINJune 1976

Introduction to Third Edition

Since this book was first published the physics of optical systems has been a major area of growth and this development is reflected in the present edition. Chapter 10 has been rewritten to form the basis of an introductory course in optics and there are further applications in Chapters 7 and 8.

The level of this book remains unchanged.

H. J. PAINJanuary 1983

Introduction to Fourth Edition

Interest in non-linear dynamics has grown in recent years through the application of chaos theory to problems in engineering, economics, physiology, ecology, meteorology and astronomy as well as in physics, biology and fluid dynamics. The chapter on non-linear oscillations has been revised to include topics from several of these disciplines at a level appropriate to this book. This has required an introduction to the concept of phase space which combines with that of normal modes from earlier chapters to explain how energy is distributed in statistical physics. The book ends with an appendix on this subject.

H. J. PAINSeptember 1992

Introduction to Fifth Edition

In this edition, three of the longer chapters of earlier versions have been split in two: Simple Harmonic Motion is now the first chapter and Damped Simple Harmonic Motion the second. Chapter 10 on waves in optical systems now becomes Chapters 11 and 12, Waves in Optical Systems, and Interference and Diffraction respectively through a reordering of topics. A final chapter on non-linear waves, shocks and solitons now follows that on non-linear oscillations and chaos.

New material includes matrix applications to coupled oscillations, optical systems and multilayer dielectric films. There are now sections on e.m. waves in the ionosphere and other plasmas, on the laser cavity and on optical wave guides. An extended treatment of solitons includes their role in optical transmission lines, in collisionless shocks in space, in non-periodic lattices and their connection with Schrödinger’s equation.

H. J. PAINMarch 1998

Acknowledgement

The author is most grateful to Professor L. D. Roelofs of the Physics Department, Haverford College, Haverford, PA, USA. After using the last edition he provided an informed, extended and valuable critique that has led to many improvements in the text and questions of this book. Any faults remain the author’s responsibility.

Introduction to Sixth Edition

This edition includes new material on electron waves in solids using the Kronig – Penney model to show how their allowed energies are limited to Brillouin zones. The role of phonons is also discussed. Convolutions are introduced and applied to optical problems via the Array Theorem in Young’s experiment and the Optical Transfer Function. In the last two chapters the sections on Chaos and Solutions have been reduced but their essential contents remain.

I am grateful to my colleague Professor Robin Smith of Imperial College for his advice on the Optical Transfer Function. I would like to thank my wife for typing the manuscript of every edition except the first.

H. J. PAINJanuary 2005, Oxford

Chapter Synopses

Chapter 1 Simple Harmonic Motion

Simple harmonic motion of mechanical and electrical oscillators (1) Vector representation of simple harmonic motion (6) Superpositions of two SHMs by vector addition (12) Superposition of two perpendicular SHMs (15) Polarization, Lissajous figures (17) Superposition of many SHMs (20) Complex number notation and use of exponential series (25) Summary of important results.

Chapter 2 Damped Simple Harmonic Motion

Damped motion of mechanical and electrical oscillators (37) Heavy damping (39) Critical damping (40) Damped simple harmonic oscillations (41) Amplitude decay (43) Logarithmic decrement (44) Relaxation time (46) Energy decay (46) Q-value (46) Rate of energy decay equal to work rate of damping force (48) Summary of important results.

Chapter 3 The Forced Oscillatior

The vector operator i (53) Electrical and mechanical impedance (56) Transient and steady state behaviour of a forced oscillator (58) Variation of displacement and velocity with frequency of driving force (60) Frequency dependence of phase angle between force and (a) displacement, (b) velocity (60) Vibration insulation (64) Power supplied to oscillator (68) Q-value as a measure of power absorption bandwidth (70) Q-value as amplification factor of low frequency response (71) Effect of transient term (74) Summary of important results.

Chapter 4 Coupled Oscillations

Spring coupled pendulums (79) Normal coordinates and normal modes of vibration (81) Matrices and eigenvalues (86) Inductance coupling of electrical oscillators (87) Coupling of many oscillators on a loaded string (90) Wave motion as the limit of coupled oscillations (95) Summary of important results.

Chapter 5 Transverse Wave Motion

Notation of partial differentiation (107) Particle and phase velocities (109) The wave equation (110) Transverse waves on a string (111) The string as a forced oscillator (115) Characteristic impedance of a string (117) Reflection and transmission of transverse waves at a boundary (117) Impedance matching (121) Insertion of quarter wave element (124) Standing waves on a string of fixed length (124) Normal modes and eigenfrequencies (125) Energy in a normal mode of oscillation (127) Wave groups (128) Group velocity (130) Dispersion (131) Wave group of many components (132) Bandwidth Theorem (134) Transverse waves in a periodic structure (crystal) (135) Doppler Effect (141) Summary of important results.

Chapter 6 Longitudinal Waves

Wave equation (151) Sound waves in gases (151) Energy distribution in sound waves (155) Intensity (157) Specific acoustic impedance (158) Longitudinal waves in a solid (159) Young’s Modulus (159) Poisson’s ratio (159) Longitudinal waves in a periodic structure (162) Reflection and transmission of sound waves at a boundary (163) Summary of important results.

Chapter 7 Waves on Transmission Lines

Ideal transmission line (173) Wave equation (174) Velocity of voltage and current waves (174) Characteristic impedance (175) Reflection at end of terminated line (177) Standing waves in short circuited line (178) Transmission line as a filter (179) Propagation constant (181) Real transmission line with energy losses (183) Attenuation coefficient (185) Diffusion equation (187) Diffusion coefficients (190) Attenuation (191) Wave equation plus diffusion effects (190) Summary of important results.

Chapter 8 Electromagnetic Waves

Permeability and permittivity of a medium (199) Maxwell’s equations (202) Displacement current (202) Wave equations for electric and magnetic field vectors in a dielectric (204) Poynting vector (206) Impedance of a dielectric to e.m. waves (207) Energy density of e.m. waves (208) Electromagnetic waves in a conductor (208) Effect of conductivity adds diffusion equation to wave equation (209) Propagation and attenuation of e.m. waves in a conductor (210) Skin depth (211) Ratio of displacement current to conduction current as a criterion for dielectric or conducting behaviour (213) Relaxation time of a conductor (214) Impedance of a conductor to e.m. waves (215) Reflection and transmission of e.m. waves at a boundary (217) Normal incidence (217) Oblique incidence and Fresnel’s equations (218) Reflection from a conductor (222) Connection between impedance and refractive index (219) E.m. waves in plasmas and the ionosphere (223) Summary of important results.

Chapter 9 Waves in More than One Dimension

Plane wave representation in 2 and 3 dimensions (239) Wave equation in 2-dimensions (240) Wave guide (242) Reflection of a 2-dimensional wave at rigid boundaries (242) Normal modes and method of separation of variables for 1, 2 and 3 dimensions (245) Normal modes in 2 dimensions on a rectangular membrane (247) Degeneracy (250) Normal modes in 3 dimensions (250) Number of normal modes per unit frequency interval per unit volume (251) Application to Planck’s Radiation Law and Debye’s Theory of Specific Heats (251) Reflection and transmission of an e.m. wave in 3 dimensions (254) Snell’s Law (256) Total internal reflexion and evanescent waves (256) Summary of important results.

Chapter 10 Fourier Methods

Fourier series for a periodic function (267) Fourier series for any interval (271) Application to a plucked string (275) Energy in normal modes (275) Application to rectangular velocity pulse on a string (278) Bandwidth Theorem (281) Fourier integral of a single pulse (283) Fourier Transforms (285) Application to optical diffraction (287) Dirac function (292) Convolution (292) Convolution Theorem (297) Summary of important results.

Chapter 11 Waves in Optical Systems

Fermat’s Principle (307) Laws of reflection and refraction (307) Wavefront propagation through a thin lens and a prism (310) Optical systems (313) Power of an optical surface (314) Magnification (316) Power of a thin lens (318) Principal planes of an optical system (320) Newton’s equation (320) Optical Helmholtz equation (321) Deviation through a lens system (322) Location of principal planes (322) Matrix application to lens systems (325) Summary of important results.

Chapter 12 Interference and Diffraction

Interference (333) Division of amplitude (334) Fringes of constant inclination and thickness (335) Newton’s Rings (337) Michelson’s spectral interferometer (338) Fabry–Perot interferometer (341) Finesse (345) Resolving power (343) Free spectral range (345) Central spot scanning (346) Laser cavity (347) Multilayer dielectric films (350) Optical fibre wave guide (353) Division of wavefront (355) Two equal sources (355) Spatial coherence (360) Dipole radiation (362) Linear array of N equal sources (363) Fraunhofer diffraction (367) Slit (368) N slits (370) Missing orders (373) Transmission diffraction grating (373) Resolving power (374) Bandwidth theorem (376) Rectangular aperture (377) Circular aperture (379) Fraunhofer far field diffraction (383) Airy disc (385) Michelson Stellar Interferometer (386) Convolution Array Theorem (388) Optical Transfer Function (391) Fresnel diffraction (395) Straight edge (397) Cornu spiral (396) Slit (400) Circular aperture (401) Zone plate (402) Holography (403) Summary of important results.

Chapter 13 Wave Mechanics

Historical review (411) De Broglie matter waves and wavelength (412) Heisenberg’s Uncertainty Principle (414) Schrödinger’s time independent wave equation (417) The wave function (418) Infinite potential well in 1 dimension (419) Quantization of energy (421) Zero point energy (422) Probability density (423) Normalization (423) Infinite potential well in 3 dimensions (424) Density of energy states (425) Fermi energy level (426) The potential step (426) The finite square potential well (434) The harmonic oscillator (438) Electron waves in solids (441) Bloch functions (441) Kronig–Penney Model (441) Brillouin zones (445) Energy band (446) Band structure (448) Phonons (450) Summary of important results.

Chapter 14 Non-linear Oscillations and Chaos

Anharmonic oscillations (459) Free vibrations of finite amplitude pendulum (459) Non-linear restoring force (460) Forced vibrations (460) Thermal expansion of a crystal (463) Electrical ‘relaxation’ oscillator (467) Chaos and period doubling in an electrical ‘relaxation’ oscillator (467) Chaos in population biology (469) Chaos in a non-linear electrical oscillator (477) Phase space (481) Chaos in a forced non-linear mechanical oscillator (487) Fractals (490) Koch Snowflake (490) Cantor Set (491) Smale Horseshoe (493) Chaos in fluids (494) Couette flow (495) Rayleigh–Benard convection (497) Lorenz chaotic attractor. (500) List of references

Chapter 15 Non-linear waves, Shocks and Solitons

Non-linear acoustic effects (505) Shock wave in a gas (506) Mach cone (507) Solitons (513) The KdV equation (515) Solitons and Schrödinger’s equation (520) Instantons (521) Optical solitons (521) Bibliography and references.

Appendix 1 Normal Modes, Phase Space and Statistical Physics

Number of phase space ‘cells’ per unit volume (533) Macrostate (535) Microstate (535) Relative probability of energy level population for statistical distributions (a) Maxwell–Boltzmann, (b) Fermi–Dirac, (c) Bose–Einstein (536) Mathematical derivation of the statistical distributions (542).

Appendix 2 Kirchhoff’s Integral Theorem

Appendix 3 Non-linear Schrödinger Equation

Index

1

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 2

l

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

ρ

, oscillating about its equilibrium position of equal levels in each limb.

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

ρ

vibrates as sound passes across the neck.

1.1(g)

is a hydrometer, a body of mass

m

floating in a liquid of density

ρ

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

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 equilibrium position. A constant value of the stiffness restricts the displacement x to small values (this is Hooke’s Law of Elasticity). The stiffness s is obviously the restoring force per unit distance (or displacement) and has the dimensions

The equation of motion of such a disturbed system is given by the dynamic balance between the forces acting on the system, which by Newton’s Law is

or

where the acceleration

This gives

or

where the dimensions of

Here T is a time, or period of oscillation, the reciprocal of v which is the frequency with which the system oscillates.

Figure 1.1 Simple harmonic oscillators with their equations of motion and angular frequencies ω of oscillation. (a) A simple pendulum. (b) A torsional pendulum. (c) A mass on a frictionless plane connected by a spring to a wall. (d) A mass at the centre of a string under constant tension T. (e) A fixed length of non-viscous liquid in a U-tube of constant cross-section. (f) An acoustic Helmholtz resonator. (g) A hydrometer mass m in a liquid of density ρ. (h) An electrical L C resonant circuit

where s/m is now written as ω2. Thus the equation of simple harmonic motion

becomes

(1.1)

(Problem 1.1)

Displacement in Simple Harmonic Motion

The behaviour of a simple harmonic oscillator is expressed in terms of its displacement x from equilibrium, its velocity , and its acceleration at any given time. If we try the solution

where A is a constant with the same dimensions as x, we shall find that it satisfies the equation of motion

for

and

Another solution