Vibrations and Waves

Citation

French, A. P. (1971). Vibrations and Waves. W. W. Norton & Company.

Chapter Summary

A volume of the M.I.T. Introductory Physics Series, a set of short, self-contained books that, according to the series preface, emphasise the interaction of experiment and intuition and can be combined in many course structures. The book has eight chapters in about 300 pages, each ending with problems, followed by a short bibliography and answers to problems. French's preface states three choices that shape it: the discussion is almost entirely confined to mechanical systems, so that the language of the subject can be developed through mechanical displacements and scalar wave equations; the treatment is quantitative and analytical rather than descriptive; and normal modes receive more emphasis than is usual in introductory courses. Theory is linked to real examples, illustrated where possible with original data and photographs. The book remains in print from CRC Press.

Chapter 1: Periodic motions

Sinusoidal vibrations and simple harmonic motion, the rotating-vector representation, and the step from rotating vectors to complex numbers and the complex exponential, which the preface says is introduced early because it is used throughout.

Chapter 2: The superposition of periodic motions

Superposed vibrations in one dimension, at equal frequency, at different frequencies (beats) and in large numbers; then combinations at right angles, including Lissajous figures, and a comparison of parallel and perpendicular superposition.

Chapter 3: The free vibrations of physical systems

The basic mass-spring problem solved with complex exponentials, then a sequence of physical oscillators: elastic solids and Young's modulus, floating objects, pendulums, water in a U-tube, torsional oscillations, "the spring of air" and massive springs. The chapter ends with the decay of free vibrations and the effects of very large damping.

Chapter 4: Forced vibrations and resonance

The undamped and damped driven oscillator, solved with the complex exponential method, with the effect of the resistive term, transients and the power absorbed. Examples of resonance range beyond mechanics to electrical, optical, nuclear and nuclear magnetic resonance, and the chapter closes with anharmonic oscillators.

Chapter 5: Coupled oscillators and normal modes

Starting from two coupled pendulums and symmetry arguments, the chapter builds the superposition of normal modes and a general analytical method for normal frequencies, then forced vibration of coupled systems. It extends to N coupled oscillators, their normal modes and properties, longitudinal oscillations, the limit of very large N and the normal modes of a crystal lattice. The preface credits this approach to Jack Tessman's teaching.

Chapter 6: Normal modes of continuous systems. Fourier analysis

Stretched strings, superposition of modes and forced vibration of a string; longitudinal vibrations of rods, air columns and the elasticity of a gas; normal modes in two and three dimensions; and Fourier analysis, applied in practice and connected to orthogonal functions.

Chapter 7: Progressive waves

What a wave is and how traveling waves relate to normal modes, wave speeds in specific media, superposition and wave pulses, dispersion with phase and group velocities, cut-off, the energy and momentum carried by mechanical waves (including mechanical radiation pressure), and waves in two and three dimensions.

Chapter 8: Boundary effects and interference

Reflection of pulses, impedance and nonreflecting terminations, longitudinal and transverse waves and polarization, the Huygens-Fresnel principle, reflection and refraction of plane waves, the Doppler effect, and interference and diffraction: double slits, gratings, single slits and real slit systems.

Key Concepts

Chapters 1–3: Oscillation

• Simple harmonic motion: sinusoidal motion produced by a restoring force proportional to displacement.
• Complex exponential: a representation of oscillation that turns differential equations into algebra.
• Beats and Lissajous figures: the patterns produced by superposing vibrations along the same line or at right angles.
• Damping: the decay of free vibrations through resistive forces.

Chapters 4–6: Resonance and normal modes

• Resonance: the large response, and maximum power absorption, of an oscillator driven near its natural frequency.
• Normal modes: patterns in which all parts of a coupled system oscillate at one frequency.
• Continuous systems: strings, rods and air columns treated as the limit of many coupled oscillators.
• Fourier analysis: representing a general motion as a sum of normal modes, using their orthogonality.

Chapters 7–8: Waves

• Phase and group velocity: the speeds of wave crests and of wave groups, which differ in dispersive media.
• Impedance: the property that governs reflection at a boundary and allows nonreflecting terminations.
• Huygens-Fresnel principle: the construction behind refraction, diffraction and interference.

Critical Analysis

Strengths

• Normal modes as the organising idea. The preface says the book stresses normal modes more than is customary, to link the dynamics of a continuum to that of one or a few particles, and with an eye to quantum mechanics, where orthogonality and completeness return. Chapters 5 and 6 carry this through from two pendulums to crystal lattices and Fourier analysis.
• Rich physical examples. Chapter 3's list of oscillators and Chapter 4's survey of resonance, from electrical circuits to nuclear magnetic resonance, connect the mathematics to real systems; the preface notes the use of original data and photographs.
• Mathematics introduced gradually. The complex exponential appears in Chapter 1, while partial differential equations are deferred until late in the book; the author considers prior work on differential equations desirable but not essential.
• Problems with answers. Every chapter ends with problems, and a short section of answers to problems follows the bibliography.

Limitations

• Mechanical systems only, by design. The preface explains that electrical and optical systems are mostly left out; electromagnetic waves and wave optics must come from other books.
• Dated presentation. First published in 1971, its notation, examples and photographs belong to its time and has had no revised edition, though it has been reprinted.
• Calculus assumed. The preface expects a fair familiarity with calculus and a good working knowledge of elementary kinematics and dynamics.

Where It Fits in the Course

In this waves and optics course, French supplies the mathematics of oscillations and waves on which optics is built. Walmsley's Light: A Very Short Introduction is a non-mathematical overview that can be read first or alongside. Hecht's Optics and the Pedrottis' Introduction to Optics take up the electromagnetic and optical waves that French deliberately leaves aside; French's Chapter 8 on Huygens-Fresnel, slits and gratings leads directly into their treatments of interference and diffraction. Saleh and Teich's Fundamentals of Photonics and Born and Wolf's Principles of Optics come much later. French's normal-mode chapters also prepare for Fourier methods in quantum mechanics, as the preface intends.

How to Use This Book

The book suits first- and second-year students who know elementary mechanics and some calculus. It works for a short course on oscillations and waves or as a companion to the waves chapters of a general physics text.

• Core: Chapters 1, 3, 4 and 5, which carry the complex exponential, damping, resonance and normal modes; then Chapters 6 and 7 for strings, Fourier analysis and travelling waves.
• Selective: Chapter 2 can be read quickly, though Lissajous figures and beats reward a look; in Chapter 4, the survey of resonance examples is optional for a first pass.
• Bridge to optics: Chapter 8, read just before an optics course begins.

Work the end-of-chapter problems and check them against the answers section where it gives them. Where possible, reproduce the demonstrations described (coupled pendulums, a stretched string) or use simulations, since the book's figures are drawn from such experiments.