THE PHYSICS OF VIBRATIONS AND WAVES
CONTENTS 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 D SUPERPOSITION OF N EQUAL SHM VECTORS OF LENGTH A WITH RANDOM PHASE SOME USEFUL MATHEMATICS DAMPED SIMPLE HARMONIC MOTION METHODS OF DESCRIBING THE DAMPING OF AN OSCILLATOR 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 X BEHAVIOUR OF DISPLACEMENT VERSUS DRIVING FORCE FREQUENCY X PROBLEM ON VIBRATION INSULATION SIGNI?CANCE OF THE TWO COMPONENTS OF THE DISPLACEMENT CURVE POWER SUPPLIED TO OSCILLATOR BY THE DRIVING FORCE VARIATION OF P AV WITH X ABSORPTION RESONANCE CURVE THE Q-VALUE IN TERMS OF THE RESONANCE ABSORPTION BANDWIDTH THE Q-VALUE AS AN AMPLI?CATION FACTOR THE EFFECT OF THE TRANSIENT TERM 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 TRANSVERSE WAVE MOTION PARTIAL DIFFERENTIATION WAVES VELOCITIES IN WAVE MOTION SOLUTION OF THE WAVE EQUATION CHARACTERISTIC IMPEDANCE OF A STRING (THE STRING AS A FORCED OSCILLATOR) RE?ECTION AND TRANSMISSION OF WAVES ON A STRING AT A BOUNDARY RE?ECTION AND TRANSMISSION OF ENERGY THE RE?ECTED AND TRANSMITTED INTENSITY COEF?CIENTS 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 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 RE?ECTION AND TRANSMISSION OF SOUND WAVES AT BOUNDARIES RE?ECTION AND TRANSMISSION OF SOUND INTENSITY WAVES ON TRANSMISSION LINES IDEAL OR LOSSLESS TRANSMISSION LINE COAXIAL CABLES CHARACTERISTIC IMPEDANCE OF A TRANSMISSION LINE RE?ECTIONS FROM THE END OF A TRANSMISSION LINE SHORT CIRCUITED TRANSMISSION LINE ÐZ L ¼ 0Þ 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 ELECTROMAGNETIC WAVES MAXWELL’S EQUATIONS ELECTROMAGNETIC WAVES IN A MEDIUM HAVING FINITE PERMEABILITY L AND PERMITTIVITY E BUT WITH CONDUCTIVITY R ¼ THE WAVE EQUATION FOR ELECTROMAGNETIC WAVES ILLUSTRATION OF POYNTING VECTOR IMPEDANCE OF A DIELECTRIC TO ELECTROMAGNETIC WAVES ELECTROMAGNETIC WAVES IN A MEDIUM OF PROPERTIES L, E AND R (WHERE R 6¼ 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 RE?ECTION AND TRANSMISSION OF ELECTROMAGNETIC WAVES AT A BOUNDARY RE?ECTION FROM A CONDUCTOR (NORMAL INCIDENCE) ELECTROMAGNETIC WAVES IN A PLASMA ELECTROMAGNETIC WAVES IN THE IONOSPHERE 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 SPECI?C HEATS RE?ECTION AND TRANSMISSION OF A THREE-DIMENSIONAL WAVE AT A PLANE BOUNDARY TOTAL INTERNAL RE?ECTION AND EVANESCENT WAVES 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 WAVES IN OPTICAL SYSTEMS LIGHT WAVES OR RAYS? FERMAT’S PRINCIPLE THE LAWS OF RE?ECTION THE LAW OF REFRACTION RAYS AND WAVEFRONTS RAY OPTICS AND OPTICAL SYSTEMS POWER OF A SPHERICAL SURFACE MAGNI?CATION 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 IN?NITY THE DEVIATION METHOD FOR (A) TWO LENSES AND (B) A THICK LENS THE MATRIX METHOD 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 FRAUNHOFER DIFFRACTION FOR TWO EQUAL SLITS ÐN ¼ 2Þ 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 WAVE MECHANICS ORIGINS OF MODERN QUANTUM THEORY HEISENBERG’S UNCERTAINTY PRINCIPLE SCHRO¨DINGER’S WAVE EQUATION ONE-DIMENSIONAL IN?NITE POTENTIAL WELL SIGNI?CANCE OF THE AMPLITUDE W NÐXÞ OF THE WAVE FUNCTION PARTICLE IN A THREE-DIMENSIONAL BOX NUMBER OF ENERGY STATES IN INTERVAL E TO E Þ DE THE POTENTIAL STEP THE SQUARE POTENTIAL WELL THE HARMONIC OSCILLATOR ELECTRON WAVES IN A SOLID PHONONS 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 Ð_X; X; T) PHASE SPACE CHAOTIC RESPONSE OF A FORCED NON-LINEAR MECHANICAL OSCILLATOR A BRIEF REVIEW CHAOS IN FLUIDS RECOMMENDED FURTHER READING REFERENCES 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