UNIVERSITY OF NAIROBI COLLEGE OF EDUCATION AND EXTERNAL STUDIES COURSE TITLE: BED (SCIENCE) UNIT TITLE: WAVES AND OPTICS UNIT CODE: SPH 103 UNIT AUTHOR: PROF. R.O. GENGA DEPARTMENT OF PHYSICS UNIVERSITY OF NAIROBI LECTURE NOTES
UNIT AIM: At the end of this unit you would be able to understand, apply and acknowledge the importance of physical processes involved in waves and optics. UNIT OUTLINE The unit shall be studied under three major topics, that is, wave motion, sound waves and wave geometry (optics), respectively. Details are given below. TOPIC 1: WAVE MOTION 1.1 INTRODUCTION................................................. 1 1.2 DESCRIPTION OF WAVE MOTION................................. 2 1.3 WAVES IN ONE-DIMENSION....................................... 7 1.4 THE GENERAL EQUATION OF WAVE MOTION....................... 11 1.5 ELASTIC (LONGITUDINAL) WAVES IN A SOLID AND IN A STRING..... 11 1.5.1 Elasticity.11 1.5.2 Elasticity and Plasticity...................................... 14 1.5.3 Elastic (Longitudinal) Waves in a Solid.......................... 20 1.5.4 Elastic (Longitudinal) Waves in a String.......................... 21 1.6 TRANSVERSE WAVES IN A STRING............................... 25 1.6.1 Transverse Travelling Waves in a String.......................... 25 1.6.2 Transverse Standing Waves in a String........................... 27 1.6.3 Resonance...31 1.6.4 Traveling Elastic Waves in a Bar.............................. 34 1.7 WAVES IN TWO-DIMENSIONS................................... 38 1.7.1 Wavefronts............................................... 39 1.7.2 Reflection of Waves......................................... 39 1.7.3 Refraction of Waves......................................... 45 1.7.4 Interference of Waves....................................... 52 1.7.5 Diffraction of Waves........................................ 56 1.8 SURFACE WAVES IN A LIQUID.................................. 61 1.9 WHAT PROPAGATES IN WAVE MOTION?......................... 66 1.10 GROUP VELOCITY.............................................69
TOPIC 2: SOUND WAVES 2.1 INTRODUCTION..73 2.2 PRESSURE WAVES IN A GAS COLUMN.74 2.3 ADIABATIC CHARACTER OF A LONGITUDINAL WAVE...76 2.4 WAVES IN THREE-DIMENSIONS.80 2.4.1 Spherical Waves in a Fluid.81 2.4.1 Spherical Harmonic Pressure Waves..82 2.5 INTENSITY OF SOUND WAVES 84 2.5.1 Intensity of Sinusoidal Waves along a Rod 84 2.5.2 Intensity of Sinusoidal waves in a gas column 88 2.5.3 Intensity of Spherical Wave.92 2.5.4 Intensity Level and Loudness.100 2.6 STANDING LONGITUDINAL ELASTIC WAVES 105 2.7 VIBRATING SYSTEMS AND SOURCES OF SOUND...105 2.7.1 Sound from Vibrating String 105 2.7.2 Sound from Vibrating Air Column 109 2.7.3 End-correction of Pipes or Tubes.. 113 2.7.4 Effects of Temperature and End-correction on Pitch of Pipes or Tubes.114 2.8 RESONANCE IN A PIPE OR TUBE..116 2.8.1 Demonstration of Phenomenon of Resonance in Air Column 117 2.8.2 Demonstration of Phenomenon of Acoustic Resonance.121 2.8.3 Applications of Resonance.121 2.8.4 Hazards of Resonance.122 2.9 CONSEQUENCES OF REFLECTION OF SOUND..125 2.10 CONSEQUENCES OF REFRACTION OF SOUND.127 2.11 CONSEQUENCES OF DIFFRACTION OF SOUND 129 2.12 CONSEQUENCE OF INTERFERENCE FO SOUND : BEATS.129 2.13 APPLICATIONS OF BEATS 130 2.14 THE DOPPLER EFFECT..132 2.15 APPLICATIONS OF THE DOPPLER EFFECT 138 2.17 APPLICATIONS OF SOUND: ULTRASOUND AND MEDICAL IMAGING 140
TOPIC 3: WAVE GEOMETRY (OPTICS) 3.1 INTRODUCTION 143 3.2 RAYS AND BEAM OF LIGHT...143 3.3 FERMAT S PRINCIPLE..144 3.4 APPLICATIONS OF FERMAT S PRINCIPLE 146 3.5 REFRACTIVE INDEX...148 3.6 THE PRINCIPLE OF REVERSIBILITY OF LIGHT.148 3.7 CONSEQUENCES (EFFECTS) OF REFRACTION..149 3.8 REAL AND APPARENT DEPTH 153 3.8.1 Expression of Refractive Index in terms of Real and Apparent Depth.. 154 3.8.2 Expression of Object distant d in terms of Medium Thickness t and refractive index n.155 3.9 TOTAL INTERNAL REFLECTION AND CRITICAL ANGLE 155 3.10 CONSEQUENCE OF TOTAL INTENAL REFLECTION..156 3.11 RELATION BETWEEN CRITICAL ANGLE AND REFRACTIVE INDEX.157 3.12 APPLICATIONS OF TOTAL INTERNAL REFLECTION.158 3.13 APLANATIC POINTS AND APLANATIC SURFACES.164 3.13.1 Aplanatic Points and Aplanatic Surface for Reflection 164 3.13.2 Aplanatic Points and Aplanatic Surface for Refraction 166 3.13.3 Conjugate Points and Conjugate planes...167 3.14 APPLICATION OF APLANATIC POINTS OF A SPHERICAL SURFACE...168 3.15 REFLECTION AT A SPHERICAL SURFACE OF SMALL APERTURE 169 3.15.1 Spherical Surfaces (Mirrors)...169 3.15.2 Reflection of Narrow Parallel Beam from Spherical Surfaces.170 3.16 BASIC DEFINITIONS..170 3.16.1 Relation between the Focal Length of a Spherical Surface and its Radius of Curvature.171 3.16.2 Sign Conventions.173 3.17 CONSTRUCTION OF RAY DIAGRAMS..174 3.18 IMAGES FORMED BY A CONCAVE SPHERICAL SURFACE..175 3.19 IMAGES FORMED BY A CONVEX SPHERICAL SURFACE..177 3.20 MAGNIFICATION FORMULA.179
3.21 DESCARTES FORMULA FOR REFLECTION AT A SPHERICAL SURFACE OF SMALL APERTURE..183 3.21.1 Focal Point.184 3.22 REFLECTION AT A SPHERICAL SURFACE OF LARGE APERTURE.188 3.23 SPHERICAL SURFACE ABERRATIONS.188 3.23.1 Spherical Aberration...188 3.23.2 Marginal (Outer Zone) Rays Parallel to the Axis of Spherical Surfaces cross the Axis inside the Focal Point F.190 3.23.3 Methods of Minimizing Spherical Aberration 191 3.23.4 Astigmatism.194 3.23.5 Astigmatism in Eye 195 3.23.6 Methods of Minimizing Astigmatism.196 3.24 APPLICATIONS OF REFLECTING SPHERICAL SURFACES..197 3.25 REFRACTION AT A SPHERICAL SURFACE OF SMALL APERTURE 198 3.25.1 Descartes Formula for Refraction at a Spherical Surface of Small Aperture..199 3.25.2 Focal Points.. 201 3.26 REFRACTION AT A SPHERICAL SURFACE OF LARGE APERTURE 205 3.27 SPHERICAL ABERRATION OF A SPHERICAL REFRACTING SURFACE.206 3.28 LENSES...208 3.28.1 Types of Lenses 208 3.28.2 Power of Lens 209 3.29 THE THIN LENS 210 3.29.1 Ray Tracing..211 3.30 BASIC DEFINITIONS 212 3.31 CONSTRUCTION OF RAY DIAGRAMS.214 3.32 FORMATION OF IMAGES BY A CONVERGING LENS..216 3.33 FORMATION OF IMAGES BY A DIVERGING LENS 218 3.34 MAGNIFICATION.. 219 3.35 DESCARTES FORMULA FOR A THIN LENS 220 3.35.1 Focal Points 222 3.36 DESCARTES FORMULAR FOR A COMBINATION OF TWO LENSES WITH POSITIONS OF THEIR FOCI SEPARATED BY DISTANCE t.222
3.37 FOCAL LENGTH OF TWO THIN LENSES IN CONTACT..224 3.38 THE LARGE LENS...227 3.39 LENS ABERRATIONS.228 3.39.1 Introduction.228 3.39.2 Spherical Aberration...228 3.39.3 Effects of Lens Shape on Spherical Aberration..230 3.39.4 Methods of Minimizing Spherical Aberration.233 3.39.5 Coma.236 3.39.6 Radius of Comatic Circle..238 3.39.7 Effects of Lens Shape and Position, on Coma..239 3.39.8 Coma and the Sine Condition 240 3.39.9 Practical Elimination of Coma from Optical Systems 241 3.39.10 Astigmatism 243 3.39.11 Astigmatism in Eye.245 3.39.12 Curvature of the Field..247 3.39.13 Methods of Minimizing Curvature of the Field.. 248 3.39.14 Distortion.250 3.39.15 Barrel Distortion..250 3.39.16 Pin-cushion Distortion.251 3.39.17 Chromatic Aberration...253 3.39.18 Methods of Minimizing Chromatic Aberration 255 3.40 THE PRISM.. 257 3.40.1 Refraction Through Prism..257 3.41 OPTICAL INSTRUMENTS.260 3.41.1 The Simple Microscope....260 3.41.2 The Compound Microscope.263 3.41.3 The Telescope 265 3.41.4 The Projector...268 4.0 SOLUTIONS TO SELF ASSESSMENT QUESTIONS.270 4.1 WAVE MOTION.270 4.2 SOUND WAVES.284 4.3 WAVE GEOMETRY (OPTICS).290
REFERENCES You may read the following textbooks to assist you understand lecture notes whenever it is necessary. 1. University Optics. Vol. 1; by D.W. Tenquist, R.M. Whittle and J. Yarwood; Published by London Ilife Books Ltd. (1969). 2. Fundamentals of Optics, Third Edition; by F.A. Jenkins and H.E. White; Published by McGraw-Hill Kogakusha, Ltd. 3. Physics, Second Edition Expanded; by H.C. Ohanian; Published by W.W. Norton and Company. (1989) 4. Physics; by C. Zafiratos; Published by John Wiley & Sons, Inc.(1975) 5. University Physics, 5 th edition; by F.W. Sears, M.W. Zamansky and H.D. Young; Published by Addison-Wesley Publishing Company. (1978).