Mechanical and electromagnetic vibrations and waves

Dealing with vibrations and waves, this text aims to provide understanding of the basic principles and methods of analysing various physical phenomena.

The content includes the general properties of propagation, a detailed study of mechanical (elastic and acoustic) and electromagnetic waves, propagation, attenuation, dispersion, reflection, interference and diffraction of waves.

It features chapters on the effect of motion of sources and observers (both classical and relativistic), emission of electromagnetic waves, standing and guided waves and a final chapter on de Broglie waves constitutes an introduction to quantum mechanics.

Author Notes

Tamer Bcherrawy received the Doctorat 3e Cycle from the University of Paris and a PhD in Theoretical Physics from the University of Rochester, New York. He has taught Physics at the Faculty of Science of the Lebanese University in Beirut (Lebanon), the University of Savoy in Chamber/, the IUFM and the University of Nancy (France). He was head of the Physics Department at the Lebanese University. He has published a number of research articles on High Energy Particle Physics.

Preface	p. xi
Chapter 1 Free Oscillations	p. 1
1.1 Oscillations and waves, period and frequency	p. 1
1.2 Simple harmonic vibrations: differential equation and linearity	p. 2
1.3 Complex representation and phasor representation	p. 5
1.4 Point mass subject to a force-Kx	p. 9
1.5 Angular oscillations	p. 12
1.6 Damped oscillations	p. 15
1.7 Dissipation of the energy of a damped oscillator	p. 19
1.8 Oscillating LCR circuits	p. 20
1.9 Small oscillations of a system with one degree of freedom	p. 22
1.10 Nonlinear oscillators	p. 25
1.11 Systems with two degrees of freedom	p. 25
1.12 Generalization to systems with n degrees of freedom	p. 29
1.13 Normal variables for systems with n degrees of freedom*	p. 32
1.14 Summary	p. 35
1.15 Problem solving suggestions	p. 38
1.16 Conceptual questions	p. 39
1.17 Problems	p. 40
Chapter 2 Superposition of Harmonic Oscillations, Fourier Analysis	p. 51
2.1 Superposition of two scalar and isochronous simple harmonic oscillations	p. 51
2.2 Superposition of two perpendicular and isochronous vector oscillations, polarization	p. 53
2.3 Superposition of two perpendicular and non-isochronous oscillations	p. 57
2.4 Superposition of scalar non-synchronous harmonic oscillations, beats	p. 58
2.5 Fourier analysis of a periodic function	p. 60
2.6 Fourier analysis of a non-periodic function	p. 65
2.7 Fourier analysis of a signal, uncertainty relation	p. 67
2.8 Dirac delta-function	p. 69
2.9 Summary	p. 71
2.10 Problem solving suggestions	p. 74
2.11 Conceptual questions	p. 75
2.12 Problems	p. 76
Chapter 3 Forced Oscillations	p. 83
3.1 Transient regime and steady regime	p. 83
3.2 Case of a simple harmonic excitation force	p. 85
3.3 Resonance	p. 87
3.4 Impedance and energy of a forced oscillator in the steady regime	p. 88
3.5 Complex impedance	p. 92
3.6 Sustained electromagnetic oscillations	p. 94
3.7 Excitation from a state of equilibrium*	p. 96
3.8 Response to an arbitrary force, nonlinear systems*	p. 97
3.9 Excitation of a system of coupled oscillators*	p. 99
3.10 Generalization of the concepts of external force and impedance*	p. 103
3.11 Some applications	p. 104
3.12 Summary	p. 105
3.13 Problem solving suggestions	p. 106
3.14 Conceptual questions	p. 107
3.15 Problems	p. 108
Chapter 4 Propagation in Infinite Media	p. 115
4.1 Propagation of one-dimensional waves	p. 115
4.2 Propagation of two- and three-dimensional waves	p. 117
4.3 Propagation of a vector wave	p. 121
4.4 Polarization of a transverse vector wave	p. 123
4.5 Monochromatic wave, wave vector and wavelength	p. 125
4.6 Dispersion	p. 127
4.7 Group velocity	p. 129
4.8 Fourier analysis for waves*	p. 130
4.9 Modulation*	p. 133
4.10 Energy of waves	p. 135
4.11 Other unattenuated wave equations, conserved quantities*	p. 137
4.12 Impedance of a medium*	p. 139
4.13 Attenuated waves	p. 140
4.14 Sources and observers in motion, the Doppler effect and shock waves	p. 143
4.15 Summary	p. 148
4.16 Problem solving suggestions	p. 150
4.17 Conceptual questions	p. 152
4.18 Problems	p. 153
Chapter 5 Mechanical Waves	p. 159
5.1 Transverse waves on a taut string	p. 159
5.2 Strain and stress in elastic solids	p. 162
5.3 Elastic waves in massive springs and rods	p. 166
5.4 Propagation of sound in a pipe	p. 168
5.5 Transverse waves on elastic membranes	p. 172
5.6 Mechanical waves in three dimensions	p. 174
5.7 Energy of mechanical waves	p. 176
5.8 Progressive waves, impedance and intensity	p. 179
5.9 Elements of physiological acoustics	p. 183
5.10 Infrasounds and ultrasounds	p. 185
5.11 Surface waves*	p. 186
5.12 Summary	p. 191
5.13 Problem solving suggestions	p. 194
5.14 Conceptual questions	p. 194
5.15 Problems	p. 195
Chapter 6 Electromagnetic Waves	p. 201
6.1 Principal results of the electromagnetic theory	p. 201
6.2 The propagation equations of the fields in vacuum and infinite dielectrics	p. 204
6.3 Electromagnetic simple harmonic plane waves	p. 205
6.4 Energy density and the Poynting vector	p. 206
6.5 Polarization of electromagnetic waves	p. 207
6.6 Momentum density and angular momentum density, radiation pressure*	p. 209
6.7 Electromagnetic waves in plasmas*	p. 212
6.8 Electromagnetic waves in Ohmic conductors*	p. 214
6.9 Quantization of electromagnetic radiation	p. 218
6.10 Electromagnetic spectrum	p. 219
6.11 Emission of electromagnetic radiations	p. 221
6.12 Spontaneous emission and stimulated emission	p. 223
6.13 Summary	p. 226
6.14 Problem solving suggestions	p. 229
6.15 Conceptual questions	p. 229
6.16 Problems	p. 231
Chapter 7 Reflection and Refraction of Waves	p. 237
7.1 Reflection of an elastic wave on two joined strings	p. 237
7.2 Reflection and transmission of a one-dimensional acoustic wave	p. 240
7.3 General laws of reflection and transmission of three-dimensional waves	p. 243
7.4 Reflection and refraction of a three-dimensional acoustic wave	p. 246
7.5 Reflection and refraction of an electromagnetic wave at the interface of dielectrics	p. 248
7.5.1 Case of linear polarization in the plane of incidence	p. 249
7.5.2 Case of linear polarization perpendicular to the plane of incidence	p. 250
7.5.3 Conservation of energy	p. 252
7.5.4 Brewster's law	p. 253
7.6 Case of attenuated waves in the second medium*	p. 255
7.7 Summary	p. 258
7.8 Problem solving suggestions	p. 260
7.9 Conceptual questions	p. 261
7.10 Problems	p. 262
Chapter 8 Interference and Diffraction	p. 269
8.1 Order and fringes of interference of two waves	p. 269
8.2 Intensity and contrast	p. 271
8.3 Interference of light waves, Young's experiment	p. 273
8.4 Multiwave interference, conditions for interference	p. 277
8.5 Holography	p. 281
8.6 Thin film interference	p. 282
8.7 The Huygens-Fresnel principle and diffraction by an aperture	p. 285
8.8 Diffraction grating	p. 290
8.9 Diffraction of X-rays	p. 295
8.10 Summary	p. 297
8.11 Problem solving suggestions	p. 299
8.12 Conceptual questions	p. 300
8.13 Problems	p. 301
Chapter 9 Standing Waves and Guided Waves	p. 307
9.1 One-dimensional standing waves	p. 308
9.2 Standing waves on a membrane and in a rectangular cavity	p. 313
9.3 Fourier analysis of standing waves*	p. 316
9.4 Resonance and standing waves	p. 319
9.5 Sound wave guided by two parallel plates	p. 320
9.6 Guided sound waves in a rectangular pipe	p. 322
9.7 Transmission lines	p. 324
9.8 Electromagnetic waveguides*	p. 326
9.9 Waveguides formed by two plane and parallel plates*	p. 328
9.10 Guided electromagnetic waves in a hollow conductor*	p. 331
9.11 Applications of waveguides	p. 335
9.12 Summary	p. 337
9.13 Problem solving suggestions	p. 340
9.14 Conceptual questions	p. 341
9.15 Problems	p. 342
Answers to the Problems	p. 349
Appendices	p. 371
Appendix A Mathematical Review	p. 373
A.1 Expansion formulas	p. 373
A.2 Logarithmic, exponential and hyperbolic functions	p. 374
A.3 Trigonometric functions	p. 374
A.4 Integrals	p. 375
A.5 Complex numbers	p. 378
A.6 Vector analysis in Cartesian coordinates	p. 380
A.7 Vector analysis in curvilinear coordinates	p. 382
Appendix B Units in Physics	p. 387
B.1 Multiples and submultiples of units	p. 387
B.2 Fundamental and derived SI units	p. 387
B.3 Mechanical units	p. 388
B.4 Electromagnetic units	p. 389
Appendix C Some Physical Constants	p. 391
C.1 Mechanical and thermodynamic constants	p. 391
C.2 Electromagnetic and atomic constants	p. 392
Further Reading	p. 393
Index	p. 395

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