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Wave Propagation

Alain Le Bot

Published May 1, 2015• 0 citations
Publisher

Abstract

Abstract This chapter is an introduction to wave phenomena and propagation in linear media. The existence of travelling waves and the dymanics of wave packets are studied. The dispersion equation, phase speed, group speed, the uncertainty principle, and energy transport relations are analysed in detail.
DOI
10.1093/acprof:oso/9780198729235.003.0005

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A physics-oriented review is first given of wave equations: examples, the Cauchy problem, solutions in various coordinates, their symmetries, how to build a wave packet. Fourier analysis and dispersion relations are then introduced, and here the key tool is the symbol of differential operators, which makes it possible to study them in terms of algebraic polynomials involving cotangent-bundle variables. Further basic material deals with geometrical optics from the wave equation, phase and group velocity (and their dual relationship in momentum space), the Helmholtz equation and the eikonal approximation for the scalar wave equation. The Schrödinger equation is then derived with emphasis on the wave packet and its relation to classical behaviour in the light of the Einstein–de Broglie relation. For this purpose, it is shown that it is possible to build a wave packet for which the wave-like properties manifest themselves for distances of the order of atomic dimensions. The Fourier transform with respect to time of the wave packet obeys the stationary Schrödinger equation, while the wave function is found to obey a partial differential equation which is of first order in time. The wave equation The wave equation occurs in several branches of classical physics, e.g. the theory of sound, electromagnetic phenomena in vacuum and in material media, and elastic vibrations of material bodies. We consider first the simplest situation, i.e. the propagation of waves in a homogeneous, isotropic and stationary medium. Propagation is characterized by the refractive index n , a quantity that is independent of the point in space-time.
Simple Harmonic Oscillation Massona Spring Simple Harmonic Oscillator Equation LC Circuits Simple Pendula Compound Pendula Damped and Driven Harmonic Oscillation Damped Harmonic Oscillation Quality Factor LCR Circuits Driven Damped Harmonic Oscillation Driven LCR Circuits Transient Oscillator Response Coupled Oscillations Two Spring-Coupled Masses Two Coupled LC Circuits Three Spring-Coupled Masses Transverse Standing Waves Normal Modes of a Beaded String Normal Modes of a Uniform String General Time Evolution of a Uniform String Longitudinal Standing Waves Spring-Coupled Masses Longitudinal Waves on a Thin Elastic Rod Sound Waves in an Ideal Gas Fourier Analysis Traveling Waves Standing Waves in a Finite Continuous Medium Traveling Waves in an Infinite Continuous Medium Wave Interference Energy Conservation Transmission Lines Normal Reflection and Transmission at Interfaces Electromagnetic Waves Doppler Effect Wave Propagation in Inhomogeneous Media Multi-Dimensional Waves Plane Waves Three-Dimensional Wave Equation Cylindrical Waves Spherical Waves Oscillation of an Elastic Sheet Polarization of Electromagnetic Waves Laws of Geometric Optics Fresnel Relations Total Internal Reflection Sound Waves in Fluids Wave Pulses Fourier Transforms General Solution of One-Dimensional Wave Equation Bandwidth Dispersive Waves Pulse Propagation Electromagnetic Waves in Unmagnetized Plasmas Faraday Rotation Electromagnetic Wave Propagation in Conductors Waveguides Pulse Propagation in Two Dimensions Gravity Waves Wave Dragon Ships Ship Wakes Capillary Waves Wave Optics Introduction Two-Slit Interference Coherence Multi-Slit Interference Thin Film Interference One-Dimensional Fourier Optics Single-Slit Diffraction Multi-Slit Diffraction Two-Dimensional Fourier Optics Wave Mechanics Introduction Photoelectric Effect Electron Diffraction Representation of Waves via Complex Numbers Schrodinger's Equation Probability Interpretation of Wavefunction Wave Packets Heisenberg's Uncertainty Principle Wavefunction Collapse Stationary States Three-Dimensional Wave Mechanics Particle in Finite Square Potential Well Square Potential Barrier Appendix A: Physical Constants Appendix B: Useful Mathematics Appendix C: Electromagnetic Theory Bibliography Index Exercises appear at the end of each chapter.
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 The propagation of light in dispersive media can be described by five different kinds of wave velocities [1-3]: • Phase velocity, which is the speed at which the zero crossings of the carrier wave move • Group velocity, at which the peak of a wave packet moves • Energy velocity, at which the energy is transported by the wave • Signal velocity, at which the half-maximum wave amplitude moves • Front velocity, at which the first appearance of a discontinuity moves Generally, all five velocities can differ from each other, although in a linear passive dispersive media, they coincide and are, usually, less than the light velocity in vacuum. Recently, experimental demonstrations [4-6] in variousmedia show that the group velocity of light canbe reducedby10-100million compared with its vacuum phase velocity. The physics is based on very steep frequency dispersion in the vicinity of narrow resonance of electromagnetically induced transparency (EIT) [7-11].
For any kind of wave phenomenon one can find ways to derive the respective dispersion relation from experimental observations and measurements. This dispersion relation determines the structure of the wave equation and thus characterizes the dynamics of the respective wave. Different wave phenomena are thus governed by different differential equations. Here we want to emphasize the experimental approach to matter waves, but before doing so we will discuss and test the procedure for other types of waves, in particular water waves.