Physics III: Vibrations and Waves

What this course covers

• Role of Waves and Vibrations in Nature
• Connecting Demonstrations to Mathematical Solutions
• Ubiquity of waves and vibrations
• Translating physics into mathematics
• Force diagram analysis for the spring-mass system
• Equation of motion for a harmonic oscillator
• General solution and initial conditions
• Universality of Hooke's law via Taylor expansion
• Linearity and superposition of solutions
• Three equivalent forms of the harmonic solution
• Complex exponential representation and Euler's formula
• Nonlinear effects and limits of the linear approximation
• Energy in simple harmonic motion
• Generalizing Hooke's law to rotational systems
• Small-angle approximation
• Introducing the damping (drag) force
• Complex exponential solution method
• Underdamped oscillation
• Critically damped oscillation
• Overdamped oscillation
• Damped harmonic oscillator review
• Equation of motion for a driven oscillator
• Complex notation solution method
• Steady-state amplitude and phase lag
• Full solution: transient plus steady-state
• Distinguishing ω₀, ω, and ω_d
• Driven pendulum as a worked example
• Limiting behaviour of amplitude and phase
• Resonance and the quality factor Q
• Resonance in physical and natural systems
• Damped driven oscillator review
• Coupled oscillators: motivation and complexity
• Normal modes: definition
• Identifying normal modes by symmetry
• General solution as superposition of normal modes
• Equations of motion for coupled systems
• Matrix formulation of coupled equations
• Normal mode ansatz in matrix form
• Eigenvalue condition: determinant equals zero
• Solving for normal mode frequencies and amplitudes
• Setting up coupled oscillator equations
• Matrix formulation of coupled oscillators
• Normal mode ansatz and complex notation
• Normal frequencies from the determinant condition
• Normal mode shapes (eigenvectors)
• General solution as superposition of normal modes
• Beat phenomenon in coupled oscillators
• Normal coordinates
• Coupled oscillators review
• Normal modes of coupled oscillators
• Superposition of normal modes
• Matrix formulation of equations of motion
• Driven coupled oscillators
• Particular (driven) solution via complex exponentials
• Cramer's rule for amplitude solutions
• Resonance in driven coupled systems
• Complete solution: homogeneous plus particular
• Symmetry and normal modes
• Normal modes and driven coupled oscillators
• Symmetry matrix (S matrix) definition
• Commutation of S and M⁻¹K matrices
• Shared eigenvectors of commuting matrices
• Infinite coupled oscillator system setup
• Translation symmetry matrix for infinite chain
• Physical constraint: |β| = 1 and wave-number k
• Dispersion relation: ω vs. k
• Normal modes as standing waves (sinusoidal amplitude profile)
• Maximum frequency mode and physical interpretation
• Eigenvalue problem for normal modes
• Space translation symmetry and the S matrix
• Eigenvectors of S and the form of Aⱼ
• Choosing β = e^{ika} and the wave number k
• Shared eigenvectors of S and M⁻¹K (commuting matrices)
• Equation of motion for a string-and-mass system
• Dispersion relation ω(k) for the discrete chain
• Boundary conditions and discrete normal modes
• Continuous limit and the wave equation
• Continuum limit of coupled oscillators
• Separation of variables
• Normal mode solutions of the wave equation
• Dispersion relation for a continuous string
• Boundary conditions and allowed wave numbers
• Determining phase from initial velocity
• Fourier series and orthogonality of sine functions
• Traveling (Progressing) Wave Solution
• Proof That f(x − v_p t) Satisfies the Wave Equation
• Direction of Wave Propagation
• Superposition and Wave Passing Through Each Other
• Decomposing a Stationary Shape into Two Traveling Waves
• Kinetic and Potential Energy of a String
• Boundary Conditions at a String Junction
• Reflection and Transmission Coefficients
• Wave reflection and transmission at a boundary
• Harmonic progressing waves
• Longitudinal waves
• Deriving the sound wave equation
• Isothermal vs. adiabatic models for sound
• Speed of sound formula
• Experimental measurement of sound speed
• Effect of molecular mass on sound speed (helium demo)
• Boundary conditions for sound in tubes
• Maxwell's equations in differential form
• Divergence and curl of vector fields
• Derivation of the EM wave equation
• Speed of light from Maxwell's constants
• Plane wave solution for the electric field
• Relationship between E and B in a plane wave
• General plane wave in three dimensions
• Boundary condition at a perfect conductor
• Standing EM waves and phase relationship
• Poynting vector and energy flow in standing waves
• Sending information using wave pulses
• Dispersion relation for an ideal string
• Non-dispersive vs. dispersive media
• Stiffness and the modified wave equation
• Fourier decomposition of a pulse
• Beat phenomenon and two-wave superposition
• Phase velocity
• Group velocity
• Normal modes in a dispersive bound system
• Phase velocity does not transmit information
• Dispersion in wave-carrying media
• Phase velocity vs. group velocity
• Fourier decomposition of an arbitrary signal
• Wave propagation formula for dispersive media
• The Dirac delta function
• Fourier inversion: extracting C(ω)
• Amplitude modulation (AM) principle
• Linear approximation of dispersion relation near ω₀
• Signal travels at group velocity without distortion
• Fourier decomposition in dispersive media
• Group velocity and phase velocity via AM radio
• Fourier transform of a decaying exponential
• Time-bandwidth trade-off
• Formal definitions of spread: Δt and Δω
• Proof of the uncertainty relation Δω·Δt ≥ 1/2
• Connection to Heisenberg's Uncertainty Principle
• Dispersion relation for light and photon mass
• Normal modes of a 2D plate via translation symmetry
• Chladni figures as nodal patterns of 2D normal modes
• Two-dimensional mass-spring lattice
• 2D dispersion relation
• 2D eigenvectors via space-translation symmetry
• Fixed boundary conditions in 2D
• 2D standing-wave normal modes (sine×sine)
• Nodal lines and Chladni figures
• Continuous 2D wave equation
• 2D plane progressive wave
• Boundary matching: Ky conservation
• Law of reflection
• Snell's law from wave boundary conditions
• Snell's Law and Refraction Index Review
• Total Internal Reflection
• Optical Fiber and Total Internal Reflection
• Electromagnetic Wave Polarization Basics
• Linear Polarization
• Circular Polarization
• Elliptical Polarization
• Unpolarized Light
• Polarizers and the Easy Axis
• Malus's Law and Intensity After a Polarizer
• Wave-Particle Duality Hint via Single-Photon Polarization
• Linear and circular polarization review
• Anisotropic refractive index in a wave plate
• Phase difference introduced by a wave plate
• Quarter-wave plate operation
• Optical activity in chiral materials
• Requirement for sustained electromagnetic radiation
• Radiation from an accelerated charge
• Dipole radiation pattern and polarization
• Electromagnetic wave radiation review
• Polarization of scattered skylight
• Electromagnetic waves in dielectric materials
• Electric displacement field D and H field
• Maxwell's equations in matter
• Refractive index from material properties
• Boundary conditions at a dielectric interface
• Fresnel reflection and transmission coefficients
• Brewster's angle and polarization of reflected light
• Wave interference and superposition
• Intensity as time-averaged Poynting vector
• Phase difference and constructive/destructive interference
• Reflection and transmission coefficients at an interface
• Phase flip upon reflection (n₁ < n₂)
• Thin-film interference in a soap film
• Optical path length and wavelength inside a medium
• Thickness dependence of reflected color
• Thin Film Interference
• Huygens' Principle
• Double-Slit Geometry and Path-Length Difference
• Phase Difference and Intensity in Two-Slit Interference
• N-Slit Interference and Principal Maxima
• Phased-Array Radar Design
• Wave–Particle Duality and Electron Double-Slit Experiment
• Single-Slit Diffraction and Intensity Envelope
• Diffraction as infinite-source interference
• 2-D Fourier transform as field calculation
• Single-slit aperture function f(x,y)
• Single-slit diffraction intensity pattern
• Minima condition and wavelength measurement
• Two-slit pattern modulated by single-slit diffraction
• Circular aperture diffraction and the Rayleigh criterion
• Optical resolution limit for display design
• Beam spreading limits directed-energy weapons
• Wave-particle duality of light
• Photoelectric effect
• Double-slit experiment with electrons
• Observer effect and which-path information
• Heisenberg Uncertainty Principle
• Probability density and wave functions
• Particle in a box (infinite potential well)
• Schrödinger equation and dispersion relation
• De Broglie matter waves and group velocity
• Gravitational waves
• LIGO and interferometric detection
• Single oscillator and damping regimes
• Driven oscillators and resonance
• Coupled oscillators and normal modes
• Translation symmetry and infinite coupled systems
• The wave equation and its solutions
• Fourier decomposition and boundary conditions
• Dispersion relation, phase velocity, and group velocity
• Fourier transform and the uncertainty principle
• Reflection, refraction, and Snell's law
• Polarization and polarizing elements
• Electromagnetic wave generation by accelerated charges
• Interference and diffraction
• Uncertainty principle and wave description
• Infinite coupled oscillators
• Symmetry arguments for simplification
• Humor as a teaching tool
• Creating a relaxed classroom atmosphere
• Reducing student stress around questioning
• In-class prediction polling
• Polarizer experiment setup
• Hand-raising over clicker technology
• Avoiding automatic graphing tools during polling
• Chalkboard vs. slide-based teaching
• Writing pace as thinking time
• Pointing to highlight key equations
• Combining chalkboard with slides
• Slides for reviewing prior content
• Slides for rich media content
• Role of recitation sections
• Q&A opportunities in recitation
• Problem-solving videos as recitation substitute
• Single oscillator behavior
• Coupled oscillators
• Infinite coupled oscillator chains
• Continuous media and wave propagation
• Unified theoretical framework for oscillations
• Role of demonstrations in learning physics
• Opening with a Big Question
• Theory-Driven Problem Solving
• Closing the Loop on the Big Question
• Comparing Calculation with Demonstration
• Summarizing Take-Home Messages
• Demonstrating Genuine Enthusiasm

Syllabus

1. Lectures

   1. 8.03SC Physics III: Vibrations and Waves Introduction
   2. 1. Periodic Oscillations, Harmonic Oscillators
   3. 2. Damped Free Oscillators
   4. 3. Driven Oscillators, Transient Phenomena, Resonance
   5. 4. Coupled Oscillators, Normal Modes
   6. 5. Beat Phenomena
   7. 6. Driven Oscillators, Resonance
   8. 7. Symmetry, Infinite Number of Coupled Oscillators
   9. 8. Translation Symmetry
   10. 9. Wave Equation, Standing Waves, Fourier Series
   11. 10. Traveling Waves
   12. 11. Sound Waves
   13. 12. Maxwell's Equation, Electromagnetic Waves
   14. 13. Dispersive Medium, Phase Velocity, Group Velocity
   15. 14. Fourier Transform, AM Radio
   16. 15. Uncertainty Principle, 2D Waves
   17. 16. 2D and 3D waves, Snell's Law
   18. 17. Polarization, Polarizer
   19. 18. Wave Plates, Radiation
   20. 19. Waves in Medium
   21. 20. Interference, Soap Bubble
   22. 21. Phased Radar, Single Electron Interference
   23. 22. Diffraction, Resolution
   24. 23. Quantum Waves and Gravitational Waves
   25. Exam Review
   26. Students' Common Misconceptions
   27. Making Time for Individual Questions in a Large Lecture
   28. Using Humor to Enhance Learning
   29. Using Demonstrations in Class
   30. Behind-the-Scenes Demo Prep
   31. Taking a Vote to Engage Learners
   32. Combining Chalk Talks and Slides in a Complementary Way
   33. Using Questionnaires to Customize Course Content
   34. The Role of Recitations
   35. Course Iteration: Incorporating Theoretical Content and Demonstrations
   36. Tips for Physics Educators