Waves & Oscillations

About this course

Fourteen lectures tracing a single idea from a mass on a spring to the unity of wave phenomena. Simple harmonic motion, damping and resonance come first, then coupled oscillators and normal modes, and the chain of oscillators whose continuum limit is the wave equation itself. From there: traveling and standing waves, boundaries and musical instruments, energy and impedance, the Doppler effect and shock waves. The final module covers interference, diffraction, dispersion and group velocity, closing on why the same mathematics describes sound, light and matter.

What you will be able to do

• Solve simple harmonic, damped and driven oscillator problems and identify resonance
• Find normal modes of coupled oscillators and explain dispersion in a chain
• Derive the wave equation and analyse traveling and standing wave solutions with boundary conditions
• Compute energy, power and impedance in waves, and apply the Doppler effect
• Explain interference, diffraction, and the distinction between phase and group velocity

What this course covers

• D'Alembert's General Solution to the Wave Equation
• Traveling Waves Preserve Shape (Non-Dispersive Strings)
• Harmonic Traveling Wave and Its Vocabulary
• Phase Velocity and the Medium's Control of Speed
• Superposition Principle from Linearity
• One Grammar for All of Module Three
• Worked Problem: Harmonic Wave Quantities
• Coupled oscillator equations of motion
• Normal modes: in-phase and out-of-phase
• Superposition and energy beats between masses
• Eigenvalue picture of normal modes
• Effect of coupling strength on mode frequencies and beat rate
• Universality of coupled-oscillator template
• Worked numerical example and course outlook
• Restoring Force and the Master Equation of SHM
• Energy Conservation in the Oscillator
• The Pendulum: Period Independent of Mass and Amplitude
• Breakdown of the Small-Angle Approximation
• Every Stable Minimum is a Parabola (Universal SHM)
• Worked Numerical Example
• Course Spine: SHM as Foundation for Damping and Resonance
• Superposition: the one rule of wave addition
• Constructive and destructive interference
• Phasors: adding waves as rotating arrows
• Beats: interference of two close frequencies
• Two-source interference fringes in space
• Worked numerical problem: speakers and forks
• Clause Eleven summary and look-ahead to diffraction
• Chain of coupled oscillators setup
• Normal modes as travelling waves; the wave-number
• Derivation of the dispersion relation
• Long-wave limit: sound and non-dispersive propagation
• Cutoff frequency and the Brillouin zone edge
• Dispersion and group velocity across the band
• Connection to solid-state physics and worked example
• Standing waves from superposition
• Boundary conditions and quantized wavelengths
• The harmonic series
• Resonance as the mode-selection mechanism
• Open and closed pipes: different mode ladders
• Timbre as the harmonic amplitude mix
• One recipe: from instruments to quantum mechanics
• Huygens' Principle and Wavefront Reconstruction
• Single-Slit Diffraction Pattern
• Diffraction Grating: Sharp Principal Maxima
• Resolution Limit and the Rayleigh Criterion
• Worked Numerical Examples
• Clause 12 Summary: One Rule, Five Results
• Simple harmonic motion review
• Deriving the damped oscillator equation
• Three damping regimes and critical damping
• Quality factor Q and energy decay rate
• Universality of the damped oscillator equation
• Damping as energy dispersal and the arrow of time
• Worked numerical example and lecture synthesis
• The Driven Oscillator Equation
• Resonance Peak and Amplitude Formula
• Damping as Resonance Ceiling
• Phase Lag, Power Absorption, and Q as Sharpness
• The RLC Circuit as a Driven Oscillator
• Resonance as Universal Frequency Selection
• Worked Example: Driven Oscillator Numbers
• The Continuum Limit: Chain to String
• Deriving the Wave Equation from the Chain
• Deriving the Wave Equation from Newton's Law
• Non-Dispersive Propagation and Shape Preservation
• Wave Speed Formula V = √(T/μ) and Its Consequences
• The Wave Equation as Universal Master Equation
• Course Synthesis and Forward Look to D'Alembert
• Dispersion relations and phase velocity
• Wave packets built from superposed wavelengths
• Group velocity VG = dω/dK
• Group velocity carries signal and energy
• Phase velocity above the speed of light
• Packet spreading from group-velocity dispersion
• The dispersion curve as a complete description
• Wave energy density and power formula
• Characteristic impedance of a medium
• Reflection and transmission coefficients at a boundary
• Fixed and free end limits of reflection
• Impedance matching: sending all energy across
• Worked numerical problem: power and energy split
• Lecture synthesis and preview of Doppler effect
• The universal wave equation for mechanical waves
• Maxwell's derivation of light as an electromagnetic wave
• The dial: one wave grammar across four systems
• The electromagnetic spectrum as one wave
• de Broglie matter waves and the Schrödinger equation
• The unity of all waves: one language
• Confinement quantizes waves: guitar strings and atoms
• Wave speed belongs to the medium
• Doppler effect — moving observer
• Doppler effect — moving source
• General Doppler formula and medium asymmetry
• Shock waves and the Mach cone
• Unifying geometry: one picture for siren and boom
• Worked numerical application of Doppler and Mach

Syllabus

1. Free & Forced Oscillations

   1. Simple Harmonic Motion
   2. Damped Oscillations
   3. Driven Oscillations & Resonance

2. Coupled Oscillators & Normal Modes

   1. Coupled Oscillators & Normal Modes
   2. The Chain of Oscillators & Dispersion
   3. From Oscillators to Waves

3. Traveling & Standing Waves

   1. The Wave Equation & Traveling Waves
   2. Standing Waves, Boundaries & Musical Instruments
   3. Energy, Power & Impedance
   4. The Doppler Effect & Shock Waves

4. Wave Phenomena & Unification

   1. Superposition, Interference & Beats
   2. Diffraction & Huygens' Principle
   3. Dispersion, Wave Packets & Group Velocity
   4. The Unity of Waves