Modes, Packets and Media

Nonlinear Oscillation, Normal Modes and Dispersion in 8 chapters — from “The anharmonic oscillator: frequency, harmonics and drift” to “Water waves and de Broglie waves”.

Chapters

  1. The anharmonic oscillator: frequency, harmonics and drift
    • Circular error and the amplitude-dependent period
    • The anharmonic potential to fourth order
    • The period as an energy integral
    • The secular term of a naive series
    • The Lindstedt-Poincare method and harmonic generation
    • Mean-position drift in a lopsided well
    • Hardening and softening of real oscillators
    • Harmonic generation read from a spectrum
    • The anharmonic potential term by term
    • Problems
  2. Driven and pumped nonlinear oscillators and the route to chaos
    • Hardening and softening under a drive
    • The driven Duffing oscillator by harmonic balance
    • Jump and hysteresis beyond the critical drive
    • Critical drive and secondary resonances
    • Parametric resonance of a pumped oscillator
    • Parametric threshold in swings, ripples and amplifiers
    • The stroboscopic section of a driven pendulum
    • The period-doubling spectrum and broadband chaos
    • The driven Duffing oscillator, settled and open
    • Problems
  3. Normal modes of many coupled oscillators: eigenvalues, symmetry and response
    • Beyond the beaded string
    • The generalised eigenvalue problem
    • Carbon dioxide as a generalised eigenvalue problem
    • Mass-weighted orthogonality and energy by mode
    • Symmetry-adapted modes from commuting operations
    • Ring modes and the zero mode
    • The localised mode of a light defect
    • Modal response of a damped, driven system
    • The vibration absorber
    • Anharmonic springs and the Fermi-Pasta-Ulam recurrence
    • The Fermi-Pasta-Ulam recurrence, still open
    • Problems
  4. The continuum limit, Fourier series and the spectrum of a measured signal
    • The discreteness correction
    • The non-uniform string
    • The hanging chain
    • Parseval's theorem for modes
    • The spectrum of a finite record
    • Spectral leakage and windows
    • The power spectral density
    • The spectrogram
    • Problems
  5. Reflection, transmission, matching and tunnelling at boundaries
    • Load impedance and reflection
    • The bead reflector
    • The input impedance of a layer
    • The anti-reflection coating
    • Broadband matching and tapers
    • The evanescent barrier
    • Frustrated reflection and resonant transmission
    • Problems
  6. Dispersion and the propagation of wave packets
    • Carrier and envelope
    • The envelope equation
    • Group-velocity dispersion in fibre
    • Stationary phase
    • Dispersive arrival
    • The energy velocity
    • Slow and fast light
    • Front velocity and barrier times
    • Problems
  7. Transmission lines and electromagnetic waves in matter, plasmas and guides
    • Dispersive arrival from a pulsar
    • The telegrapher's equations
    • The lossy line and Heaviside's condition
    • Line terminations and time-domain reflectometry
    • Dielectrics and the coaxial line
    • The plasma dispersion relation
    • The ionosphere and the dispersion measure
    • The hollow waveguide
    • One dispersion relation, three media
    • Problems
  8. Water waves and de Broglie waves
    • Dispersive arrival across an ocean
    • Water-wave regimes
    • Tsunami speed and shoaling
    • Ocean swell and storm distance
    • The Kelvin wake
    • The Faraday wavelength
    • The solitary wave
    • The de Broglie dispersion relation
    • The relativistic matter wave
    • Group velocity in every medium
    • Problems