Oscillations, Waves and Acoustics

Physics in 18 chapters — from “Simple harmonic motion” to “Pipes, horns, waveguides, Helmholtz resonators and acoustic filters”.

Chapters

  1. Simple harmonic motion
    • Linear restoring force and the equation of motion
    • Amplitude and phase constant
    • Rotating vector picture of simple harmonic motion
    • Energy of the harmonic oscillator
    • Small oscillations about an equilibrium
    • Linearity and superposition
    • Problems
  2. Damped oscillation and the quality factor
    • Viscous damping force and the damped oscillator equation
    • The complex exponential method
    • Underdamped, critical and overdamped motion
    • Energy decay and the quality factor
    • Problems
  3. Forced oscillation and resonance
    • Transient and steady state
    • The complex amplitude
    • Resonance curve and phase lag of the response
    • Absorbed power and resonance width
    • Mechanical impedance
    • Phase lag of the response in two devices
    • Problems
  4. Superposition of oscillations: beats, modulation and Lissajous figures
    • Phasor addition of equal frequencies
    • Beats
    • Amplitude modulation and sidebands
    • The phasor sum of N equal oscillations
    • Perpendicular oscillations and Lissajous figures
    • Problems
  5. Two coupled oscillators and normal coordinates
    • Coupled equations of motion for two carts
    • Symmetric and antisymmetric normal modes
    • Determinant condition for the mode frequencies
    • Normal coordinates decouple the equations
    • General motion as a sum of modes
    • Weak coupling and energy exchange
    • Energy in normal coordinates
    • Problems
  6. Normal modes of many coupled oscillators
    • Matrix form of the coupled equations
    • Eigenvalue problem for normal modes
    • Zero-frequency mode of a free molecule
    • Modes of the beaded string
    • Wavenumber of a chain mode
    • Dispersion relation of the chain and its cutoff
    • Symmetry classification of modes
    • Orthogonality and completeness of modes
    • Problems
  7. The continuum limit, Fourier series and the spectrum of a signal
    • The continuum limit and the wave equation
    • Normal modes of a string fixed at both ends
    • Fourier sine series on an interval
    • The spectrum of a plucked string
    • Fourier series of a periodic signal
    • Harmonic spectrum and timbre
    • Problems
  8. The wave equation and travelling waves
    • The wave equation from a string element
    • D'Alembert's travelling-wave solution
    • Harmonic travelling waves and phase
    • Longitudinal waves in a bar and a fluid column
    • Wave speed from restoring and inertial properties
    • Problems
  9. Wave energy, power flow and characteristic impedance
    • Energy per unit length of a string wave
    • Power transmitted along a string
    • Characteristic impedance of a string
    • Intensity of a wave
    • Problems
  10. Reflection, transmission and impedance matching at boundaries
    • Boundary conditions at a junction
    • Amplitude reflection and transmission coefficients
    • Reflection at fixed and free ends
    • Power reflection and transmission coefficients
    • Partial standing wave and the standing wave ratio
    • Impedance matching by a quarter-wave section
    • Evanescent waves on a chain
    • Problems
  11. Standing waves and the normal modes of strings and pipes
    • Standing wave from two travelling waves
    • Boundary conditions for strings and pipes
    • Harmonic series of open and closed pipes
    • Resonant response of a bounded medium
    • Problems
  12. Waves from moving sources: the Doppler effect, Mach cones and sonic booms
    • Doppler effect for a moving source
    • Doppler effect for a moving observer
    • General Doppler formula and its low-speed limit
    • Mach cone and sonic boom
    • Problems
  13. Dispersion, phase and group velocity, and wave packets
    • Dispersion relations and the phase velocity
    • Group velocity from two waves
    • Wave packets and the bandwidth relation
    • Spreading of a wave packet
    • Dispersion of deep-water waves and stiff strings
    • Problems
  14. Waves in two and three dimensions: wave vectors, refraction and polarisation
    • Wave equation in two and three dimensions
    • Modes of a rectangular membrane and box
    • Plane wave and wave vector
    • Spherical and cylindrical waves
    • Huygens' construction, reflection and refraction
    • Refraction in a continuously varying medium
    • Polarisation of transverse waves
    • Problems
  15. Interference, coherence and diffraction
    • Two-source interference
    • Coherence of two sources
    • The N-source array and the grating
    • Single-slit diffraction
    • Angular spread and resolution
    • Problems
  16. The acoustic wave equation in fluids
    • Acoustic variables and the small-amplitude assumption
    • Acoustic pressure from compression
    • Linearised equation of motion for a fluid slab
    • Acoustic wave equation and the speed of sound
    • Adiabatic compression of a gas and sound speed
    • Pressure and particle velocity in a plane wave
    • Liquids, solids and the spherical sound wave
    • Problems
  17. Sound intensity, sound pressure level and specific acoustic impedance
    • Specific acoustic impedance
    • Acoustic intensity of a plane wave
    • Sound pressure level and the decibel
    • Level and distance from a source, and adding sources
    • Acoustic energy density and the smallness of sound
    • Sound at an impedance step
    • Problems
  18. Pipes, horns, waveguides, Helmholtz resonators and acoustic filters
    • Lumped acoustic elements: acoustic mass and compliance
    • The Helmholtz resonator
    • Input impedance of a pipe
    • Horn as an impedance transformer
    • Modes of a rigid duct and their cutoff
    • Acoustic filters: expansion chambers and side branches
    • Problems