PU-103 · Vibrations & Waves
The unit builds the entire theory of waves from a single idea — a mass on a spring — by adding damping and driving, coupling many oscillators together, and taking a continuum limit that produces the wave equation. From that equation the student derives standing modes, Fourier decomposition, energy transport, impedance, dispersion and the Doppler shift, arriving at the field concept that underpins electromagnetism and quantum mechanics.
Lectures
| L01 | — |
| L02 | |
| L03 | |
| L04 | |
| L05 | — |
| L06 | |
| L07 | |
| L08 | |
| L09 | |
| L10 | |
| L11 | |
| L12 | |
| L13 | |
| L14 | |
| L15 | |
| L16 | |
| L17 | |
| L18 | |
| L19 | |
| L20 | |
| L21 | |
| L22 | |
| L23 | |
| L24 | — |
Derivations homed in this unit
Simple Harmonic Motion from a Linear Restoring Force
Any smooth potential near a stable minimum yields sinusoidal motion with angular frequency omega = sqrt(k/m).
Energy Exchange and Conservation in SHM
Total energy of an undamped oscillator is constant while kinetic and potential parts trade at twice the oscillation frequency.
The Damped Harmonic Oscillator
Adding linear friction gives under-, critically, and over-damped regimes classified by the discriminant of the characteristic equation.
Quality Factor and Exponential Energy Decay
The stored energy of a lightly damped oscillator decays as exp(-gamma t), and Q = omega0/gamma counts radians per e-fold.
Driven Oscillator: Amplitude, Phase and Resonance
The steady-state response to sinusoidal forcing peaks near omega0 with a phase lag sweeping from 0 to pi through resonance.
Power Absorption and the Lorentzian Lineshape
The time-averaged power delivered to a driven oscillator is a Lorentzian of width gamma, linking absorption bandwidth to Q.
Coupled Oscillators and Normal Modes
Diagonalizing the coupled linearized equations of motion gives normal modes as eigenvectors and normal frequencies as eigenvalues.
Normal Coordinates for N Coupled Oscillators
Simultaneous diagonalization of the mass and stiffness matrices reduces any small-oscillation system to independent normal coordinates.
The Wave Equation as a Continuum Limit
Taking the spacing to zero in a chain of coupled masses yields the wave equation with speed c = sqrt(T/mu).
d'Alembert's General Solution
Every solution of the 1D wave equation is a sum of an undistorted right- and left-moving profile f(x-ct)+g(x+ct).
Standing Waves and Boundary Quantization
Fixed-end boundary conditions select a discrete spectrum of standing modes with frequencies omega_n = n pi c / L.
Fourier Decomposition into Normal Modes
Any string configuration expands over the orthogonal standing modes, giving each mode's amplitude by projection.
Energy Density and Power Transport
A travelling wave carries energy density and a flux whose time average scales as amplitude squared times frequency squared.
Reflection, Transmission and Impedance Matching
Continuity at a junction of differing media fixes reflection and transmission coefficients set by the impedance mismatch.
Longitudinal Sound Waves in a Fluid
Conservation of mass and momentum in a compressible fluid give a wave equation with c = sqrt(gamma P / rho).
Dispersion, Phase Velocity and Group Velocity
A frequency-dependent dispersion relation separates the phase speed omega/k from the signal-carrying group speed domega/dk.
Wave Packets, Group Velocity and Spreading
A superposition of waves forms a localized packet that travels at the group velocity and broadens under dispersion.
The Doppler Effect and Shock Fronts
Relative motion of source and observer shifts the observed frequency, with a shock cone forming above the wave speed.