Waves - Quick Revision
Nature of waves
- A wave is a moving disturbance that transports energy and information without bulk transfer of matter.
- Mechanical waves (string, sound, water, seismic) need a material medium; they depend on the elastic and inertial properties of the medium.
- Transverse wave: particles oscillate perpendicular to the direction of propagation (only in media that sustain shear, i.e. solids and strings).
- Longitudinal wave: particles oscillate along the direction of propagation (possible in solids, liquids and gases, e.g. sound).
Progressive wave
- y(x, t) = a sin(kx - omega t + phi) for a wave moving in +x; replace -omega t with +omega t for -x.
- k = 2 pi/lambda (angular wave number); omega = 2 pi/T = 2 pi nu.
- Speed: v = omega/k = nu lambda = lambda/T. The medium fixes v; the source fixes nu; lambda then follows.
Wave speeds
- String: v = sqrt(T/mu), mu = mass/length.
- Sound in a fluid: v = sqrt(B/rho); in a solid bar: v = sqrt(Y/rho).
- Sound in a gas (Laplace): v = sqrt(gamma P/rho); Newton's isothermal v = sqrt(P/rho) is ~15% low.
Superposition, reflection, standing waves
- Superposition: net displacement = algebraic sum of individual displacements.
- Reflection at a rigid boundary -> phase change of pi; at an open boundary -> no phase change.
- Standing wave: y = 2a sin(kx) cos(omega t); fixed nodes (zero amplitude) and antinodes; node-to-node spacing = lambda/2.
Normal modes
- String fixed at both ends / open pipe: nu_n = n v/(2L), n = 1, 2, 3, ... (all harmonics).
- Pipe closed at one end: nu_n = (2n - 1) v/(4L) (only odd harmonics).
Beats
- Two close frequencies give waxing/waning intensity: nu_beat = |nu1 - nu2|.