🌊 Waves

1. Introduction to Waves

A wave is a disturbance that travels through a medium, transferring energy and momentum without transferring matter.

Transverse Waves: Particles oscillate perpendicular to the direction of wave propagation (e.g., waves on a string, light).
Longitudinal Waves: Particles oscillate parallel to the direction of wave propagation (e.g., sound waves). They travel in compressions and rarefactions.

2. Wave Equation

The displacement equation for a progressive (traveling) wave moving in the positive x-direction is:

y(x,t) = A sin(kx - ωt + φ) A = amplitude, k = wave number (2π/λ), ω = angular frequency (2π/T)

Wave Speed: v = ω/k = λf (where f is frequency).

3. Principle of Superposition

When two or more waves overlap, the resultant displacement is the algebraic sum of their individual displacements (y = y₁ + y₂ + ...).

4. Standing (Stationary) Waves

Formed by the superposition of two identical waves traveling in opposite directions.

🌊 Nodes and Antinodes

Nodes: Points of zero amplitude (destructive interference).

Antinodes: Points of maximum amplitude (constructive interference).

Organ Pipes: In closed pipes, only odd harmonics are present. In open pipes, all harmonics (even and odd) are present.

5. Beats and Doppler Effect

Beats: Periodic variation in sound intensity caused by the superposition of two waves of slightly different frequencies. Beat frequency = |f₁ - f₂|.

Doppler Effect

The apparent change in frequency of sound due to the relative motion between the source and the observer. (e.g., pitch of an ambulance siren drops as it passes by).