Practice question

**Reflection at boundaries** follows phase change rules: rigid boundary (fixed end) introduces π phase shift, inverting displacement y → -y, while free boundary reflects without phase change. Reflected wave derived by reversing propagation direction kx → -kx and applying phase shift, preserving k = 2π/λ and ω = 2πf. At a free end, the wave reflects without inversion (phase change of 0 radians) because the boundary allows maximum displacement, aligning the reflected wave with the incident wave. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Remains unchanged, illustrating frequency-length-speed interdependence and quantization by boundaries.