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 boundary is unconstrained, allowing maximum displacement. The reflected wave reinforces the incident wave in the same direction, maintaining phase. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Due to maximum displacement, illustrating frequency-length-speed interdependence and quantization by boundaries.