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 rigid boundary, phase changes by π . Incident: y_i = 0.07 sin (20x - 60t) . Reflected: y_r = -0.07 sin (20x + 60t) . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields y = -0.07 sin (20x + 60t), illustrating frequency-length-speed interdependence and quantization by boundaries.