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. Newton’s formula: v = √((P/rho)) = 280 m/s . Laplace correction: v = √((gamma P/rho)) = 331 m/s . Divide: (331/280) = √(gamma) . √(gamma) = 1.182 , so gamma = (1.182)² ≈ 1.4 . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 1.4, illustrating frequency-length-speed interdependence and quantization by boundaries.