Practice question

**Quantization due to boundaries** leads to discrete harmonic series. Frequency difference between harmonics is f₁, so f₃ - f₁ = 2f₁ = v/L. Understanding node-antinode pattern explains resonance and overtones in strings. At an antinode, particle displacement is maximum, oscillating perpendicular to wave propagation. Particle velocity (rate of displacement change) is perpendicular to wave velocity (along the string). Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields They are perpendicular, illustrating frequency-length-speed interdependence and quantization by boundaries.