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. In longitudinal waves, displacement nodes (zero displacement) occur where particles are maximally compressed or rarefied, leading to maximum pressure variation (pressure antinodes). Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields They occur at the same points, illustrating frequency-length-speed interdependence and quantization by boundaries.