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 a standing wave, nodes occur where the displacement is zero, which happens when sin(kx) = 0 . This implies kx = nπ , where k = (2π/λ) , so x = (nλ/2) (n = 0, 1, 2, ..). Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Displacement is zero, illustrating frequency-length-speed interdependence and quantization by boundaries.