Practice question

**Standing wave in fixed string** has nodes at ends, quantizing modes. Fundamental n=1 has λ₁ = 2L, higher harmonics multiples of fundamental f₁. Third harmonic n=3 has three half-wavelengths in length L, f₃ = 3v/(2L), illustrating standing wave condition and boundary enforcement. For a standing wave, v_n = (n v/2L) . Halving the length L to L/2 gives v_n' = (n v/2 · (L/2)) = (n v/L) = 2 · (n v/2L) , doubling the frequency. Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Doubles, illustrating frequency-length-speed interdependence and quantization by boundaries.