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 fixed ends: v_n = (n v/2L) . Third harmonic ( n = 3 ): v₃ = (3 × 90/2 × 2) = (270/4) = 67.5 Hz . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 67.5 Hz, illustrating frequency-length-speed interdependence and quantization by boundaries.