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 string fixed at both ends, fundamental wavelength is λ₁ = 2L , and second harmonic is λ₂ = L . Thus, λ₁ = 2 λ₂ , or λ₂ = λ₁ / 2 . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields Second harmonic is half the fundamental, illustrating frequency-length-speed interdependence and quantization by boundaries.