Physics · Principle of superposition of waves, reflection of waves, standing waves in strings and organ pipes, fundamental mode and harmonics, beats
If are the wavelength of the waves giving resonance to the fundamental, first an
If \( \lambda_{1}, \lambda_{2}, \lambda_{3} \) are the wavelength of the waves giving resonance to the fundamental, first and second overtone modes respectively in a string fixed at both ends. The ratio of the wavelengths \( \boldsymbol{\lambda}_{1}: \boldsymbol{\lambda}_{2}: \boldsymbol{\lambda}_{3} \) is
- A. 1: 2: 3
- B. 1: 3: 5
- C. \( 1: \frac{1}{2}: \frac{1}{3} \)
- D. \( 1: \frac{1}{3}: \frac{1}{5} \)
Step-by-step solution
For a string fixed at both ends, the wavelength of the nth harmonic is λ_n = 2L/n. Fundamental (n=1): λ1 = 2L; first overtone (n=2): λ2 = L; second overtone (n=3): λ3 = 2L/3. Taking λ1 as reference, ratio λ1:λ2:λ3 = 1:1/2:1/3.