Standing Waves: Resonant Frequencies and Harmonics

Resonant Frequencies and Standing Waves

• Standing waves exist only at certain frequencies, called resonant (natural) frequencies.
• Some objects have a single resonant frequency (e.g., a tuning fork, a pendulum, a mass bouncing on a spring).
• Others have many resonant frequencies (e.g., a rope, a stretched spring, or air in an air column).
• For objects with multiple resonances, the resonant frequencies are whole-number multiples of the lowest frequency, the fundamental.
• Only waves with resonant frequencies persist; others decay due to interference.

Fundamental Frequency and Harmonics

• The fundamental frequency is the first harmonic; the resonant frequencies are harmonics.
• For the fundamental, the end-to-end distance equals one-half a wavelength: L = \frac{\lambda1}{2}, hence \lambda1 = 2L.
• The fundamental frequency is f1 = \frac{v}{\lambda1} = \frac{v}{2L}.
• All higher resonant frequencies are multiples of the fundamental: fn = n f1 = \frac{n v}{2L}, \quad n = 1,2,3,… or equivalently fn = n f0 where f0 = f1.
• The fundamental has only one loop or antinode.

Standing Wave on a String (Fixed Ends)

• Ends are nodes (nodes at the fixed ends).
• Allowed wavelengths: \lambda_n = \frac{2L}{n}, \quad n = 1,2,3,…
• Corresponding frequencies: fn = \frac{v}{\lambdan} = \frac{n v}{2L} = n f_1.
• The fundamental corresponds to one half-wavelength fitting in the length, producing one antinode (one loop).