Harmonic Frequencies - Waves, Oscillations, and Resonance Step by Step

Each harmonic's frequency is the speed over its wavelength, giving evenly spaced frequencies.

waves With v = 12 m/s and L = 6 m the harmonics come out to a clean 1, 2, 3 hertz.

Example

Each harmonic's frequency is the speed over its wavelength, giving evenly spaced frequencies — here a clean 1, 2, 3 hertz. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Each harmonic has its own frequency

Every harmonic obeys speed equals frequency times wavelength, so its frequency is the speed over its wavelength. Since the n-th wavelength is twice the length over n, the n-th frequency is n times the speed over twice the length.

f_n = \frac{v}{\lambda_n} = \frac{n\,v}{2L}

Clean harmonics: 1, 2, 3 hertz

With a speed of 12 metres per second and a length of 6 metres, twice the length is 12, which equals the speed — so the n-th frequency is simply n hertz. The harmonics are 1, 2, 3 hertz, and so on.

f_n = \frac{n \,\cdot\, 12}{2 \,\cdot\, 6} = \hlmath{n}\ \text{Hz}

Shorter wavelength means higher frequency

Hold the wave speed fixed. Higher harmonics have shorter wavelengths, so the same wave speed gives higher frequencies.

\begin{array}{c|c|c}n & \lambda_n & f_n \\ \hline 1 & 12\ \text{m} & 1\ \text{Hz} \\ 2 & 6\ \text{m} & 2\ \text{Hz} \\ 3 & 4\ \text{m} & 3\ \text{Hz}\end{array}

Same pattern, faster wave, higher pitch

Hold the second-harmonic wavelength fixed at six metres. If waves travel faster on the string, more cycles pass each second and the frequency rises.

\begin{array}{c|c|c}v & \lambda & f \\ \hline 6\ \text{m}/\text{s} & 6\ \text{m} & 1\ \text{Hz} \\ 12\ \text{m}/\text{s} & 6\ \text{m} & 2\ \text{Hz} \\ 18\ \text{m}/\text{s} & 6\ \text{m} & 3\ \text{Hz}\end{array}