Harmonic Frequencies - Waves & Sound Step by Step

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

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}

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