Standing Waves

Harmonic Frequencies

a clean 1, 2, 3

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.

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.

fn=vλn=n v2Lf_n = \frac{v}{\lambda_n} = \frac{n\,v}{2L}
The third harmonicThe third standing wave: three bulges with nodes at the ends and two more inside.

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.

fn=n ⋅ 122 ⋅ 6=n Hzf_n = \frac{n \,\cdot\, 12}{2 \,\cdot\, 6} = \hlmath{n}\ \text{Hz}
waves With v = 12 m/s and L = 6 m the harmonics come out to a clean 1, 2, 3 hertz.