# What is 'harmonic likeness'?

While reading about sympathetic vibrations/resonance, I came across the term 'harmonic likeness'.

Sympathetic resonance or sympathetic vibration is a harmonic phenomenon wherein a formerly passive string or vibratory body responds to external vibrations to which it has a harmonic likeness.

What does it exactly mean?

## 1 Answer

It means that both systems have some normal modes (resonant frequencies) with same frequency. Then by letting one of the systems vibrate it induces a vibration in the second system by the phenomena of resonance.

For example, if you pluck the fifth string (bottom to top) of a guitar it vibrates in a superposition of numerous modes (harmonics). The frequency of the nth mode is $n\cdot 110$, where $110\, \mathrm{Hz}$ is the frequency of the fundamental mode. Then this will make the 6th string to vibrate as well. This string has a fundamental frequency of $82.4\, \mathrm{Hz}$ so its 4th mode will be in resonance with the 3th mode of the 5th string since both have approximately the same frequency, namely $330\, \mathrm{Hz}$. This is called harmonic likeness. The effect is a bit hard to notice in an acoustic guitar since those third and fourth harmonics have small amplitudes which means small volume. It is easier to hear if the sound is amplified such as in an electric guitar.

• Correct, except the 5th string is 110Hz and the 6th is 82.4Hz. Then the 3rd harmonic on the 5th is 330Hz and the 4th harmonic on the 6th is 329.6Hz. Close enough to resonate. – safesphere Nov 1 '17 at 16:25
• @safesphere Thanks for pointing this out! I don't know why but I was sure that the A string had a fundamental mode at 440 Hz. – Diracology Nov 1 '17 at 16:33
• 440Hz is the accepted standard for A of the Second Octave, which is on the 5th fret on the 1st string. The 5th string is also A, so easy to get confused. No big deal. The concept is valid, just a couple octaves shifted. In fact, I tune a guitar exactly by matching these (and other) harmonics :) – safesphere Nov 1 '17 at 16:54
• Please check out my answer on the guitar body being a bass reflex system inverting the phase :) physics.stackexchange.com/questions/365557/… – safesphere Nov 1 '17 at 19:38
• @safesphere Already checked that one. Great answer, by the way =) – Diracology Nov 1 '17 at 19:45