## Equation #13: Resonant frequencies/Harmonics

vn = nth harmonic
n = integer
v0 = fundamental frequency

We’ve all seen those opera singer references which claim that at a particular frequency, the singer’s voice can break a wine glass. The physics behind the trick is that of resonance.
Basically, anything that can vibrate has some fundamental frequency of vibration which depends on its physical properties. A fixed, vibrating string is the best way to visualize this. If you pluck it at just the right point, it starts to vibrate. The string vibrates in a manner that the fixed ends and the point you plucked(approximately) stay at their position while the rest of the string oscillates up and down. If it is one of those resonant frequencies, the vibrations will last for some time(ideally they should last forever). Otherwise they’ll dampen quickly.
The string prefers to vibrate at these resonant frequencies, the lowest of which is called the fundamental frequency. Every object has a fundamental frequency. If you put another object near it vibrating at one of the resonating frequencies, the first object will pick up the vibration as well. This is called resonance (e.g. a tuning fork and a pipe).

Some examples: