Overtones and Harmonics in Standing Waves
The lowest frequency standing wave possible in a given system is called the fundamental frequency (also known as the first harmonic). In many systems - a taut string, for example - the next lowest frequency standing wave is called the first overtone (also known as the second harmonic). And so on, with the nᵗʰ overtone being the (n+1)ᵗʰ harmonic. Initially, the harmonic/overtone distinction might seem to be merely an irritatingly trivial difference in terminology, but in fact it's an important distinction, especially in musical contexts.
For example, in a pipe open at one end, the even-numbered harmonics are the wrong wavelength to form stable standing waves, so the fundamental is still the first harmonic, but the first overtone is actually the third harmonic, the second overtone is the fifth harmonic and so on.
If this is sounding a bit tricky, here are some definitions which might help:
- The fundamental frequency of a system is the lowest frequency standing wave that can form in it.
- A harmonic is any integer multiple of the fundamental frequency (whether or not the system can form a standing wave of this frequency)
- An overtone is any frequency that the system is actually capable of resonating at (whether or not it has a convenient mathematical relationship to the fundamental frequency).
There are examples of systems where:
- all overtones are harmonics (e.g. on a guitar string), or
- some harmonics are not overtones (e.g. in a pipe closed at one end but open at the other), and
- not all overtones are harmonics (e.g. unpitched percussion instruments produce non-harmonic overtones which is why we perceive them as being unpitched).
Hyperphysics has a usefully clear explanation of this whole area which I found helpful.
Standing waves in pipes are even more potentially confusing, because the particle motion and pressure variation are 'out of phase' with each other, so a pressure node is a motion antinode and vice versa. I wrote an explanation of this for my pupils many years ago in response to a particular unclear textbook explanation. It includes everything above, plus some extension work on end corrections, which your musical pupils might appreciate, and a practical you can do by measuring Boomwhacker tubes.