Harmonics on a String

Much like "pumping a swing," a guitar string (or any vibrating string) will vibrate will certain "modes of vibration". That is, it will vibrate in certain specific configurations that satisfy the geometry of the string - fixed at both ends.

Since the string is fixed at both ends, it must have nodes (points of NO disturbance) at both ends. However, other possible modes of vibration can satisfy this condition. The wavelength must be satisfied by:

wavelength (lambda) = 2L / n

where n is the so-called "harmonic number," or if you prefer, the number of HALF-waves.

So, the wavelengths can be given by:

n wavelength
1 2L
2 L
3 2L/3
4 2L/4, or L/2
5 2L/5
6 2L/6, or L/3

The frequencies that correspond to these "harmonics" are given by:

v = f * lambda

or...

f = v/lambda

It is also important to note that the frequencies increase LINEARLY. That is, the frequency for n=2 is twice that of n=1. The frequency for n=3 is three times that of n=1. Got it?

This turns out to have musical significance.

Doubling a frequency generates something called and OCTAVE. For those of you who do NOT speak music, an octave is the difference between DO and DO, if you sing:

DO RE MI FA SO LA TI DO

Note that there are 8 notes here, thus the term octave.

Tripling the frequency also gives something of musical significance - it is 3/2 times greater than n=2. In music, a frequency multiple of 3/2 is defined as a "fifth", so named since it is the difference between DO and SO (5 notes).

But don't worry about that business, please - I mention it for your interest.

Play with this:

http://zonalandeducation.com/mstm/physics/waves/standingWaves/understandingSWDia1/UnderstandingSWDia1.html

This one is ok, but the pictures are a little misleading:

http://zonalandeducation.com/mstm/physics/waves/standingWaves/standingWaveDiagrams1/StandingWaveDiagrams1.html

A word on terminology:

The lowest harmonic (n=1) is called the fundamental. Any other n-value above 1 is called an overtone. So, the second harmonic (n=2) is called the first overtone.

Got it?

Next up.... how to translate these ideas to sounds in tubes/pipes, and how to construct musical scales.