You really have to see and hear it done in person to appreciate it, but I'll try to outline it here. If you know a guitarist, have them show you... Then get them to do it for your class!

When a guitar string is plucked, the result is a very complex sum of sine waves - along with a very cool sound. 8-)

There are places along the string that can be touched while plucking that create a perfect tone -- which means a nice, clean, simple sine wave. This is called a harmonic.

During tuning, the guitarist can pluck a harmonic in a special spot on the top string and pluck a harmonic on the next string down in a different special spot (further up the neck)... If the strings are in tune with each other, the same tone should be produced. Same tone = the same exact sine wave.

If the string are NOT in tune with each other, the harmonic tones will not be the same. If the tones are different, the periods of the two sine waves will be different! AND YOU CAN HEAR IT!! I mean REALLY hear it! Especially if an electric guitar is used and you crank up the distortion!!

When the strings are in tune, the two harmonic tones sound like one louder constant tone... 

When the strings are out of tune, you hear a pronounced "wobble!" It sounds like "WAAA ooo WAAA ooo WAAA ooo." (You really have to hear it - I'll probably put in on Coolmath.com as a lesson one of these days.)  You can see the wobble in the graphs too!!  Right before I pull out the guitar, we visit this Interference of Sinusoidal Waveforms website.
(http://www.journey.sunysb.edu/ProjectJava/WaveInt/home.html)

The cool thing is: The further out of tune they are, the more frequent the wobble! This is how the guitarists tune it... They adjust the tuning on the second string and listen for the wobbles to get further apart and, then disappear.

It is VERY cool to do in class and the kids dig it! With the electric guitar, of course, it is pretty loud... but only for a few minutes. After the demo, you can always break into a rousing Hendrix version of The Star Spangled Banner. I do, however, discourage you from lighting the guitar on fire.

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