Ah, now I see your question more clearly. Thanks for clarifying!
A good place to start is with sound waves, since the sympathetically vibrating strings are responding to sound waves generated by the plucked string. This website provides a nice discussion of sound waves, written for at an upper middle school level: http://www.school-for-champions.com/sci ... rYsAvRDuSo
This website has several activities that explore sound and sound waves: http://www.discoveryeducation.com/teach ... -waves.cfm
Every string vibrates at a fundamental frequency, which is determined by factors like the string's length and thickness. When you pluck a guitar string without any fingers on the fret, the pitch (or frequency) of the sound wave you hear is the natural frequency of the string. Whether or not plucking one string causes another string to vibrate sympathetically depends on whether the two strings share any overtones
, which are multiples of the strings' fundamental frequencies. These following two Wikipedia articles discuss resonances. The second article specifically addresses resonances in string instruments.http://en.wikipedia.org/wiki/Acoustic_resonance
(up to the end of the section called "string resonance in musical instruments")http://en.wikipedia.org/wiki/String_resonance
The equations in the first article can be used to calculate the fundamental frequency and overtones of a string. Since this is a middle school project (or, it's posted in the middle school forum), you can probably leave out the detailed math, unless math is a strong point of your student.
If you have questions about anything in these articles, post back and I'll try to clear things up for you.
Let us know if we can help you with anything else!