Understanding standing waves is key to learning how musical instruments work. Here I'll talk a little bit about how guitars work. The high E string of a guitar is 64.0cm long (as are all the other strings), and has a fundamental frequency (or first harmonic frequency) of 330Hz (if tuned correctly). This is the frequency of the E note. When you pluck a guitar string you are exciting the first harmonic*: Think about how the first harmonic looks (a big lump); when you pluck the string, you simply are creating a big lump that oscillates up and down (a standing wave). For the not so musical among you, the high E string is the string that causes the highest-pitched sounds in a guitar. When you press down on the first fret of the guitar, the note you play is an F, which has a frequency of 350Hz.
a. (Take your time for this one) How far away is the first fret from the nut of the guitar? The nut is the thing that supports the strings at the end closest to the headstock. In other words, it's one of the endpoints on which you get a node. When the guitar is pressed at some fret, you shorten the distance through which the string vibrates.
b. (Easy!) When musicians tune their guitar they turn the tuning keys located on the headstock. What physical quantity of the string are they changing when they turn the keys? This should explain why turning the keys makes the frequency of the sounds change!
c. (Easy!) Guitars have six strings, each one producing different frequencies. What physical quantity is different between the six strings of the guitar that makes them sound distinctively? Assume they all have the exact same length and tension.