Here is the graph.
You can see that there are about, well, 25.6 waves during one tenth of a second. If we show you one second in the same amount of space, it just looks like a blob.
If we double the frequency of any tone, we go up exactly one octave. So 512 vibrations per second is c above middle c.
Here is the graph.
We get more complex sounds by adding waves of different frequencies and loudnesses together. A violin string or an oboe cavity will vibrate with more than one frequency at once, usually at multiples of a fundamental frequency. Here is a wave with components at 512 Hertz, 1024 Hz, 1536 Hz and 2048 Hz (1 Hz = 1 vibration per second).
The higher components are called harmonics of the fundamental tone. If the fundamental is c at 512 Hertz, then the second harmonic is twice the frequency, c an octave higher at 1024 Hertz, the third harmonic is g at 1536 Hertz, and the fourth harmonic is c at 2048 Hertz. A musical tone consists of the fundamental and many harmonics added together.
If we pluck a string in the middle, we only get the odd-numbered harmonics. This is because the center part can't be at rest (we plucked it!) so the string can't vibrate in two halves. A sample graph and sound follow.
For more information on the mathematics and physics of music, a good book to read is Science and Music, by Sir James Jeans. Also try The Acoustical Foundations of Music, by John Backus.