Suppose we have a tuning fork, we know it will vibrate at a constant pitch(frequency). Let's say the tuning fork is of 440hz, so I imagine the prongs of the fork to be vibrating 440 times per second, but how does an object that will have a fundamental frequency plus added overtones vibrate physically?
Bonus question : Why are overtones added to the original fundamental frequency anyway?
When a string fixed at both ends is plucked (like a guitar) or hit (like a piano) this generates waves in both directions along the string at many different frequencies. These waves are reflected back from the fixed ends and so travel very rapidly to and fro along the string. Most of these waves interfere with each other and die away very quickly.
If, however, the wavelength of a wave is twice the length of the string $L$, or divides into $2L$ a whole number of times, then it will reinforce itself and will persist for a long time. So the tones that we hear the string produce are its resonant frequency, with wavelength $2L$, and multiples of this frequency, with wavelengths $L, \frac {2L} 3, \frac L 2 \dots$ etc. These multiples are overtones.
In most string instruments the player alters the vibrating length of each string by pressing it against a fingerboard. The string will produce a different note depending on where it is pressed. Some instruments (such as the piano or harp) instead have a separate string or strings for each note.
The body of each string instrument is designed so as to amplify certain tones and overtones, which is what gives that instrument its particular quality or timbre.
A tuning fork is a special case, since it is instead designed to suppress overtones and produce a very pure tone at its resonant frequency - see this Wikipedia article for details.
