Normal modes of a string thought experiment

[Nikitin]

Hey! So If I have a stretched string of length L fastened at one end, and I am moving the other end sinusoidally, will a standing wave appear ONLY if I move the other end with one of the normal-mode frequencies of the string? If not, will the resulting wave be a moving wave which is a superposition of the string's normal modes?

 

[Simon Bridge]

Do it and see :)

Shaking the string from one end is a bit different from plucking it.
You get a node a little distance x from your fingers - and you will have excited a normal mode of a string length L-x which is fixed at both ends. Otherwise you have to move your hand with the frequency of the normal modes of your string (work them out for a string length L with one end free).

When you drive a string like that - the system is a "driven harmonic oscillator" and the solutions can get quite complicated. You can represent the result as a superposition of normal modes - just as the normal modes can be represented as a superposition of traveling waves.

Real strings can get more complicated still.