Is there no way to create an arbitrary wave on a string fixed at two ends?

Question

My question concern standing waves on a string with 2 fixed end. As there is one wave created, it will be reflected when it reaches one end and create another identical wave with opposite direction standing wave on the string. 2 fixed end is 2 nodes so the wave length must be $2L/n$ for L to be the length of the string.

As the wave speed is fixed by tension and linear density value and so the frequency must be in harmonic with wave length so any wave created on that string must meet the required wave length and frequency. I want to know if it is really true there is no possible way to create any wave whose frequency and wavelength are different from requirement?

For string fixed on 1 end, is it possible to create an arbitrary wave whose frequency don't meet above requirement. For such 2 waves that's different in wave length and amplitude, can they create a standing wave?

Answer 1

A standing wave appears from the superposition of two running waves. A running wave is reflected by the wall to which it is fixed at one end, and the reflected wave meets the direct wave.

You can create a running wave of the desired wave-length, but not a standing wave of arbitrary wavelength. That for the simple reason that at both walls the amplitude of vibration has to be zero (the string doesn't jump from its fixed points).

For a string fixed at one end we have a well-known example: monochromatic light sent to a perfectly reflecting mirror forms near the mirror interference fringes. So the answer is yes.