Standing waves arbitary frequency

Question

Suppose I have a string that is fixed rigidly at one end. Now when I start making wave, it will get reflected and should lead to standing waves (Mathematically and by visualization), but then we have a fixed condition that the end of the rope at the rigid wall has to be a node, while the free end has to be an antinode. That limits the wavelength to a finite possibilities, thereby defining the frequency. But, isn't frequency dependent on us? Like, I can decide how fast I will move my hand thus I decide time period and frequency, though I can't decide 'k'. So what will happen if I do so at a random frequency? Will a standing wave not be formed? What will happen? My calculations:

EDIT - I just realized that the velocity of the wave is not constant and can be changed upon change in tension. So then, will it be that For any frequency, the wave will be a standing wave, and that the only difference is that now wave velocity and tension in the rope will be different?

Answer 1

You can shake your hand at any arbitrary frequency you like but standing waves only form for particular frequencies
$$f_n = n\frac{v}{2L}=\frac{n}{2L}\sqrt{\frac{T}{\mu}}$$
where $n$ is an integer. At other frequencies travelling waves move both left and right along the string. For standing waves the nodes are stationary, for travelling waves they move along the string.

The reason (as you know) is that the length of the string has to be a multiple of the distance between antinodes. The distance between antinodes depends only on wave speed $v$ and frequency $f$. If these are fixed then the distance between antinodes is fixed.

If you can change $T$ this changes $v$. Then you can get a standing wave at any frequency you choose.