Why varying the frequency of a propagation wave in a non-dispersive media doesn’t change it’s propagation velocity?
Before anything, I want to note that I’ve seen already similar questions on this site but none have the perspective I give it to it. It may sound as a dumb question since I’ve learnt by heart that the propagation velocity in travelling waves in non-dispersive media don’t get affected by variations of frequency, but only to the media characteristics, such as tension. I can believe it but at the same time it sounds weird mathematically, since for me $$v=\sqrt{\frac{T}{\mu}}$$ and since $T=\frac{1}{f}$, it makes sense to me that $$v=\sqrt{\frac{1}{f\mu}}$$ so then say we have a wave through a chord that propagates at a frequency $f$ and then through that chord, we decelerate so that the frequency is now $f’=2f$, it’s totally legal for me mathematically that then $$v’=\sqrt{\frac{1}{2f\mu}}=\frac{v}{\sqrt{2}}$$ so it does vary, of course I know this doesn’t actually happen, but I’m wondering why it is still legal mathematically to obtain this contradictory result.
Best Answer
Per @Farcher's comment (answer?), $T$ is not the period.
When confused by an equation: do a dimensional analysis. Were $T$ a period, then the dimensions of:
$$ \sqrt{\frac T {\mu}} $$
would be
$$ \Big(\frac{[T]}{[M]/[L]}\Big)^{\frac 1 2}$$
which is an inverse velocity per mass, square-rooted. Not a velocity.
Try it with $T$ having dimensions of tension and see what happens.