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We know that $$ v = \sqrt\frac{T}{\mu} $$ meaning that increase in the tension of a string increases the velocity of the traveling wave. But how exactly does this happen? If we consider that the travelling wave is just a certain amount of energy and momentum ($C$) which is propagating then, I think that increasing tension (along one direction) stretches the string in that direction hence decreasing the density (along the other two directions) therefore for a fixed amount of momentum $C=mv$ to travel less $m$ means more $v$ for a fixed $C$. However, I am not sure of this interpretation.

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Increasing the string tension effectively reduces the remaining elastic capacity.

A "wave" or mechanical signal (such as a force or impulse) propagates through a perfectly rigid material at the speed of sound. If the material is not rigid but elastic, then for each particle along the string, that particle first must move a bit before the elastic force has been established to the next particle. This will take a longer time, and then you see a delayed propagation.

Elastic forces are delayed in their very nature - just try to hang a spring vertically and then let go of the top. The bottom will keep hanging stationary in its spot even while the top of the spring is rushing down towards it. The spring force in a properly "soft" of flexible/elastic spring takes a longer time to propagate than the speed that the top is falling with.

By adding tension to a string you are actually "pre-stretching" it. Try to pre-stretch a spring and then you'll feel that it is much harder to stretch it further - you have used some of its elastic capacity. Each particle along the string is now "less loose" so we have effectively reduced the elasticity and thus reduced the elastic behaviour.

Your own interpretation as a density change along the string is also correct, as far as I can see. I think you can use that as well.

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    $\begingroup$ The material does not have to be rigid for the waves to propagate with the speed of sound. Sound waves propagate in all materials and they do this with the speed of sound, of course. If the medium were rigid the speed of sound were infinite. The notion that you reduce th ellasticity by stretching the string is misleading. What quantity do you mean when you say "ellasticity" in this context? $\endgroup$
    – nasu
    May 11, 2021 at 13:43
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Tension determines the vertical force (perpendicular to wave motion) on molecules of string and hence determines the speed of perpendicular motion. Faster the perpendicular motion, faster the wave has passed by.

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