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.
Forces – How to Increase Tension in a String to Boost the Speed of Travelling Waves
densityforcesmomentumstringwaves
Related Solutions
Your intuition is right: the density of the string goes down a little bit when you increase the tension.
HOWEVER: the wave in a string is a transverse wave which depends on the tension and the mass per unit length. If you double the tension the mass per unit length goes down by a small amount (the string gets a bit "thinner" because it gets longer) . Both these things increase the velocity of the transverse wave which is given by
$$v = \sqrt{\frac{T}{\rho}}$$
Where T is the tension and $\rho$ the mass per unit length. Finally, the fundamental frequency is determined as the reciprocal of the round trip time of the wave along the string:
$$T = \frac{2 \ell}{v}$$
so that
$$f = \frac{1}{T} = \frac{v}{2\ell} = \frac{\sqrt{\frac{T}{\rho}}}{2\ell}$$
So to raise the frequency by an octave you need four times the tension (don't try this - you might break the neck) or a string that's half the diameter (one quarter of the area - so one quarter of the mass per unit length). This explains why different strings on the guitar have different gage - you would need too much tension to get the high range from a long thick string.
The short answer is that the elasticity does affect the wave speed. However, when people typically talk about the wave speed on a taut string they are referring to very small disturbances. In the limit that the disturbance is infinitesimal, these phenomena you are referring to become negligible, and it is in this limit that the wave speed is defined.
I found a dissertation on nonlinear waves on a string with inhomogeneous properties that provides plenty of mathematical and physical detail on how to account for the elasticity of the string. From this dissertation we find that the first set of equations that account for elasticity (you need two because there is both vertical and horizontal displacement of the string) may be written as \begin{gather} u_{tt} - c_\lambda^2u_{XX} = 0, \\ v_{tt} - c_\tau^2v_{XX} = (c_\lambda^2-c_\tau^2)\left[ v_Xu_{XX} + v_{XX}u_X \right], \end{gather} where $u$ is the horizontal displacement of the string, $v$ is the vertical displacement, $X$ is the horizontal position of the string at rest, $t$ is time, subscripts denote partial differentiation with respect to the subscripted quantity, $c_\lambda^2=EA/\rho$ is the square of the longitudinal wave speed, $c_\tau^2=T_0/\rho$ is the square of the transverse wave speed, $E$ is the elastic modulus, $\rho$ is the mass density per unit length, and $T_0$ is the initial tension. The terms on the right-hand sides are all proportional to the product of two displacements. If we assume the displacements are small, then these nonlinear terms become very small. If we were to neglect them, returning to the linear equations, the two equations would decouple, and the vertical displacement equation would become the standard wave equation on a string people refer to.
Best Answer
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.