Velocity of a Mechanical Wave on a String I recently read a derivation for an equation which governs how quickly a wave is transmitted along a string, $v = \sqrt{\frac{T}{\mu}} $, where T is the tension in the string, and $\mu$ is the mass per unit length along the string. The derivation makes sense but gives a more mathematical and geometrical account as to why this is the case.
Could someone please explain more qualitatively why an increase in density of the string would reduce the velocity of transmission, and why an increase in tension would increase the velocity? 
I can see, in very vague terms why an increase in tension would cause neighbouring elements of the string to more quickly follow the motion of preceding elements when for instance, a pulse is sent down the wave as follows:
 
though I cannot visualise the effects of density. I can imagine each element of the string having more mass and their movements becoming more 'sluggish,' but would this not affect the frequency of their oscillations? 
 A: More mass means more inertia.  Thus, it takes more force to move a differential mass on the string, and that differential mass responds more slowly than a "lighter weight" string because the acceleration of that differential mass must follow Newton's 2nd law (a=F/m).  This means that the wave speed on a string of large linear density will be lower than the wave speed on a string of small linear density, assuming the same force applied to each differential mass on the strings as the wave passes by.
A: If you assume that tension and density are the only factors then dimensional analysis will lead you straight to that formula except for an undetermined constant. If you want an answer without such an assumption, then maybe you have to accept that the math is needed after all.
A: A mechanical wave requires a medium, the string in this case, which is composed of particles.
These particles oscillate about a mean position whilst interacting with one another.
You may have noticed that the formula for speed of a number of mechanical wave has the form 
$\text{speed} =\sqrt {\left( \dfrac{\text{a term to do with a restoring force or "springiness" of the medium}}{\text{a term to do with the mass of the medium}}\right)}$  
For example $\sqrt {\left( \dfrac{\gamma \text{ pressure of gas}}{\text{density of gas}}\right)}$ for sound waves in a gas and
$\sqrt {\left( \dfrac{\text{Young's modulus of solid}}{\text{density of solid}}\right)}$ for longitudinal waves in a solid rod.
For the transverse waves on  a string how much restoring force a particle will be subjected to is related to the tension in the string.
A greater tension results in a greater restoring force and hence a greater acceleration of the particle.
A greater tension thus results in a particle returning to its equilibrium position faster which in turns means that the speed of the wave (transfer of information between particles) is faster.  
However a given restoring force a string with greater mass (per unit length)  will undergo a smaller acceleration and hence return to the equilibrium position in a longer period of time.
This results in the speed of the wave being lower.
