Question

A long string is pulled so that the tension in it increases by a factor of four. If the change in length is negligible, by what factor does the wave speed change?

Answer 1

To solve this problem we will apply the concepts related to wave velocity as a function of the tension and linear mass density. This is

$v = \sqrt{\frac{T}{\mu}}$

Here

v = Wave speed

T = Tension

$\mu$ = Linear mass density

From this proportion we can realize that the speed of the wave is directly proportional to the square of the tension

$v \propto \sqrt{T}$

Therefore, if there is an increase in tension of 4, the velocity will increase the square root of that proportion

$v \propto \sqrt{4} = 2$  

The factor that the wave speed change is 2.