Question

Use dimensional analysis to express the wave velocity in terms of the tension in the string and mass per unit length of the string.

Answer 1

Let the wave velocity is dependent on the tension and mass per unit length. Then

$$v=F^{\alpha}\mu^{\beta}$$

Since dimensions of velocity, force and mass per unit length are as follows,

$$[v]=LT^{-1}$$

$$[F]=MLT^{-2}$$

$$[\mu]=ML^{-1}$$

we obtain

$$LT^{-1}=\left(MLT^{-2}\right)^{\alpha}(ML^{-1})^{\beta}$$

$$LT^{-1}=M^{\alpha+\beta}L^{\alpha-\beta}T^{-2\alpha}$$

So

$$\left\{\begin{matrix} \alpha+\beta &=& 0 \\ \alpha-\beta &=& 1\\ -2\alpha &=& -1 \end{matrix}\right.$$

Solution

$$\left\{\begin{matrix} \alpha &=& 1/2\\ \beta &=& -1/2 \end{matrix}\right.$$

Finally

$$v=F^{1/2}\mu^{-1/2}=\sqrt{\frac{F}{\mu}}$$