Obtain the equation of speed of transverse wave on tensed (stretched) string.
Obtain the equation of speed of transverse wave on tensed (stretched) string.
The speed of transverse waves on a string is determined by two factors, $(i)$ the linear mass density or mass per unit length $\mu$ and $(ii)$ the tension $\mathrm{T}$.
The linear mass density, $\mu$ of a string is the mass $m$ of the string divided by its length $l$. Therefore its dimension is $\left[\mathrm{M}^{1} \mathrm{~L}^{-1}\right]$. The tension $\mathrm{T}$ has the dimension of force - namely $\left[\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\right] .$
We have to combine $\mu$ and $\mathrm{T}$ in such a way as to generate $v$ [dimension (LT $^{-1}$ )].
It can be seen that the ratio $\frac{T}{\mu}$ has the dimension $\left[\mathrm{L}^{2} \mathrm{~T}^{-2}\right]$.
$\left[\frac{T}{\mu}\right]=\frac{\left[M^{1} L^{1} T^{-2}\right]}{\left[M^{1} L^{-1}\right]}=\left[L^{2} T^{-2}\right]$
Therefore, if $v$ depends only on $\mathrm{T}$ and $\mu$, the relation between them must be,
$v=C \sqrt{\frac{T}{\mu}}$
Here $\mathrm{C}$ is a dimensionless constant and constant $\mathrm{C}$ is indeed equal to unity.
The speed of transverse waves on a stretched string is therefore given by,
$v=\sqrt{\frac{\mathrm{T}}{\mu}}$
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