On the basis of dimensional consideration …

Speed of a transverse wave on a stretched string. The wave velocity through a medium depends on its inertial and elastic properties. So the transverse wave through a stretched string is determined by two factors:

1. Tension T in the string is a measure in the string. Without tension no can propagate in the string. of elasticity disturbance Dimensions of T = [Force] = [MLT ²]
2. Mass per unit length or linear mass density m of the string so that the string can store kinetic energy.
   Dimensions of  $m=\frac{[\text { Mass }]}{[\text { Length }]}=\left[\mathrm{ML}^{-1}\right]$
   Now, dimensions of ratio $\frac{T}{m}=\frac{\left[\mathrm{MLT}^{-2}\right]}{\left[\mathrm{ML}^{-1}\right]}=\left[\mathrm{L}^{2} \mathrm{~T}^{-2}\right]$
   As the speed v has the dimensions [LT^-1] so we can express v in terms of T and m as $v=C \sqrt{\frac{T}{m}}$
   From detailed mathematical analysis! or from experiments, the dimensionless constant C = 1. Hence the speed of transverse waves on a stretched string is given by
   $v=\sqrt{\frac{T}{m}}$