Laws of transverse vibrations of string: Please refer - State the Laws of transverse vibration in a stretched string.

A string is a flexible wire of a uniform area of cross-section. It is observed that the velocity of transverse waves in a stretched string depends on

1. The Tension (T)
2. The Mass (m), and 
3. The Length (l)

Here, \(v\propto T^x,v\propto m^y,v\propto l^z\)

∴ \(v=kT^xm^yl^z\) ---------- (i) ,

where k, x, y, and z are numbers.

Since, the dimension of velocity, \(v=\left[LT^{-1}\right]\),

the dimension of tension, \(T=\left[MLT^{-2}\right]\)

the dimension of mass, \(m=\left[M\right]\)

On substituting we get,

\(\left[LT^{-1}\right]=\left[MLT^{-2x}M^yL^z\right]\)
Or, \(\left[LT^{-1}\right]=\left[M^{x+y}L^{x+z}T^{-2x}\right]\)

Equating the indices on the both sides, we get

For M, x + y = 0
For L, x + z = 1
For T, -2x = -1

∴ \(x=\frac12\),    \(y=-\frac12\)    and \(z=\frac12\)

Thus, equation (i) becomes,

\(v=kT^{1/2}m^{-1/2}L^{1/2}\)
∴ \(v=k\sqrt{\frac{Tl}m}=k\sqrt{\frac T{m/l}}\)

Since, \(\frac ml=\mu\), which is the mass per unit length of the string, then

\(v=k\sqrt{\frac T\mu}\)

Experimentally it is observed that, k = 1

∴ \(v=\sqrt{\frac T\mu}\)

Thus, the speed of propagation of transverse waves depends only on the tension and mass per unit length.