We have studied earlier that due to disturbance in stretched string transverse waves are generated. Hence, to calculate the velocity of transverse waves, let us assume a string, whose unit length mass is m, and tension is T. A disturbance is in motion from left towards right with velocity v.We can also assume that the observer is moving in the direction of disturbance with the same velocity v, then the vibration will appear stable to the observerand the string will appear moving in the opposite direction. Now analyse the small part δl; due to disturbance if there is small displacement in the string then this small part δl; can be considered as a part of the circle with radius R and the mass of this small part will be m δ l.
On this small part as shown in the diagram, the component of tension towards the center is 2Tsinθ which will provide the necessary centripetal force.
Therefore;
2 T sinθ = \(\frac{Mv^2}{R}\) where M = mass = m δ l
When, θ is very small then sinθ ≅ θ
Therefore; 2 T θ = \(\frac{m\delta l b^2}{R}\) ...........(1)
Again from the figure, formula for arc PQ.
Angle = Arc/Radius
2 θ = \(\frac{\delta l}{R}\) ...........(2)
\(\frac{\delta l}{R}\times T = \frac{m\delta l}{R}\times v^2\)
or \(v^2 = \frac{T}{m}\)
This is the equation for velocity of a transverse wave which shows that the velocity of the wave in a string is dependent on the tension and the mass of unit length and is not dependent on amplitude of the wave and its wavelength. In the above situation string is considered to be ideal (that is totally elastic and same density) and while doing vibration motion there is no change in the length of the string.
