Velocity of Transverse Wave in a Stretched String

We can generate a transverse wave on a stretched string. The speed of transverse wave set up in the string depends upon several factors like tension applied (T), mass per unit length (μ) of the string etc.

We can determine the velocity (v) of the transverse wave on the stretched string by means of dimensional method as follows:

As we knew that wave in string depends upon the tension applied (T) and mass per unit length (μ) of the string. We can write:

V ∝ Txμy

or, V = K. Txμy……(i)

Where K is the proportionality constant and x and y are the coefficient to be determined.

Now,

Dimension of V = [MoL1T-1]

Dimension of T = [M1L1T-2]

Dimension of μ = [M1L-1To]

Now, putting these dimensional value in equation (i), we get:

[MoL1T-1] = [M1L1T-2]x . [M1L-1To]y

or, [MoL1T-1] = [MxLxT-2x] . [MyL-y]

Equation the dimension of both sides,

x + y = 0

x – y = 1

-2x = -1

On solving the above equation, we get the value of x = 1/2 and y = -1/2.

Therefore, v = KT1/2μ-1/2 = K $\sqrt{ \frac{T}{ \mu }}$

Experimentally, the value of K is found to be 1. So, the equation becomes,

$V = \sqrt{ \frac{T}{ \mu }}$

Here, $μ = M/l = \frac{M}{V}.A = ρA =  \frac{ \pi d^2}{4}  \rho$

Where, D is the diameter and  ρ is the density of the string.

The velocity of the stretched string can also be determined by analytical method.

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