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Speed of Transverse Wave | Speed of Transverse Wave on a Straight Wire

Speed of Transverse Wave | Speed of Transverse Wave on a Straight Wire :- In this article, we focus on determining the speed of a transverse pulse traveling on a taut string. Consider a pulse moving on a taut string to the right with a uniform speed v measured relative to a stationary frame of reference.

To obtain the expression for speed v it is convenient to describe the motion of pulse with respect to a reference frame that moves along with the pulse with the same speed as the pulse, so that the pulse appears to be at rest within the frame(as Newton’s laws are valid in either a stationary frame or a frame moving with constant velocity, so it is not violating the laws of motion).

Now from the moving frame of reference although the pulse appears to be at rest but the small segment of length Δl appears to be moving to the left with speed v. This small element of the string of length Δl forms an approximate arc of a circle of radius R and has a centripetal acceleration(a_c) equal to v²/R, which is supplied by radial components of the tension force T in the string. The horizontal components of T cancel, and each vertical component T sinθ acts radially toward the center of the arc.

The total radial force on the element 2T sinθ acts as centripetal force, i.e,

$\displaystyle {{F}_{c}}=2T\sin \theta \approx 2T\theta$ …..(1)

Here we have assumed the approximation sinθ ≅ θ for small θ.

If μ is the mass per unit length of the string, the mass of the segment of length Δl is

m = μΔl = μ(2θR) = 2μRθ …..(2)

From Newton’s second law,

$\displaystyle {{F}_{c}}=m{{a}_{c}}$

$\displaystyle 2T\theta =2\mu R\theta \times \left( \frac{{{v}^{2}}}{R} \right)$

$\displaystyle \Rightarrow v=\sqrt{\frac{T}{\mu }}$ …..(3)

If D is the diameter of the string, L is its length and ρ is its density, then

$\displaystyle \mu =\frac{\pi {{\left( \frac{D}{2} \right)}^{2}}L\rho }{L}=\frac{\pi {{D}^{2}}\rho }{4}$

$\displaystyle \Rightarrow v=\sqrt{\frac{T}{\frac{\pi {{D}^{2}}\rho }{4}}}$

$\displaystyle \Rightarrow v=\frac{2}{D}\sqrt{\frac{T}{\pi \rho }}$ …..(4)

Also

μ = mass per unit length of the string

$\displaystyle \mu =\frac{m}{L}=\frac{mA}{LA}=\left( \frac{m}{V} \right)A=\rho A$

Here A is the cross sectional area of the string and V is the volume of the string. Hence the above expression can also be written as

$\displaystyle v=\sqrt{\frac{T}{\rho A}}$ …..(5)

Note:-

We have derived the equations (3), (4) and (5) assuming that the pulse height is small relative to the length of the string. Using this assumption, we were able to use the approximation sinθ ≅ θ.
This model assumes that the tension T is not affected by the presence of the pulse, hence T is the same at all points on the string.
This derivation does not assume any particular shape for the pulse. Therefore, we conclude that a pulse of any shape travels along the string with speed given by equations (3), (4) and (5).

Speed of Transverse Wave | Speed of Transverse Wave on a Straight Wire

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