Name: Modes of Vibration of a String
Uploaded: 2019-07-19
Duration: 8 min 37 s
Description: You must have heard soothing music coming out of a guitar and a violin. Do you know all these musical instruments work on the principle of standing waves? Wa...

Question

Write the formula for the fundamental frequency of a transverse wave in a stretched string and explain with its help how the frequency of the string depends on the length and radius of the string. Write the frequency of the first overtone produced along with fundamental frequency $n$ of the string.

Accepted Answer

When a wire clamped to rigid supports at its ends is plucked in the middle, transverse progressive waves travel towards each end of the wire. The speed of these waves is

$v = \sqrt{\frac{T}{m}} . . . . (1)$

where $T$ is the tension in the wire and $m$ is the mass per unit length of the wire. These waves are reflected at the ends of the wire.

We know that the distance between two consecutive nodes is $λ / 2$ where $λ$ is wavelength. Hence, $l$ be the length of the wire between clamped ends, then

$l = \frac{λ}{2} λ = 2 l .$

If $n$ be the frequency of vibration of the wire, then

$n = \frac{v}{λ} = \frac{v}{2 l}$ .

Subsituting the value of $v$ from eq. (1), we have

$n = \frac{1}{2 l} \sqrt{\frac{T}{m}}$

This is frequency of the sound note emitted by the wire. The frequency of the sound emitted from a stretched string can be changed in two ways : by changing the length of the string or by changing the tension in the string.

Second mode of vibration : Let

$λ_{2} = \frac{2 l}{2} l = \frac{2 λ_{2}}{2} .$

In the case, the string vibrates in two segments, as shown in fig(b). If $n_{2}$ be the frequency of vibration, then

$n_{2} = \frac{v}{λ_{2}} = \frac{2 v}{2 l} \sqrt{\frac{T}{m}} = 2 n_{1}$

that is, in this case the frequency of the tone emitted from the string is twice the frequency of the fundamental tone. this tone is called the 'first overtone'.

G L Mittal and TARUN MITTAL Solutions for Chapter: Vibrations of Stretched Strings, Exercise 1: QUESTIONS

G L Mittal Physics Solutions for Exercise - G L Mittal and TARUN MITTAL Solutions for Chapter: Vibrations of Stretched Strings, Exercise 1: QUESTIONS

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