Name: Waves and Their Phases
Uploaded: 2019-07-19
Duration: 6 min 51 s
Description: In this video, we will learn how two or more waves can be studied in terms of their phases. Equations of two waves and graphs of two waves. Learn more!

Question

Draw the diagram of the first three modes of transverse vibration of stretched string clamped at both ends. Show the position of nodes and anti-nodes in the diagrams.

Accepted Answer

When a string of mass per unit length (m) under the action of tension; T; is set in to transverse vibration the velocity of the wave propagation through the string is given by
$m / l = T_{m}$
In such a vibration of the string the disturbance produced at one fixed end. Travels along the string and gets reflected back at the other end, since the original wave and reflect wave have the same frequency and amplitude, they superimpose to produce stationary transverse wave.

Different modes of Vibration in stretched string:-

a) First mode of vibration:-

In this mode of vibration, the string vibration in one segment. This there are two nods at fixed ends and an anti-node in between them. If $λ$ be the length of string and $λ_{\circ}$ be the wave length of wave in this mode of vibration.

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

If $V$ be the velocity of wave and to be the frequency of wave in this mode of vibration

$f_{\circ} = \frac{V}{λ_{\circ}} = \frac{V}{2 l}$ and hence, $f_{\circ} = \frac{1 \sqrt{\frac{T}{M}}}{2 l}$

This is the fundamental frequency or frequency of 1^st harmonics.

Second mode of vibration:

In this mode of vibration the string vibration in two segments. Thus, there are three nodes and two anti nodes between of string and $λ_{1}$ be the wavelength of wave in this mode of vibration.
$l = λ_{1}$
If u be the velocity of wave and $f_{1}$ be the frequency of wave in this mode of vibration.

$f_{1} = \frac{V}{λ_{1}} = \frac{2 V}{2 l} f_{1} = \frac{1 \sqrt{\frac{T}{M}}}{l} f_{1} = 2 f_{\circ}$

This is the frequency of 1^st overtone and 2^nd harmonics.

Third mode of vibration:

In this case there are 4 nodes & 3 anti-nodes.

$l = \frac{3 λ_{2}}{2} λ_{2} = \frac{2 l}{3}$

Hence,

$f_{2} = \frac{2 \sqrt{\frac{T}{M}}}{3 l} f_{2} = 3 f_{\circ}$

This is the frequency of 2^nd overtone or 3^rd harmonics
In this way, for the 4^th, 5^th …… modes of vibration in the stretched string the frequency
For the fifth, harmonics are emitted.
The general equation of frequency of wave in stretched string is given by,

$f_{n} = \frac{(n + 1) \sqrt{\frac{T}{M}}}{2 l}$

Draw the diagram of the first three modes of transverse vibration of stretched string clamped at both ends. Show the position of nodes and anti-nodes in the diagrams.

Important Questions on Waves

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