Question

A composition string is made up by joining two strings of different masses per unit length ⟶μ and 4μ. The composite string is under the same tension. A transverse wave pulse: \[Y = (6mm){\rm{ sin(5t + 40x)}}\], where 't' is in seconds and 'x' in metres, is sent along the lighter string towards the joint. The joint is at x=0. The equation of the wave pulse reflected from the joint is
A. \[(2\,mm){\rm{ sin(5t - 40x)}}\]
B. \[(4\,mm){\rm{ sin(40x - 5t)}}\]
C. \[ - (2\,mm){\rm{ sin(5t - 40x)}}\]
D. \[(2\,mm){\rm{ sin(5t - 10x)}}\]

Answer 1

Hint:In the question two strings are joining at a some point. At this point there is a change in medium. The wave must reflect or transmit whenever a change in medium takes place. To calculate the equation of the wave pulse reflected from the joint, first we calculate the velocity of the wave in lighter string and heavier string then amplitude can be easily calculated.

Formula used:
The amplitude of transmitted wave is given as,
\[{A_t} = \dfrac{{{v_1} - {v_2}}}{{{v_1} + {v_2}}}A\]
Where \[{v_1}\] is the velocity of the wave in the first medium, \[{v_2}\] is the velocity of the wave in the second medium and A is the initial amplitude.

Complete step by step solution:
Given a transverse wave pulse is
\[Y = (6\,mm){\rm{ sin(5t + 40x)}}\]
This equation shows the pulse is travelling in a negative direction. The pulse is travelling from lighter string towards the heavier string at a joint x=0. At this point the medium of the wave is changing.

The velocity of the wave in lighter strings and heavier strings.
As we know that \[v = \sqrt {\dfrac{T}{\mu }} \]
As it is given that tension(T) in both cases is same
Velocity for lighter string,
\[v = \sqrt {\dfrac{T}{\mu }} \]
Velocity for heavier string,
\[v = \sqrt {\dfrac{T}{{4\mu }}} \]
So here we get the velocity of lighter and heavier strings is v and 2v. The wave is reflected from low to denser medium and travels back from the origin so 40x will become -40x.

Now as we know the amplitude of transmitted wave is
\[{A_t} = \dfrac{{{v_1} - {v_2}}}{{{v_1} + {v_2}}}A\]
Substituting the values,
\[{A_t} = - \dfrac{v}{{3v}} \times 6\]
\[\therefore {A_t} = - 2\,mm\]
As the transmitted wave is travelling in the same phase as the incident wave. So only the amplitude is changing here. Therefore, the equation of the wave pulse reflected from the joint is \[ - (2\,mm){\rm{ sin(5t - 40x)}}\].

Hence option C is the correct answer.

Note: As the wave is partially transmitted and partially reflected. The transmission occurs at a joint point of the string. So the second medium is denser for reflection to take place and velocity will be decreased.