Question

A wave is travelling along a string. At an instant, the shape of the string is as shown in figure. At this instant, point A is moving upwards. Which of the following statements is/are correct?

(A) The wave is travelling to the right.
(B) Displacement amplitude of the wave is equal to displacement of B at this instant.
(C) At this instant velocity of C is also directed upwards.
(D) Phase difference between A and C may be equal to $\dfrac{\pi }{2}$

Answer 1

Hint: To find the correct options out of the given, you need to go through all of the options and check them one by one by analyzing the given situation to reach at the conclusions regarding all the four given statements. Remember that the question may be multiple-correct.

Complete step by step solution:
We will approach the solution to the question exactly as explained in the hint section of the solution. Let us first have a look at the statement given in the option (A):
The statement claims that the wave is travelling to the right. Let’s have a look at what the situation really is. We can see that the particle at point A is travelling upwards, we can clearly see that the particles on the right side of A are at a higher position than A while the particles on the left side are below the particle at A. We know that the following particles follow the motion of their preceding particles in a wave, so if the particle at A is not following the particles on the left side, it must mean that the wave is travelling to the left and not to the right. Hence, option (A) is wrong.
Now, let us have a look at the option (B):
This statement claims that displacement amplitude of the wave is equal to displacement of B at this instant. We can see that B is actually at the highest point and we know that the amplitude of all the particles is exactly the same in a wave So, the amplitude of all the particles of the string is exactly the same as the amplitude of the particle at B, and thus, is equal to the displacement of B at the given instant in the diagram.
So, we can safely say that the option (B) is a correct option.
If we have a look at the option (C), we can see that it claims that at the given instant, the velocity of C is directed upwards. Since the particles are following the particles on their right side, its velocity at this instant should be directed downwards, and thus, we can confidently say that the given statement in the option (C) is false, and thus, is not correct.
Now, if we check the option (D), we can see that the position difference between both the particles A and C is roughly around one-fourth of the total wavelength of the wave. This can be given as:
$\Delta x = \dfrac{\lambda }{4}$
From this, we can safely compute that the phase difference between A and C will be:
$\Delta \phi = \dfrac{{2\pi }}{\lambda }\Delta x$
Substituting in the value of the path difference, we get:
$\Delta \phi = \dfrac{{2\pi }}{\lambda }\left( {\dfrac{\lambda }{4}} \right)$
Upon solving, we get:
$\Delta \phi = \dfrac{\pi }{2}$
Hence, we can see that the option (D) is also correct.

So, the final correct options to the question are: option (B) and (D).

Note: Many students do not remember the fact that there may be multiple correct options in the question and leave the solution of the question only to the point when they find the first correct option and thus, lose important and possibly easy marks. You are never supposed to do so.