Question

Three coherent waves having amplitudes$12mm$, $6mm$ and $4mm$ arrive at a given point with successive phase differences of $\dfrac{\pi }{2}$. Then the amplitude of the resultant wave is
(A) $7mm$
(B) $10mm$
(C) $5mm$
(D) $4.8mm$

Answer 1

We are given with the amplitudes of the three waves which are classified as coherent and that they arrive at a given point with a successive phase difference. Thus, we will visualize the given situation and find the positions of the wave when they arrive at the given point. Then we will find the resultant of the three in a step by step manner.

Complete step by step solution:
Here,
It is said that the waves are in a phase difference of $\dfrac{\pi }{2}$ successively. Thus, the second wave is in a phase difference of $\dfrac{\pi }{2}$ with the first wave and the third wave is in a phase difference of $\dfrac{\pi }{2}$ with the second wave.
Now,
Clearly,
The third wave will be in a phase difference of $\pi$ with the first wave, which means the third wave will have a direction opposite to that of the first wave.
Thus, the resultant amplitude of the first and the third wave will be the difference between their individual amplitudes.
Now,
${A_1} = 12mm$
${A_3} = 4mm$
Thus,
Resultant amplitude, ${A_{R1}} = (12 - 4)mm = 8mm$
Now,
The direction of the resultant amplitude will be in the direction of the fundamental wave which had higher amplitude.
Thus, the direction will be in the direction of the first wave.
Again,
Clearly,
The second wave will be at a phase difference of $\dfrac{\pi }{2}$ with the resultant of the first and the third wave.
Now,
${A_3} = 6mm$
Thus,
The resultant will be:
${A_R} = \sqrt {{A_{R1}}^2 + {A_3}^2}$
Putting in the values in the equation,
${A_R} = \sqrt {{8^2} + {6^2}}$
After further calculation, we get
${A_R} = 10mm$

Hence, the correct option is (B).

Note: In this case it was told that the waves were at phase difference of $\dfrac{\pi }{2}$ successively and thus we had that the second wave was flowing $\dfrac{\pi }{2}$ ahead of the first one and the third wave was flowing $\dfrac{\pi }{2}$ ahead of the second one. But if it was told that the waves were at a phase difference of $\dfrac{\pi }{2}$ with each other, then the case would have been that the second wave would have been flowing with a phase difference of $\dfrac{\pi }{2}$ with the first wave as well as the third wave and so on. Thus, the total calculation process would have been different.