Question

If the phase difference between two component waves of different amplitudes is $ 2\pi $ , their resultant amplitude will become
(A) sum of the amplitudes
(B) difference of the amplitudes
(C) product of the amplitudes
(D) ratio of their amplitudes

Answer 1

Hint:
From the formula for the resultant amplitude we need to substitute the phase difference between the two component waves as $ 2\pi $ . Then by calculating and removing the square root, we can find the resultant amplitude.
Formula Used: In this solution we will be using the following formula,
 $\Rightarrow I = \sqrt {I_1^2 + I_2^2 + 2{I_1}{I_2}\cos \theta } $
where $ I $ is the resultant wave amplitude and $ {I_1} $ , $ {I_2} $ are the component wave amplitudes, $ \theta $ is the phase difference between the two waves.

Complete step by step answer:
In the question we are given two waves having a phase difference between them as $ 2\pi $ . So let us consider the amplitudes of the component waves as, $ {I_1} $ and $ {I_2} $ . Now the resultant amplitude of two component waves is given by the formula,
 $\Rightarrow {I^2} = I_1^2 + I_2^2 + 2{I_1}{I_2}\cos \theta $
Now here the angle $ \theta $ is the phase difference between the two waves and we can substitute it with $ 2\pi $ . So we have,
 $\Rightarrow {I^2} = I_1^2 + I_2^2 + 2{I_1}{I_2}\cos 2\pi $
Now the value of $ \cos 2\pi $ is 1. So substituting this we get,
 $\Rightarrow {I^2} = I_1^2 + I_2^2 + 2{I_1}{I_2} $
Now here we can see that the RHS of the equation is the formula for the whole square of 2 variables.
Therefore we can write,
 $\Rightarrow {I^2} = {\left( {{I_1} + {I_2}} \right)^2} $
Now taking square root on both the sides we get,
 $\Rightarrow I = \sqrt {{{\left( {{I_1} + {I_2}} \right)}^2}} $
Hence we get,
 $\Rightarrow I = {I_1} + {I_2} $
Therefore, we can see that the resultant amplitude is the sum of the amplitudes of the two component waves.
Hence, the correct option will be A.

Note:
The phase difference or the phase shift between two waves can be defined as the difference between the two waves in radian or degrees when they reach their maximum or their zero values. In other words we can describe it as the lateral shift between two or more waveforms along a common axis.