Question

The displacement \[y\;\] of a particle executing periodic motion is given by \[y = 4co{s^2}\left( t \right)sin\left( {1000t} \right)\]. This expression may be considered to be a result of the superposition of waves
A.\[5\]
B.\[4\]
C.\[3\]
D.\[2\]

Answer 1

Hint: We can define waves by their frequency, amplitude, and phase. If two or three waves collide with each other they mix and produce a new wave. This interference or mixing of waves is governed by the superposition principle. This superposition principle states that the wave function describing the resulting wave in the interference of two or more waves is equal to the sum of the two or more wave functions of the individual waves.

Complete answer:
Given that the displacement of a particle as \[y = 4co{s^2}\left( t \right)sin\left( {1000t} \right)\]
Solving the above equation,
\[y = 4co{s^2}\left( t \right)sin\left( {1000t} \right)\]
\[y = 2(2{\cos ^2}t)\sin 1000t\] ……. (1)
By the identity,
\[\cos 2\theta = 2{\cos ^2}\theta - 1\]
\[ \Rightarrow 1 + \cos 2\theta = 2{\cos ^2}\theta \]
We can write equation (1) as,
\[y = 2(1 + cos2t)sin1000t\]
\[ \Rightarrow y = 2\sin 1000t + 2\cos 2t\sin 1000t\]…….. (2)
Now by using the identity,
\[2\sin A\cos B = \sin (A + B) + \sin (A - B)\]
We can write the equation (2) as,
\[ \Rightarrow y = \sin 1000t + \sin 998t + \sin 1002t\]
Therefore the three independent harmonic motions are superposed for given periodic motion.

Therefore the correct option is C.

Note:
When two or more waves arrive at the same place they tend to superimpose themselves on one another. We can say this more specifically when the disturbance of the waves is superimposed and the phenomenon is called a superposition. Each one of the disturbances corresponds to one force. And these forces will be added together. If we consider the disturbances of the waves to be in the same line then the resulting wave will be the simple addition of the disturbances of the individual waves. In the other words, their amplitude will be added together. If two identical waves arrive at the same point exactly in the same phase then the superposition of these waves produces pure constructive interference. And if the two identical waves arrive exactly in a position where the two waves are out of phase to each other then this superposition produces pure destructive interference.