Three one dimensional mechanical waves in an elastic medium is given as and${y_1} = 3A\sin (\omega t - kx)$
${y_{2 = A\sin (\omega t - kx + \pi )}}$
${y_{3 = 2a\sin (\omega t + kx)}}$
Are superimposed with each other. The maximum displacement amplitude of the medium particle would be
A) $4A$
B) $3A$
C) $2A$
D) $A$

Hint: The principle of superposition may be applied to waves whenever two or more waves travelling through the same medium at same time. The waves pass through each other without being disturbed. According to this principle net displacement of medium at any point in space or time, is simply the sum of the individual waves displacements.

Complete step by step answer:
It is clear from the question that the waves 1 and 2 will always interfere destructively since they are completely out of phase with respect to each other. Therefore, the resultant wave will always have an amplitude of $3A - A = 2A$ Now, this wave has to interact with the third wave, constructively to produce a wave that has maximum amplitude this would only be possible if the resultant wave has an amplitude of $2A + 2A = 4A$.

So, the correct answer is “Option A”.

Note:
By the equation we know that we have used the principle of superposition of waves which states that, for all the linear systems, the net response at a given place and time is caused by two or more. Stimuli is the sum of the response which would have been caused by each stimulus individually. The amplitude of the wave is the distance from the centre line or still position to the top of the crest or to the bottom of through. amplitude is measured in meters.