Question

if two waves with equal amplitudes and wavelengths travel through a medium in such a way that a particular particle of the medium is at the crest of one wave and at the trough of the other wave at the same time, what will happen to that particle?

Answer 1

If two waves with equal amplitude and wavelength travel through a medium such that particular particle is at the crest of one wave and at the trough of another wave, then destructive interference will take place.Both waves will cancel each other out at that point such that a node will form in which the particle will remain at its rest position only and will not move. 

Answer 2

Answer:

As per the law of superposition we know that when two waves superimpose at a point on the medium then the resultant displacement of the medium particle is vector sum of the displacement due to each wave.

So here by superposition principle we can write

$y_{net} = y_1 + y_2$

now here we know that one of the wave reaches with crest while other wave reaches at trough position.

So we will have

$y_1 = +A$

$y_2 = -A$

so here by superposition we will have

$y_{net} = A - A = 0$

so after superposition the medium particle will come to its mean position.