So you have a piece of string. Two people create a half wave at either end, one up and one down of equal magnitude. When they reach each other they will superimpose to create the sum of their amplitudes, X and -X producing 0. Freeze the string at this instant in time and it's completely flat, there is no longer a wave. So where does the energy go? And how can the two initial waves continue? The string isn't moving so what causes the wave to continue?
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The displacement might be zero but the string is still moving.
Have a look at this animation which shows this.
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