When two equal and opposite waves meet at a point, destructive interference occurs. But afterwards, waves go about their own ways. But for waves on a string for example, two opposite waves have momentums of the same magnitude and opposite (in both horizontal and vertical) directions at the instance and point of encounter so shouldn't the waves simply cancel?

Is it because the at the point of contact happens a sort of elastic collision (where incoming wave from +x bounces off to equal and opposite -x direction)? If so, what are the condition that enforces it? That is, what property that arises from the geometry of wave conserved this property despite the physical nature of its medium (air, water, string)? 

If the conservation of momentum of waves at the point of encounter of propagating medium cannot be seen in terms of elastic collision, what would be an intuitive way to view it?