Conservation of linear momentum in destructive interference 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?
 A: I am using two gaussian bell curves as pulses to make the pictures

The green pulse travels towards right and red one towards left ,note that the points on the front part of wave have a tendency to move up and the back parts have tendency to go down(when the pulse points in positive y axis,you can work out the other conventions)

When the pulses collide in my diagram ,the portion of string towards left of $x=0$ have a tendency to go up while the portion towards right of $x=0$ wants to go down ,this creates a pull in the actual string ,the left of $x=0$ is pulled down while right is pulled up,note that the back portions of the pulses are still trying to maintain their original state.

The amplitude of colliding pulses is becoming smaller due to the pull.

After a certain time pulses flip because the back portion was pulling in the origianl direction (downwards for green and upwards for red)and the pull created in the collision created a pull opposite to the direction of original pull (When the pulses were travelling)

The back portions still maintain directions but left of $x=0$ is now being pulled up while right is being pulled down ,this creates the scenario for travelling pulses again i.e front and back portion of waves have opposite tendencies of motion ,the pulses grow bigger and leave unaffected.

A: I think what you have done very well is highlight a term which is confusing.  Destructive is probably a poor choice of words because it implies that the two travelling waves are interacting with each other in a way that would cause say a loss of energy when in fact that is not really the case, they are passing each other and temporarily creating zero amplitude in the medium.  Destructive interference is counter intuitive to our sense of the world, just like extra dimensions are.  Destructive interference can create points of silence where we would otherwise expect a cumulative loud sound, when considering for example interfering sound waves
