Why do waves add in contructive interferrence and continue to propagate through a medium? I don't fundamentally understand why waves add to make a bigger wave in constructive interference. The explanation I have seen everywhere is just that they add to form something bigger. But what happens in the molecules, for example, a rope, for those two waves to add? The classical intuitive explanation is not enough for me to understand this. I'm looking for an explanation as to WHY this happens. Do the molecules moving up at some velocity cause neighbouring molecules to move up as well on both sides? When they meet is the net momentum when they overlap just both sides imparting their shared momentum on the same molecule, causing the amplitude of the wave to increase? Im a little confused.
Also, just a side thought, why don't the molecules moving up and the molecules travelling down cancel each other out and cause no movement? Why does the disturbance continue and would it ever stop given an infinite amount of time?
 A: The fact that individual wave solutions can be "added" to obtain another wave solution is called the superposition principle. It is only valid for strictly linear wave equations. As soon as there is only a minute nonlinear term in the equations, the superposition principle breaks down. As you can imagine, nature is almost always nonlinear.
Take your example of the rope: the forces between adjacent elements of the rope (better not call them molecules, because that would bring with it a lot of additional complications) are linear for small deformations, which is called Hooke's law. But if the deformations become larger, Hooke's law must be violated, for the simple reason that any rope will break if force becomes big enough. There is just no such thing as infinite strength. And, as you might know from pulling a rubber band, the material resists more (Hooke's stiffness increases instead of being constant, as for linearity) if it approaches it's ultimate strength (just before tearing).
So think about waves adding up to get "bigger" waves, that do not influence each other, as a very special property of very special equations. More generally, waves are a more or less complicated result of adjacent continuum elements acting on each other, without them being easily attributable to a sum of something.
I don't know of any intuitive explanation for superposition. It all boils down to the equations, of which the 1+1 dimensional scalar linear wave equation is the easiest to understand (details of propagation velocity is suppressed by choosing appropriate units for x/t)
$$\frac{\partial^2 \psi}{\partial t^2}=\frac{\partial^2 \psi}{\partial x^2}$$
This tells you that the acceleration of a continuum element (the left hand side) is just a linear function of the curvature in the neighbourhood of that continuum element (the right hand side). For the example of the rope, the curvature $\frac{\partial^2 \psi}{\partial x^2}$ is the inverse or something (sorry I don't remember the exact dependency at the moment) of the local bending radius of the rope. If the rope is locally straight somewhere (infinite bending radius) in some instant, the curvature is zero, and the rope will not experience any acceleration at that point and in that moment.
If you have found two solutions $\psi_1$ and $\psi_2$, then due to linearity, any combination $\psi=\beta_1\psi_1+\beta_2\psi_2$ is also a solution:
$$\frac{\partial^2 \psi}{\partial t^2}=\beta_1\frac{\partial^2 \psi_1}{\partial t^2}+\beta_2\frac{\partial^2 \psi_2}{\partial t^2}=\beta_1\frac{\partial^2 \psi_1}{\partial x^2}+\beta_2\frac{\partial^2 \psi_2}{\partial x^2}=\frac{\partial^2 \psi}{\partial x^2}$$
That's it. No more to understand about this.
