Why do peaks and troughs of a wave cancel each other out? And why peaks and peaks or troughs and troughs add up? I have thought that the cancellation of peaks and troughs is a consequence of Newton's third law of motion that equal and opposite forces cancel each other out.
Or it has something to do with conservation of energy or momentum.But I have never truly understood it correctly.
I believe there is an obvious and clear explanation that I am simply not aware of.
In other words, how does interference work?
 A: If you flick a taught rope from both ends, you will see waves forming and moving.

*

*A rising peak happens because the particles were flicked upwards by an upwards force by the person, and this upwards force propagates from particle to particle.


*A lowering valley happens likewise due to a downwards initial force which propagates.
When waves meet, then both forces act on the same particles. If they are equal, then a particle feels an upwards and a downwards force that are equal at the same time. It thus doesn't move due to Newton's 1st law.
This same mechanism happens with sound waves where the moving media is air molecules.
In other types of waves, such as electromagnetic radiation travelling through space with no medium, we can think of the rising radiation wave as a positive value of the electric or magnetic field or the like. Two meeting waves add together, and if one is upwards and one is downwards, then we have a positive and a negative value added together, causing them to mathematically cancel out.
A: Mathematically it can be seen as a consequence of waves obeying a linear second order differential equation.
Physically, for mechanical waves, it’s the fact that forces are additive that result in the addition (superposition) of waves. This means that if two forces are acting at a point, then the net force that acts is the resultant force.
Similarly for EM waves, the fields themselves are additive in nature and the net outcome is the resultant of all the fields at that point.
