The superposition principle for linear waves As far as i've seen, the proof for that principle is to show that, the equation representing linear waves has the perk of being linear, thus if y(x,t) and z(x,t) are solutions of the linear equation,naturally follows y(x,t)+z(x,t) is a solution.
But what i'm struggling to understand is why is it sufficient to imply, if y(x,t) and z(x,t) are coexisting in space and time then they are going to behave according to the waves in interference analysis model.
Just because y(x,t)+z(x,t) is a valid state of existing it still doesn't imply that if we super-positioned both waves that we'll end up with the state y(x,t)+z(x,t).
 A: The existence of anything implies that it has affects on other things. For example, water waves on the surface of a pond will cause a floating duck to bob up and down. If two such waves impinge upon the duck [1] at the same time, then there must be some combined effect that is different from the effect of a single wave. Otherwise, what would it mean for the second wave to exist?
It is an empirical observation that superimposed waves add together. The height of two water waves superimposed is the sum of the individual waves heights. Electromagnetic fields from two sources behave according to their vector sum, causing interference patterns in electromagnetic waves. Because of these observations, the equations describing waves had better have valid solutions corresponding to linear combinations of waves.
It's not so much that the existence of mathematical superposition solutions imply that they exist in reality, but that the superposition behavior of waves in reality requires the equations describing them to have such solutions.
[1] I call dibs on the indie shoegaze band name "Impinge Upon the Duck!"
