Why do multiple forces or wave act together in such a way that one wave/force does not affect the action of other? In case of force I am referring to principle of physical independence of forces. What is the theoretical proof of it? Is it something just empirical?
In case of waves, we get to observe that after superposition two waves propagate in such a way that nothing happened between them. Why does it happen and what is its proof(theoretical)?
I am just curious about these. I basically asked two questions in one because I think these two questions are almost the same and perhaps have pretty answers. I have looked for answers to these questions on internet but couldn't find anything related by far.
Any link regarding answers to this will also be very helpful
Thank you.
 A: "Why" questions often don't have an answer in physics. Physics describes how the universe behaves. It does not say why it does so.
Sometimes you can answer. A complex law might be derived from a simple law. Why is the complex law true? Because the simple law is. But that just makes a new question. Why is the simple law true? Eventually you come to a law that is not derived from anything.
For these laws, and all laws of physics, the ultimate thing that decides if they are true is if they agree with experiment. So yes, they are empirical.
Keep in mind there are two things. There is the way the universe behaves. And there is physics, a mathematical description of the way the universe behaves. The math is chosen to match the behavior. Physicists get used to deriving one mathematical law from another, and then doing experiments to verify the math matches the universe. They quickly start to think of the math and behavior as the same thing.
For your question about multiple forces, the relevant laws are Newton's laws. The second law describes the relationship between force and acceleration. Both of these are vectors.
People usually think of vectors as little arrows. But really they are things with a size and direction that behave reasonably when added together. Little arrows are the most common example of this.
So a physicist watches a tug of war where the two teams pull equally hard. He sees that neither team moves. He says the two teams exert equal forces in opposite directions. Forces are vectors. Equal and opposite vectors add to $0$. $F = ma$, so $a = 0$. Objects at rest stay at rest unless a (non-$0$) force acts on them.
Physicists think about the universe in terms of a mathematically consistent set of laws. The universe behaves this way because these laws say so.
But the part you are asking about is why do both teams stay still when each pulls equally hard. They just do. That is just the way the universe works.
You can see that adding the math makes it a lot easier to talk about the behavior and verify it. You use numbers to say how hard each team is pulling and how they are accelerating. Without numbers, it is hard to be precise about things. And that is why physicists think the way they do.
We owe this way of thinking to Galileo. He showed that the behavior of the universe could be precisely described by mathematical laws. He laid down the first simple laws of motion. Not long afterward, Newton greatly expanded his ideas, and revolutionized how we think about nature.
Before this, the behavior of the universe was a branch of philosophy called natural philosophy. People thought of the reason the wind blows in terms clouds with faces blowing. Aristotle said a moving rock comes to rest because it is the nature of rocks (earth) to be at rest. A disturbance might temporarily change this, but the nature of the rock will assert itself. This is different from celestial object like the sun, planets, and stars (fire, air). Their nature is to move.
For more on this, see Sean Carroll's first episode of The Biggest Ideas in the Universe | 1. Conservation
