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I've just learned the superposition of waves. It got me thinking.... For example, if we're talking about waves in a string, can't we express all the things that happens on the rope using just $F=ma$ and some other mechanic equations? Then shouldn't we be able to prove the superposition of a wave in this manner? Using basic Newtonian mechanics? I don't think that this will be possible to be done on waves like light....

All this above was for one thing; how do I prove the superposition of a wave? Is it provable?

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    $\begingroup$ The proof of superposition is that the wave equation describing the wave propagation is a linear differential equstion. That insures that the sum of two solutions is also a solution. That is true for light waves as well as long as they are not propagating in a nonlinear optical medium. $\endgroup$ – Lewis Miller Aug 16 '16 at 14:59
  • $\begingroup$ Are you familiar with differential equations? The answer you want is quite simple if you are familiar with differential equations, but may require more hand waving otherwise. $\endgroup$ – zeldredge Aug 16 '16 at 15:39
  • $\begingroup$ I'm just a highschool student. I've learned most of Newtonian mechanics involving single variable calculus. But not differential equation... I could ask my dad about it though.. My dad teaches differential equation in college. (But not physics!!) $\endgroup$ – Danny Han Aug 16 '16 at 15:44
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    $\begingroup$ You know, in high school textbooks, they just state this as 'a principal'.... Which was very unsatisfactory for me $\endgroup$ – Danny Han Aug 16 '16 at 15:45
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    $\begingroup$ Here is an example of superposition physicsclassroom.com/class/waves/Lesson-3/Interference-of-Waves and simple linear differential equations tutorial.math.lamar.edu/Classes/DE/Linear.aspx $\endgroup$ – user108787 Aug 16 '16 at 15:52
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For example, if we're talking about waves in a string, can't we express all the things that happens on the rope using just F=ma and some other mechanic equations?

Wave equations do come from Newton's law; in particular, if you apply such equations of motion to each infinitesimal piece of the string and include the tensions in all directions, you exactly end up with the wave equation $$ \frac{1}{v^2}\frac{\partial^2 f}{\partial t^2}-\frac{\partial^2 f}{\partial x^2} = g(x,t). $$

how do I prove the superposition of a wave? Is it provable?

A mathematical property of the above equation is that, given any two solutions, their sum is a solution too. You can just plug the sum in and check that it still fulfills the equation provided the two initial solutions do.

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