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In sinusoidal wave equations that produce interference we simply add their displacements by superposition principle, however superposition position principle can be applied to only linear equations. So what's the deal?

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What allows superposition is that the wave equation

$$\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2}$$

is linear in the “waving” variable $u$... for example, the string displacement for a string stretched along $x$. This linearity means that if $u_1(x,t)$ and $u_2(x,t)$ are solutions, then so is their sum. The fact that this linear differential equation has sinusoidal (and many other) solutions $u(x, t)$ that, for fixed $t$, are not linear functions of the coordinate $x$ is irrelevant.

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  • $\begingroup$ By u do you mean y displacement?? If yes then let me just ask this for confirmation linearity should be in the main variable whose change I am considering?? $\endgroup$ – naruto_022 Sep 28 '19 at 17:52
  • $\begingroup$ If the string is vibrating in the $xy$ plane, then the displacement $u$ is $y$. Yes, linearity of the dynamical equation in the variable that is changing is what matters. $\endgroup$ – G. Smith Sep 28 '19 at 18:06
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The wave equation is a linear operator, ie: $L(x+y)=L(x)+L(y)$.
Define the wave equation operator $$L(u)=u_{xx}-\frac{1}{c^2}u_{tt}$$ where u is the solution and c is the speed of propagation. Then the wave equation is $$L(u)=0$$ and since its linear$$L(u_1+u_2)=0$$ where $u_1$ and $u_2$ are any functions that have first and second derivatives.

So to answer your question you can add the sinusoids because the wave equation is a linear operator--re. G Smiths answer.

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  • $\begingroup$ What is a linear operator? I forgot to mention that I am in high school and this seems like something from higher calculus or something. Can you provide something that i can refer to?? Thanks in advance $\endgroup$ – naruto_022 Sep 28 '19 at 17:54
  • $\begingroup$ Google 'linear operator'. 'operator' is a function that operates on functions instead of numbers--like a normal function does. Linearity is one of the most important concepts you will run into. It applies to both functions and operators--and is basically defined by my equation for $L(x+y)=L(x)+L(y)$. $\endgroup$ – user45664 Sep 28 '19 at 19:03
  • $\begingroup$ A simple linear example would be: apples= 1 dollar ea and pears=2 dollars ea so one apple plus one pear=3.dollars This could be made non linear by adding that you get the second fruit for half price, then one apple plus one pear=2 dollars. $\endgroup$ – user45664 Sep 28 '19 at 19:11

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