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I was studying waves on a string and there was this 'Principle of Superposition of Waves' which states that the net displacement of a point is equal to the vector sum of displacements caused to the point by the individual waves. So, basically if two waves are meeting at a point then net displacement of that point is given by

$\vec y_{\rm net}=\vec y_1 + \vec y_2$

where $\vec y_1$ is the displacement caused by the first wave alone and $\vec y_2$ is the displacement caused by the second wave alone. But what is the proof of principle of superposition? How do we know that this is true? There is literally no explanation given in my textbook as to why this is true. Please someone explain. I am so confused. Please give a rigorous proof of this.

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One way to look at this is to consider the wave equation. The wave equation is a differential equation given by $$\frac 1{v^2}\frac{\partial^2 y}{\partial t^2}-\frac{\partial^2 y}{\partial x^2}=0.$$ Where $v$ is the speed of the waves.You can check easily check that when $y_1,y_2$ are both solutions to the wave equation the result $y_{\text {net}}=y_1+y_2$ obeys the wave equation as well. In fact any linear differential equation that only involves derivatives (so no terms proportional to $y$) obeys a superposition principle.

Now to make it more clear what's happening consider the case that you have two wavepackets really far apart that are headed towards eachother. Because they are so far apart these wavepackets don't interact with eachother and it is easy to predict their motion. For each wavepacket imagine the other wavepacket isn't there and evolve the wavepacket according the equations of motion. For the wave equation this means to translate each wavepacket to the left/right. When the wavepackets meet their behaviour is hard to predict. What will happen? The superposition tells us that something really easy happens: they pass right through eachother. More precisely to compute the state after a time $t$ has passed, first split the wavefunction $y$ into a left- and right moving wavepacket called $y_L$ and $y_R$, then move each wavepacket to the left/right according to the wave equation to obtain the wavefunctions at a time $t$ and finally add these two functions together to get the final result.

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    $\begingroup$ "any differential equation that only involves derivatives (so no terms proportional to y) obeys a superposition principle."Do non-linear differential equations with only derivatives obey a superposition principle? $\endgroup$ – Bill N yesterday
  • $\begingroup$ @BillN Good catch, a bit sloppy on my part. I updated my answer. $\endgroup$ – AccidentalTaylorExpansion 17 hours ago
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Consider the wave equation on a string that is located in the $xy$–plane:

$$L_d(y)=\frac{{{\partial ^2}y}}{{\partial {x^2}}} - \frac{1}{{{v^2}}}\frac{{{\partial ^2}y}}{{\partial {t^2}}} = 0,$$ where, $L_d$ is a linear differential operator with the explicit form of

$${L_d} = {\frac{{{\partial ^2}}}{{\partial {x^2}}} - \frac{1}{{{v^2}}}\frac{{{\partial ^2}}}{{\partial {t^2}}}}.$$

This linear operator has two important properties:

$${\bf{Additivity}}\,:\,\,\,{L_d}({y_1} + {y_2}) = {L_d}({y_1}) + {L_d}({y_2}),$$ and $${\bf{Homogeneity}}:{L_d}(ay) = a{L_d}(y).$$

This means that if both $y_1$ and $y_2$, separately, are the solutions of the string wave equations, so $y_{net}=a_1 y_1+a_2 y_2$ is also a solution ($a_1$ and $a_2$ are two real numbers). These (i.e., additivity and homogeneity) together define the superposition principle.

Systems that satisfy both homogeneity and additivity are considered to be linear systems and a linear theory could describe a linear system (For example, consider Maxwell's theory of electromagnetism). So, these kinds of systems obey the superposition principle. It means the net amplitude caused by two waves is simply the sum of the amplitudes that would have been produced by the individual waves separately.

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I would like to try to give an intuitive, physical explanation of the principle of superposition, an alternative to the other valid answers which use the Wave Equation. Unfortunately, I can't prove the principle of superposition this way, but I hope to give some physical justification.

Consider the simple spring-mass system. When the spring is extended by a distance $2x$, the force that the spring exerts is double the force when the spring is extended by a distance $x$. You could think of this as a superposition of 2 spring-mass systems where each mass is extended a distance of $x$.

Superposition of two spring-mass systems

Now, what does this have to do with waves? Well, string waves are basically just a lot of tiny spring-mass systems arranged side by side in the transverse direction. The physics behind waves is based on the spring-mass system, so intuitively you would expect some results to carry forward. So, if you superpose two wave pulses, what you get is the following:

Superposition of two waves

The "extension lengths" of the resultant wave should intuitively be the sum of the "extension lengths" of the original waves, just like with the spring-masses systems!

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because displacement is caused by the momentum of each point of the string and momentum is additive like that

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