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From my understaning, a simple progressive wave is written in the form: $y(x,t)=A \sin(kx \pm wt)$ (or its cossine equivalent) and it is a solution to the equation: $$\frac{\partial^2 y}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 y}{\partial t^2}$$

In simple examples, the equation can usually be written as a sum of solutions $y(x,t)=A \sin(kx \pm wt)$, and in that case, since the wave equation is linear, the outcome is still a wave (progressive or standing). But what about more complex equations, such as $y(x,t)=A \sin(kx - wt)^2$ or $y(x,t) = \exp(kx - wt)^2$?

Is there any "method" to determine wether they are progressive or stationary waves? Is it sufficient to show that they solve the wave equation and whether they have nodes/antinodes? Or is there a better way to identify them?

user2320292
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Rye
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    Does this answer your question? How to determine whether a wave is travelling or standing? – jsloan Jan 06 '20 at 17:23
  • Not really... In that question, they give an example of equations you can write the as a sum of progressive waves, but when that's not the case (like in the examples I wrote down), I don't know what to do – Rye Jan 06 '20 at 17:36
  • What is your definition of "wave" here, and what is the 'sen' function? – ACuriousMind Jan 06 '20 at 17:52
  • I don't really know what you mean with my "definition of wave", could you elaborate on that? As for the function, it's a sine - but as you'd write it in portuguese. I always forget it's written differently in english, I'll correct that. – Rye Jan 06 '20 at 17:56
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    The examples you give are of the form $f(kx \pm \omega t)$ so they are solutions of the wave equation - either progressive (travelling) or standing. It is not necessary for $f(x,t)$ to be a sine/cosine function. – sammy gerbil Jan 07 '20 at 03:26
  • The general solution to the wave equation is $f(x-ct)+g(x+ct)$ were f and g are any functions with at least first and second derivatives. (c is the speed of propagation) – user45664 Jan 07 '20 at 17:37
  • I see... so that would mean that if the equation was of the form $y=Asin(kx^2 - wt^2)$, iIt's not a progressive wave because it's in the form $f(kx^2\pm wt^2)$, right? so I can't separate the function in $g(kx\pm wt)$ in that case – Rye Jan 07 '20 at 19:20

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