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I have seen that in the derivation of wave equation, they always use the periodic property of waves in the derivation. But what about non-periodic waves? Do they have some different wave equation?

Is the general wave equation $$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$$ a periodic wave equation?

Please help.

EDIT: If the question is difficult to understand as what it now stands, I would add that I am looking for non-linear equations with solitons as solutions.

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    $\begingroup$ Linear wave equations, including the one above, can have perfectly non-periodic solutions. I am not sure what you mean. Do you mean that it's homogeneous? Are you looking for non-linear equations with solitons for solutions? $\endgroup$
    – CuriousOne
    Apr 18, 2016 at 17:13
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    $\begingroup$ In fact, no derivation of $\partial_x^2\psi=\partial_{ct}^2\psi$ I've ever seen has used any periodic property of waves. $\endgroup$
    – AccidentalFourierTransform
    Apr 18, 2016 at 17:17
  • $\begingroup$ @CuriousOne Yes, that's it. $\endgroup$
    – SchrodingersCat
    Apr 18, 2016 at 17:25
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    $\begingroup$ You should edit your question to make clear what you are really up to. $\endgroup$
    – garyp
    Apr 18, 2016 at 17:29
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    $\begingroup$ The wave equation that you quote, $\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$, is a linear equation. $\endgroup$
    – jim
    Apr 18, 2016 at 17:46

2 Answers 2

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Solutions to the wave equation need not be periodic. In fact, any profile with the argument $x \pm c t$ will satisfy the wave equation.

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No. It is an equation of a moving function. Take any general function dependent on position and time such that the function is f(x+ct). Now, partially differentiate it by applying chain rule. You will see that you end up with the same differential equation as above.

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