# Is the wave equation a periodic wave equation?

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?

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.

• 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? Commented Apr 18, 2016 at 17:13
• In fact, no derivation of $\partial_x^2\psi=\partial_{ct}^2\psi$ I've ever seen has used any periodic property of waves. Commented Apr 18, 2016 at 17:17
• @CuriousOne Yes, that's it. Commented Apr 18, 2016 at 17:25
• You should edit your question to make clear what you are really up to. Commented Apr 18, 2016 at 17:29
• 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.
– jim
Commented Apr 18, 2016 at 17:46

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.