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What does it mean for a function to satisfy the wave equation? Does it necessarily mean that it is a wave equation?

Take for instance the function $f(t,z)=(at+b z)^2$.

This function satisfies the wave equation. Does this mean that it could describe a wave?

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    $\begingroup$ It is hard to tell what you are asking here. I think the first time you say "equation" you mean "function," and what do you mean by "Does it necessarily mean that it is a wave equation?" And what is a "potential wave"? A function of the form you describe obeys a certain kind of wave equation (describing waves with velocity $-a/b$), but whether it is physical or not depends entirely on the context of what you're trying to describe. $\endgroup$
    – Michael
    Commented Nov 2, 2013 at 10:59
  • $\begingroup$ @MichaelBrown: Thank you very much for your comment. I have reprhased my question. I hope my point is clear now. $\endgroup$
    – Thanos
    Commented Nov 2, 2013 at 11:34

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Physical problems also have boundary conditions. In the case of waves, we usually want a well-defined flux of energy or current or whatnot going to or coming from the boundary of the region, or infinity. The wave function you gave diverges as $z \to \infty$ so it might not be physically realistic. Placed inside a box, you might want to demand that no energy or current flow through the walls of the box, which is probably not satisfiable using your example. The equations for these fluxes are given by conservation laws, Noether currents.

A note on terminology, an equation is something that equates, and therefore has an equal sign in it. A wave equation usualy contains operators that act on functions, and so can be satisfied by a wave function.

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