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Any function that depends on space and time through the combination $\vec{k}\cdot\vec{r}-\omega t $, namely a function $$f(\vec{r},t)=g(\vec{k}\cdot\vec{r}-\omega t)$$ where g is an arbitrary function of a single real variable, represents a perturbation that propagates in the direction of $\vec{k}$ with a velocity $v=\omega/k$. For sure this tipe of function can be considered a wave, if the broadest definition of wave is adopted: "wave = moving perturbation".

But do all types of wave have such space-time dependence? Is this the only way to implement a function that describes a moving perturbation?

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  • $\begingroup$ when you describe various systems from each level mathematically / algoritgmically, what do they all have in common? $\endgroup$
    – Engineer
    Commented Oct 13, 2022 at 13:20

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But do all types of wave have such space-time dependence?

Certainly not. General form of wave equation is :

$$ \Box u=0, ~~~(1)$$

where,

$$ \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2} ~~~(2)$$ is a d'Alembert operator .

So any function $u(\vec r,t)$ which satisfies eq. 1, has wave properties. Such functions should be many.

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    $\begingroup$ But all solutions of the wave equation are linear combinations of the ones given by the OP. For example, we can take them to be monochromatic plane waves. $\endgroup$
    – megaleo
    Commented Oct 14, 2022 at 5:14
  • $\begingroup$ But do they have to be linear ? On the other hand most basic requirement for a wave is periodicity of some amplitude over space and time. Usually this is seen as a plane wave, i.e. $\sin(x)$ function. But does most common trigonometric functions are the only periodical functions? I feel there should be a huge set of them too. Overall,- it's not the point what common linear wave equation solutions do we have now, but it's about what kind of different solutions we can have at all ? I feel such question is not analyzed much. $\endgroup$ Commented Oct 14, 2022 at 6:25
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    $\begingroup$ I changed the text to be more clear. Note that waves, from what I know, can be solution of different differential equations other than d'Alambert equation. Moreover the definition "wave = moving perturbation" does not imply a space or time periodicity of the perturbation itself: think to a moving single light pulse that moves in space. $\endgroup$ Commented Oct 14, 2022 at 15:13
  • $\begingroup$ About a single moving perturbation - you are right,- that periodicity is not a necessary attribute. But, about a different general wave differential equations I suppose you may be wrong. As far as I know d'Alembert operator is most general description of waves. And Btw this operator equally well suits to single pulse traveling, i.e. for example for stress pulse in a bar : $$ \ddot u(x,t)={\frac {E}{\rho }} \nabla ^{2} u(x,t). $$ So I think that this operator can define any perturbation in space-time. What can be more general? $\endgroup$ Commented Oct 14, 2022 at 20:11
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    $\begingroup$ An example, from what I understood, could be the Klein-Gordon equation: $(\Box + m^2)\phi = 0$ . $\endgroup$ Commented Oct 16, 2022 at 14:08

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