# Waves and linear dependence on space and time

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?

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

## 1 Answer

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.

• 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. Commented Oct 14, 2022 at 5:14
• 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. Commented Oct 14, 2022 at 6:25
• 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. Commented Oct 14, 2022 at 15:13
• 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? Commented Oct 14, 2022 at 20:11
• An example, from what I understood, could be the Klein-Gordon equation: $(\Box + m^2)\phi = 0$ . Commented Oct 16, 2022 at 14:08