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?