This could be extremely trivial but I am having problems figuring it out.
I think I understand properly the difference between waves and fields. A field is a function valued on space or spacetime which takes values that can be scalars, vectors, tensors and so on. While waves can be understood as a consequence of fields' perturbations that brings the fields out of their equilibrium state. For example a static mass generates a gravitational field and if the mass oscillates, then it forms gravitational waves.
My problem is why in QFT (and not only) the fields ends up being solutions of wave equations, for example the free scalar field ends up being solution of the Klein gordon equation (a relativistic wave equation)
$$(\Box + m^2)\phi=0$$
Similarly Maxwell equations can be rewritten in the Coulomb Gauge for the potential vector field $A$ as $$\Box A=0$$
What to me seems to be a problem is that we name fields objects that are solutions of wave equations, while these are different concepts even thoguh are related. In a not rigorous way at first I thought I could partially solve my doubts thinking to make a perturbation of the field $A \rightarrow A+\delta A$ through the presence of a source $\rho$ (or through an interaction) and considering this
$$\Box A=0$$ $$\Box(A+\delta A)=\rho \rightarrow \Box \delta A=\rho $$ and so I could see the perturbation as a wave related to the source that generates the perturbation. Even if this is correct (which I am absolutely not sure) the problem stands even from a mathematical point of view as fields and waves are different objects. Fields are sections that goes from a manifold to a bundle while waves I guess are just solutions of certain differential equations.