First of all, I know there's a much alike question here but this is not duplicate since I couldn't find there the answer I'm seeking. My problem is the following: I know that intuitively we have a wave when we have some quantitiy (that as I see can be anything) oscilating at each point in space. So for instance, electromagnetic waves are composed of electric and magnetic fields oscilating on each point of space.
Now, this is vague and imprecise. It is not clear at first how to model this mathematically and what properties this thing should have. However there's the wave equation:
$$\nabla^2 \psi = \dfrac{1}{v^2} \dfrac{\partial ^2}{\partial t^2}\psi,$$
but asking some physicists they told me that not every wave obeys that equation. That anything obeying that equation is really a wave, but that there are waves which evolve differently, some that are nonlinear and all of that.
In that case it seems everything is totally vague. A wave is something that moves like a wave, some of those things obeys a certain equation and the others can obey equations totally different. In that case it becomes a little bit difficult to grasp what really is a wave and how do we treat waves with precision.
So, what is a wave and how waves are preciselly dealt with in some mathematical framework?
Best Answer
A propagating disturbance in a material medium, e.g., air, or immaterial 'medium', e.g., the electromagnetic field.
A wave function is a mathematical description of the propagating disturbance and is a solution to some partial differential equation involving spatial and time derivatives.
One can quite simply construct a wave function by taking an ordinary function of one variable, e.g.
$$f(\theta) = \cos (\theta)$$
and replacing the argument with a function of the space and time coordinates, e.g.
$$\theta = \vec k \cdot \vec x - \omega t$$
so that the wave function is
$$f(x,t) = \cos(\vec k \cdot \vec x - \omega t)$$
This particular wave function is a solution to the wave equation in your question if
$$\frac{\omega^2}{k^2} = v^2$$
and is a sinusoid that propagates in the $\vec k$ direction with a phase velocity of $v$.
(From the Wikipedia article "Wave")
But, of course, there are other wave equations that some wave functions solve. A somewhat famous one is
$$-\frac{\hbar^2}{2m}\nabla^2\psi + V(x)\psi = i\hbar\frac{\partial}{\partial t}\psi$$
And it's still not clear precisely what or where the 'medium' is for these waves.