What is the interpretation of speed of a travelling wave, based on $\frac {\partial x}{\partial t}$ formula
Motion satisfies the wave equation if the following holds true $$\nu^2 * \frac {\partial ^2 \psi} {\partial x^2} = \frac {\partial ^2\psi} {\partial t^2}$$ where $\psi$ is the vertical displacement of the medium (if a transverse wave), $x$ is the position of each particle whose displacement is being considered / the position at which the displacement is considered, $t$ is the time elapsed, and $\nu$ is the velocity of the wave in the transverse direction.
Also, $$\nu^2 =\left(\frac {\partial x}{\partial t}\right)^2$$
from most general considerations of velocity.
My question is this:
I think that the velocity of the wave can be interpreted as 'the speed at which the $x$-value at which a certain displacement can be found changes'. That would make sense, given that the velocity is also $\frac {\partial x}{\partial t}$, and this is, literally, the rate of change of a position on a string / in a medium, wrt time. Why is it that this, the velocity, is commensurate with $\frac {\partial x}{\partial t}$, when $\frac {\partial x}{\partial t}$ implies, I think, that position of a particle / part of the medium is changing with respect to time, i.e that $x$ and $t$ are not independent? They seem from prior consideration to be independent.
In the wave equation, $\psi$ is varying with respect to time and position, which is why the derivatives make sense when taken wrt those values. The positions $x$, are positions in the medium (say, along a string), and they are fixed points, which are used as reference points for calculating amplitude ($\psi$).
So how does it come that there is a formula, and we work with it, where $x$ and $t$ don't seem to be independent variables, because we take the derivative of one with respect to the other, and it has meaning?
Is it just a mathematical formality, because we have something that can be written in terms of $x$ and in terms of $t$, and so they can be written in terms of each other (and the function is differentiable), so a derivative exists?
How comes it (if it does) to have that meaning I ascribed to it earlier, of the change in position, $x$ with respect to $t$, where a particular amplitude of the wave can be found (which is what the velocity is)? 453